Groupprops - User contributions [en]

Groups of order 56

2022-02-17T23:44:12Z

Anarchic Fox: Added list of order-56 groups.

{{groups of order|56}}

This article gives basic information comparing and contrasting groups of order 56. The prime factorization of 56 is <math>56 = 2^3 \cdot 7</math>.

==Statistics at a glance==

{| class="sortable" border="1"
! Quantity !! Value
|-
| Number of groups of order 56|| [[count::13]]
|-
| Number of abelian groups || [[abelian count::3]]
|-
| Number of nilpotent groups || [[nilpotent count::5]]
|-
| Number of solvable groups || [[solvable count::13]]
|-
| Number of simple groups || 0
|}

==The List==

===The list===

There are 13 groups of order 56:

{| class="sortable" border="1"
! Group !! Second part of GAP ID !! Abelian !! Nilpotent || Direct Product
|-
| [[SmallGroup(56,1)]] || 1 || no || no || no
|-
| [[SmallGroup(56,2)]] || 2 || yes || yes || no
|-
| [[SmallGroup(56,3)]] || 3 || no || no || no
|-
| [[SmallGroup(56,4)]] || 4 || no || no || yes
|-
| [[SmallGroup(56,5)]] || 5 || no || no || no
|-
| [[SmallGroup(56,6)]] || 6 || no || no || yes
|-
| [[SmallGroup(56,7)]] || 7 || no || no || no
|-
| [[SmallGroup(56,8)]] || 8 || yes || yes || yes
|-
| [[SmallGroup(56,9)]] || 9 || no || yes || yes
|-
| [[SmallGroup(56,10)]] || 10 || no || yes || yes
|-
| [[SmallGroup(56,11)]] || 11 || no || no || no
|-
| [[SmallGroup(56,12)]] || 12 || no || no || yes
|-
| [[SmallGroup(56,13)]] || 13 || yes || yes || yes
|}

==GAP implementation==

{{this order in GAP|order = 56|idgroup = yes}}

<pre>gap> SmallGroupsInformation(56);

There are 13 groups of order 56.
They are sorted by their Frattini factors.
1 has Frattini factor [ 14, 1 ].
2 has Frattini factor [ 14, 2 ].
3 - 7 have Frattini factor [ 28, 3 ].
8 - 10 have Frattini factor [ 28, 4 ].
11 - 13 have trivial Frattini subgroup.

For the selection functions the values of the following attributes
are precomputed and stored:
IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup,
LGLength, FrattinifactorSize and FrattinifactorId.

This size belongs to layer 2 of the SmallGroups library.
IdSmallGroup is available for this size.</pre>

Groups of order 30

2022-02-17T05:13:05Z

Anarchic Fox: Added list of order-30 groups.

{{groups of order|30}}

==Statistics at a glance==

The number 30 has prime factors 2, 3, and 5. The prime factorization is:

<math>30 = 2^1 \cdot 3^1 \cdot 5^1</math>

[[Square-free implies solvability-forcing]], so all groups of order 30 are [[finite solvable group]]s. Moreover, [[every Sylow subgroup is cyclic implies metacyclic]], so all groups of order 30 are in fact [[metacyclic group]]s.

===The list===

There are 4 groups of order 30:

{| class="sortable" border="1"
! Group !! Second part of GAP ID !! Abelian !! Nilpotent || Direct Product
|-
| [[SmallGroup(30,1)]] || 1 || no || no || yes
|-
| [[SmallGroup(30,2)]] || 2 || no || no || yes
|-
| [[dihedral group:D30]] || 3 || no || no || no
|-
| [[cyclic group:Z30]] || 4 || yes || yes || yes
|}

==GAP implementation==

{{this order in GAP|order = 30|idgroup = yes}}

<pre>gap> SmallGroupsInformation(30);

There are 4 groups of order 30.
1 is of type S3x5.
2 is of type D10x3.
3 - 3 are of types 3:2+5:2.
4 is of type c30.

The groups whose order factorises in at most 3 primes
have been classified by O. Hoelder. This classification is
used in the SmallGroups library.

This size belongs to layer 1 of the SmallGroups library.
IdSmallGroup is available for this size.</pre>

Dicyclic group:Dic12

2022-02-17T03:30:56Z

Anarchic Fox: Added another presentation.

{{particular group}}

This group, sometimes denoted <math>\operatorname{Dic}_{12}</math> and sometimes denoted <math>\Gamma(3,2,2)</math>, is defined in the following equivalent ways:

* It is the [[member of family::dicyclic group]] (i.e., the binary dihedral group) of order <math>12</math>, and hence of degree <math>3</math>.
* It is the [[member of family::binary von Dyck group]] with parameters <math>(3,2,2)</math>.

A presentation for the group is given by:

<math>\langle a,b,c \mid a^3 = b^2 = c^2 = abc \rangle</math>.

It also has the presentation:

<math>\langle a,x \mid a^6 = x^4 = e, a^3 = x^2, xax^{-1} = a^{-1} \rangle </math>

==GAP implementation==

{{GAP ID|12|1}}

===Other definitions===

The group can also be defined using its presentation:

<pre>F := FreeGroup(3);
G := F/[F.1^3 * F.2^(-2), F.2^2 * F.3^(-2), F.1 * F.2 * (F.3)^(-1)];</pre>

SmallGroup(16,3)

2022-02-17T02:43:11Z

Anarchic Fox: /* Definition */

{{particular group}}

==Definition==

This group is a [[semidirect product]] <math>(Z_4 \times Z_2) \rtimes Z_2</math>, and can be defined using the following presentation:

<math>G := \langle a,b,c \mid a^4 = b^2 = c^2 = e, ab = ba, bc = cb, cac^{-1} = ab \rangle</math>

Note that <math>G</math> is generated by <math>a,c</math> alone, because the final relation allows us to write <math>b</math> in terms of <math>a</math> and <math>c</math>.

The subgroup <math>\langle a,b \rangle</math> is isomorphic to the [[direct product of Z4 and Z2]], and the element <math>c</math> is an element of order two that acts on the subgroup <math>\langle a,b \rangle</math> by conjugation by fixing <math>b</math> and sending <math>a</math> to <math>ab</math>.

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 16}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|16}} ||
|-
| {{arithmetic function value order p-log etc|4}}
|-
| {{arithmetic function value given order|exponent of a group|4|16}} ||
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|derived length|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|rank of a p-group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|number of conjugacy classes|10|16|4}} ||
|-
| {{arithmetic function value given order and p-log|number of conjugacy classes of subgroups|17|16|4}} ||
|}

==Group properties==

{{quotation|Want to compare with other groups of the same order? Check out [[groups of order 16#Group properties]].}}

{| class="wikitable" border="1"
!Property !! Satisfied !! Explanation
|-
|[[Dissatisfies property::abelian group]] || No ||<math>a,c</math> don't commute
|-
|[[Satisfies property::group of prime power order]] || Yes ||
|-
|[[Satisfies property::nilpotent group]] || Yes || [[prime power order implies nilpotent]]
|-
|[[Satisfies property::group of nilpotency class two]] || Yes ||
|-
|[[Satisfies property::supersolvable group]] || Yes ||
|-
|[[Dissatisfies property::T-group]] || No || The subgroup <math>\langle a \rangle</math> is [[2-subnormal subgroup|2-subnormal]], not [[normal subgroup|normal]]
|-
|[[Dissatisfies property::monolithic group]] || No ||
|-
|[[Dissatisfies property::one-headed group]] || No ||
|-
|[[Dissatisfies property::ambivalent group]] || No ||
|-
|[[Dissatisfies property::rational group]] || No ||
|-
|[[Dissatisfies property::rational-representation group]] || No ||
|}

==Subgroups==

{{further|[[subgroup structure of SmallGroup(16,3)]]}}

# The [[subgroup::trivial group]]. (1)
# The subgroup <math>\langle b \rangle</math>, which is the unique characteristic subgroup of order <math>2</math>. Isomorphic to [[subgroup::cyclic group:Z2]]. It is the [[commutator subgroup]], and can also be described as the unique group of order two containing an element that is ''not'' a square but is a product of squares. The quotient group is isomorphic to [[quotient group::direct product of Z4 and Z2]]. (1)
# The subgroups <math>\langle a^2 \rangle</math> and <math>\langle a^2b \rangle</math>, which are both normal subgroups related by an outer automorphism. Isomorphic to [[subgroup::cyclic group:Z2]]. The quotient group for each is isomorphic to [[quotient group::dihedral group:D8]]. (2)
# The subgroups <math>\langle c \rangle</math>, <math>\langle bc \rangle</math>, <math>\langle a^2c \rangle</math> and <math>\langle a^2bc \rangle</math>. Neither is normal, and they come in two conjugacy classes of size two each. Isomorphic to [[subgroup::cyclic group:Z2]]. (4)
# The subgroup <math>\langle a^2,b \rangle</math>, which is the [[center]], [[agemo subgroups of group of prime power order|first agemo subgroup]], and [[Frattini subgroup]]. Isomorphic to [[subgroup::Klein four-group]]. The quotient group is isomorphic to [[quotient group::Klein four-group]]. (1)
# The subgroups <math>\langle b,c \rangle</math> and <math>\langle b, a^2c\rangle</math>. Both are [[normal subgroup]]s and are related by an [[outer automorphism]]. Isomorphic to [[subgroup::Klein four-group]]. The quotient group is isomorphic to [[quotient group::cyclic group:Z4]]. (2)
# The subgroups <math>\langle a^2,c \rangle</math>, <math>\langle a^2,bc \rangle</math>, <math>\langle a^2b,c\rangle</math>, and <math>\langle a^2b, bc \rangle</math>. Two conjugacy classes of size two each. Isomorphic to [[subgroup::Klein four-group]]. The quotient group is also isomorphic to a [[quotient group::Klein four-group]]. (4)
# The subgroups <math>\langle a \rangle</math>, <math>\langle ab \rangle</math>, <math>\langle ac \rangle</math>, and <math>\langle abc \rangle</math>. All of them are related by outer automorphisms, and they form two conjugacy classes of subgroups of size two each: <math>\langle a \rangle</math> is conjugate to <math>\langle ab \rangle</math>, while <math>\langle ac \rangle</math> is conjugate to <math>\langle abc \rangle</math>. Isomorphic to [[subgroup::cyclic group:Z4]]. (4)
# The subgroups <math>\langle a,b \rangle</math> and <math>\langle ac, b \rangle</math>. These are both [[normal subgroup]]s related by an outer automorphism. Isomorphic to [[subgroup::direct product of Z4 and Z2]]. The quotient group is isomorphic to [[quotient group::cyclic group:Z2]]. (2)
# The subgroup <math>\langle a^2,b,c \rangle</math>. Isomorphic to [[subgroup::elementary abelian group:E8]]. The quotient group is isomorphic to [[quotient group::cyclic group:Z2]]. (1)
# The whole group. (1)

==Subgroup-defining functions==

{| class="wikitable" border="1"
! Subgroup-defining function !! Subgroup type in list !! Page on subgroup embedding !! Isomorphism class !! Comment
|-
| [[Center]] || (4) || [[subgroup-defining function value as embedding::center;center of SmallGroup(16,3)| ]][[center of SmallGroup(16,3)]] || [[subgroup-defining function value::center;Klein four-group| ]][[Klein four-group]] ||
|-
| [[Commutator subgroup]] || (2) || [[subgroup-defining function value as embedding::commutator subgroup;commutator subgroup of SmallGroup(16,3)| ]][[commutator subgroup of SmallGroup(16,3)]] || [[subgroup-defining function value::commutator subgroup;cyclic group:Z2| ]][[cyclic group:Z2]] ||
|-
| [[Frattini subgroup]] || (4) || [[subgroup-defining function value as embedding::center;center of SmallGroup(16,3)| ]][[center of SmallGroup(16,3)]] || [[subgroup-defining function value::center;Klein four-group| ]][[Klein four-group]] ||
|-
| [[Socle]] || (4) || [[subgroup-defining function value as embedding::center;center of SmallGroup(16,3)| ]][[center of SmallGroup(16,3)]] || [[subgroup-defining function value::center;Klein four-group| ]][[Klein four-group]] ||
|-
| [[omega subgroups of group of prime power order|first omega subgroup]] || (7) || || [[elementary abelian group:E8]] ||
|}

==GAP implementation==

{{GAP ID|16|3}}

===Other descriptions===

The group can be constructed using the following GAP commands:

<pre>gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^4, F.2^2, F.1*F.2*F.1^(-1)*F.2^(-1),F.3^2,F.3*F.2*F.3^(-1)*F.2^(-1),F.3*F.1*F.3^(-1)*F.2^(-1)*F.1^(-1)];
<fp group on the generators [ f1, f2, f3 ]></pre>

SmallGroup(16,3)

2022-02-17T02:42:56Z

Anarchic Fox: Small typo

{{particular group}}

==Definition==

This group is a [[semidirect product]] <math>(Z_4 \times Z_2) \rtimes Z_2</math>, and can be defined using the following presentation:

<math>G := \langle a,b,c \mid a^4 = b^2 = x^2 = e, ab = ba, bc = cb, cac^{-1} = ab \rangle</math>

Note that <math>G</math> is generated by <math>a,c</math> alone, because the final relation allows us to write <math>b</math> in terms of <math>a</math> and <math>c</math>.

The subgroup <math>\langle a,b \rangle</math> is isomorphic to the [[direct product of Z4 and Z2]], and the element <math>c</math> is an element of order two that acts on the subgroup <math>\langle a,b \rangle</math> by conjugation by fixing <math>b</math> and sending <math>a</math> to <math>ab</math>.

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 16}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|16}} ||
|-
| {{arithmetic function value order p-log etc|4}}
|-
| {{arithmetic function value given order|exponent of a group|4|16}} ||
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|derived length|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|rank of a p-group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|number of conjugacy classes|10|16|4}} ||
|-
| {{arithmetic function value given order and p-log|number of conjugacy classes of subgroups|17|16|4}} ||
|}

==Group properties==

{{quotation|Want to compare with other groups of the same order? Check out [[groups of order 16#Group properties]].}}

{| class="wikitable" border="1"
!Property !! Satisfied !! Explanation
|-
|[[Dissatisfies property::abelian group]] || No ||<math>a,c</math> don't commute
|-
|[[Satisfies property::group of prime power order]] || Yes ||
|-
|[[Satisfies property::nilpotent group]] || Yes || [[prime power order implies nilpotent]]
|-
|[[Satisfies property::group of nilpotency class two]] || Yes ||
|-
|[[Satisfies property::supersolvable group]] || Yes ||
|-
|[[Dissatisfies property::T-group]] || No || The subgroup <math>\langle a \rangle</math> is [[2-subnormal subgroup|2-subnormal]], not [[normal subgroup|normal]]
|-
|[[Dissatisfies property::monolithic group]] || No ||
|-
|[[Dissatisfies property::one-headed group]] || No ||
|-
|[[Dissatisfies property::ambivalent group]] || No ||
|-
|[[Dissatisfies property::rational group]] || No ||
|-
|[[Dissatisfies property::rational-representation group]] || No ||
|}

==Subgroups==

{{further|[[subgroup structure of SmallGroup(16,3)]]}}

# The [[subgroup::trivial group]]. (1)
# The subgroup <math>\langle b \rangle</math>, which is the unique characteristic subgroup of order <math>2</math>. Isomorphic to [[subgroup::cyclic group:Z2]]. It is the [[commutator subgroup]], and can also be described as the unique group of order two containing an element that is ''not'' a square but is a product of squares. The quotient group is isomorphic to [[quotient group::direct product of Z4 and Z2]]. (1)
# The subgroups <math>\langle a^2 \rangle</math> and <math>\langle a^2b \rangle</math>, which are both normal subgroups related by an outer automorphism. Isomorphic to [[subgroup::cyclic group:Z2]]. The quotient group for each is isomorphic to [[quotient group::dihedral group:D8]]. (2)
# The subgroups <math>\langle c \rangle</math>, <math>\langle bc \rangle</math>, <math>\langle a^2c \rangle</math> and <math>\langle a^2bc \rangle</math>. Neither is normal, and they come in two conjugacy classes of size two each. Isomorphic to [[subgroup::cyclic group:Z2]]. (4)
# The subgroup <math>\langle a^2,b \rangle</math>, which is the [[center]], [[agemo subgroups of group of prime power order|first agemo subgroup]], and [[Frattini subgroup]]. Isomorphic to [[subgroup::Klein four-group]]. The quotient group is isomorphic to [[quotient group::Klein four-group]]. (1)
# The subgroups <math>\langle b,c \rangle</math> and <math>\langle b, a^2c\rangle</math>. Both are [[normal subgroup]]s and are related by an [[outer automorphism]]. Isomorphic to [[subgroup::Klein four-group]]. The quotient group is isomorphic to [[quotient group::cyclic group:Z4]]. (2)
# The subgroups <math>\langle a^2,c \rangle</math>, <math>\langle a^2,bc \rangle</math>, <math>\langle a^2b,c\rangle</math>, and <math>\langle a^2b, bc \rangle</math>. Two conjugacy classes of size two each. Isomorphic to [[subgroup::Klein four-group]]. The quotient group is also isomorphic to a [[quotient group::Klein four-group]]. (4)
# The subgroups <math>\langle a \rangle</math>, <math>\langle ab \rangle</math>, <math>\langle ac \rangle</math>, and <math>\langle abc \rangle</math>. All of them are related by outer automorphisms, and they form two conjugacy classes of subgroups of size two each: <math>\langle a \rangle</math> is conjugate to <math>\langle ab \rangle</math>, while <math>\langle ac \rangle</math> is conjugate to <math>\langle abc \rangle</math>. Isomorphic to [[subgroup::cyclic group:Z4]]. (4)
# The subgroups <math>\langle a,b \rangle</math> and <math>\langle ac, b \rangle</math>. These are both [[normal subgroup]]s related by an outer automorphism. Isomorphic to [[subgroup::direct product of Z4 and Z2]]. The quotient group is isomorphic to [[quotient group::cyclic group:Z2]]. (2)
# The subgroup <math>\langle a^2,b,c \rangle</math>. Isomorphic to [[subgroup::elementary abelian group:E8]]. The quotient group is isomorphic to [[quotient group::cyclic group:Z2]]. (1)
# The whole group. (1)

==Subgroup-defining functions==

{| class="wikitable" border="1"
! Subgroup-defining function !! Subgroup type in list !! Page on subgroup embedding !! Isomorphism class !! Comment
|-
| [[Center]] || (4) || [[subgroup-defining function value as embedding::center;center of SmallGroup(16,3)| ]][[center of SmallGroup(16,3)]] || [[subgroup-defining function value::center;Klein four-group| ]][[Klein four-group]] ||
|-
| [[Commutator subgroup]] || (2) || [[subgroup-defining function value as embedding::commutator subgroup;commutator subgroup of SmallGroup(16,3)| ]][[commutator subgroup of SmallGroup(16,3)]] || [[subgroup-defining function value::commutator subgroup;cyclic group:Z2| ]][[cyclic group:Z2]] ||
|-
| [[Frattini subgroup]] || (4) || [[subgroup-defining function value as embedding::center;center of SmallGroup(16,3)| ]][[center of SmallGroup(16,3)]] || [[subgroup-defining function value::center;Klein four-group| ]][[Klein four-group]] ||
|-
| [[Socle]] || (4) || [[subgroup-defining function value as embedding::center;center of SmallGroup(16,3)| ]][[center of SmallGroup(16,3)]] || [[subgroup-defining function value::center;Klein four-group| ]][[Klein four-group]] ||
|-
| [[omega subgroups of group of prime power order|first omega subgroup]] || (7) || || [[elementary abelian group:E8]] ||
|}

==GAP implementation==

{{GAP ID|16|3}}

===Other descriptions===

The group can be constructed using the following GAP commands:

<pre>gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^4, F.2^2, F.1*F.2*F.1^(-1)*F.2^(-1),F.3^2,F.3*F.2*F.3^(-1)*F.2^(-1),F.3*F.1*F.3^(-1)*F.2^(-1)*F.1^(-1)];
<fp group on the generators [ f1, f2, f3 ]></pre>

SmallGroup(32,28)

2021-03-01T21:26:41Z

Anarchic Fox: /* Definition */ Correction.

{{particular group}}

==Definition==

This groups is a [[semidirect product]] <math>(Z_4 \times Z_2 \times Z_2) \rtimes Z_2</math>, with the following [[presentation]]:

<math>\langle a,b,c,d \mid a^4 = b^2 = c^2 = x^2 = e, ab = ba, ac = ca, bc = cb, xc = cx, xax^{-1} = a^{-1}, xbx^{-1} = bc\rangle</math>

==Arithmetic functions==

{{compare and contrast arithmetic functions|order=32}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| underlying prime || [[arithmetic function value::underlying prime;2|2]] || ||
|-
| {{arithmetic function value order|32}} ||
|-
| {{arithmetic function value order p-log|5}} ||
|-
| {{arithmetic function value given order|exponent of a group|4|32}} ||
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|derived length|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|rank of a p-group|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|3|32|5}} ||
|}

==GAP implementation==

{{GAP ID|32|28}}

SmallGroup(36,13)

2021-02-26T07:04:04Z

Anarchic Fox: Created group page.

{{particular group}}

==Definition==

This [[group]] is a [[semidirect product]] of the form <math>((Z_3 \times Z_3) \rtimes Z_2) \times Z_2</math>. Explicitly, it is given by:

<math>\langle a,b,c,x \mid a^3 = b^3 = c^2 = x^2 = e, ab = ba, ac = ca, bc = cb, xc = cx, xax^{-1} = a^{-1}, xbx^{-1} = b^{-1} \rangle</math>

where <math>e</math> denotes the identity element.

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 36}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| {{arithmetic function value order|36}} ||
|-
| {{arithmetic function value given order|exponent of a group|6|36}} ||
|}

==GAP implementation==

{{GAP ID|36|13}}

SmallGroup(36,12)

2021-02-26T06:55:41Z

Anarchic Fox: Created group page.

{{particular group}}

==Definition==

This [[group]] is a [[direct product]] of the form <math>(Z_6 \times S_3)</math>. Explicitly, it is given by:

<math>\langle a,r,s \mid a^6 = r^3 = s^2 = e, ar = ra, as = sa, srs^{-1} = r^{-1} \rangle</math>

where <math>e</math> denotes the identity element.

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 36}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| {{arithmetic function value order|36}} ||
|-
| {{arithmetic function value given order|exponent of a group|6|36}} ||
|}

==GAP implementation==

{{GAP ID|36|12}}

SmallGroup(36,6)

2021-02-26T06:44:28Z

Anarchic Fox: Created group page.

{{particular group}}

==Definition==

This [[group]] is a [[semidirect product]] of the form <math>(Z_3 \rtimes Z_4) \times Z_3</math>. Explicitly, it is given by:

<math>\langle a,b,x \mid a^3 = b^3 = x^4 = e, ab = ba, bx = xb, xax^{-1} = a^{-1} \rangle</math>

where <math>e</math> denotes the identity element.

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 36}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| {{arithmetic function value order|36}} ||
|-
| {{arithmetic function value given order|exponent of a group|12|36}} ||
|}

==GAP implementation==

{{GAP ID|36|6}}

Holomorph of Z8

2021-02-26T04:38:00Z

Anarchic Fox: /* Definition */ Correction.

{{particular group}}

==Definition==

This group (which we shall call <math>G</math> throughout) can be defined in either of these ways:

* It is the [[holomorph]] of the cyclic group on eight elements. In other words, it is the [[semidirect product]] of the cyclic group on eight elements, with its automorphism group.
* It is the [[holomorph of a ring|holomorph]] of the ring <math>\Z/8\Z</math>. In other words, it is the [[member of family::general affine group]] <math>GA(1,\Z/8\Z)</math>.

The group has the following presentation (with <math>e</math> denoting the identity element):

<math>\! G := \langle a,x,y \mid a^8 = x^2 = y^2 = (xy)^2 = e, xax^{-1} = a^{-1}, yay^{-1} = a^5 \rangle</math>

==Arithmetic functions==

{{compare and contrast arithmetic functions|order=32}}
{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|32}} || The group is an [[external semidirect product]] of [[cyclic group:Z8]] (order 8) and it automorphism group, which is a [[Klein four-group]] (order 4)
|-
| {{arithmetic function value order p-log etc|5}}
|-
| {{arithmetic function value given order|exponent of a group|8|32}} ||
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|3|32|5}} || The derived subgroup is <math>\langle a^2 \rangle</math>, the next member of the lower central series is <math>\langle a^4 \rangle</math>
|-
| {{arithmetic function value given order and p-log|derived length|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|rank of a p-group|3|32|5}} || The subgroup <math>\langle a^4,x,y\rangle</math> is an elementary abelian subgroup of maximum rank
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|2|32|5}} ||
|}

==GAP implementation==

{{GAP ID|32|43}}

===Other definitions===

The group can be defined using GAP's [[GAP:AutomorphismGroup|AutomorphismGroup]] and [[GAP:SemidirectProduct|SemidirectProduct]] functions. Here is a full code snippet:

<pre>gap> C := CyclicGroup(8);
<pc group of size 8 with 3 generators>
gap> SemidirectProduct(AutomorphismGroup(C),C);
<pc group with 5 generators></pre>

This can be compressed by coding a function [[GAP:Holomorph|Holomorph]] for computing the holomorph of a group. With this function coded, we can use:

<pre>Holomorph(CyclicGroup(8))</pre>

SmallGroup(32,27)

2021-02-26T03:30:44Z

Anarchic Fox: /* Definition */

{{particular group}}

==Definition==

This group is a [[semidirect product]] of [[elementary abelian group:E8]] and [[Klein four-group]] where the latter acts faithfully by transvections relative to a particular plane, or as a semidirect product of the [[elementary abelian group:E16]] and <math>Z_2</math>. It is given by the following [[presentation]]:

<math>\langle a,b,c,d,x \mid a^2 = b^2 = c^2 = d^2 = x^2 = e, ab = ba, ac = ca, ad = da, bc = cb, bd = db, cd = dc, xax^{-1} = ab, xb = bx, xcx^{-1} = cd, xd = dx \rangle</math>

It can also be described as the subgroup of [[upper-triangular unipotent matrix group:U(4,2)]] given by matrices with the <math>(1,2)</math>-entry equal to zero, i.e., matrices of the form:

<math>\begin{pmatrix} 1 & 0 & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & * \\ 0 & 0 & 0 & 1 \\\end{pmatrix}</math>

It can also be defined as the 2-[[Sylow subgroup]] of the [[automorphism group]] of the [[homocyclic group]] given as the [[direct product of Z4 and Z4]].

Another group that occurs as a faithful semidirect product of the elementary abelian group of order eight and the Klein four-group is [[SmallGroup(32,49)]].

==Position in classifications==

{{quotation|Get more information about groups of the same order at [[Groups of order 32#The list]]}}

{| class="sortable" border="1"
! Type of classification !! Position/number in classification
|-
| GAP ID || <math>(32,27)</math>, i.e., <math>27^{th}</math> among groups of order 32
|-
| Hall-Senior number || 33 among groups of order 32
|-
| Hall-Senior symbol || <math>32\Gamma_4a_1</math>
|}

==Arithmetic functions==

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|32}} ||
|-
| {{arithmetic function value order p-log|5}} ||
|-
| {{arithmetic function value given order|exponent of a group|4|32}} ||
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|derived length|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|4|32|5}} ||
|-
| {{arithmetic function value given order and p-log|rank of a p-group|4|32|5}} ||
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|4|32|5}} ||
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|4|32|5}} ||
|}

==Group properties==

{| class="sortable" border="1"
! Property !! Satisfied? !! Explanation
|-
| [[Dissatisfies property::Cyclic group]] || No ||
|-
| [[Dissatisfies property::Abelian group]] || No ||
|-
| [[Dissatisfies property::Metacyclic group]] || No ||
|-
| [[Satisfies property::Metabelian group]] || Yes || Has elementary abelian maximal subgroup
|-
| [[Satisfies property::Group of nilpotency class two]] || Yes || Derived subgroup is the plane of translation, which is in the center
|}

==GAP implementation==

{{GAP ID|32|27}}

SmallGroup(32,27)

2021-02-26T03:30:28Z

Anarchic Fox: /* Definition */

{{particular group}}

==Definition==

This group is a [[semidirect product]] of [[elementary abelian group:E8]] and [[Klein four-group]] where the latter acts faithfully by transvections relative to a particular plane, or as a semidirect product of the [[elementary abelian group:E16]] and <math>Z_2</math. It is given by the following [[presentation]]:

<math>\langle a,b,c,d,x \mid a^2 = b^2 = c^2 = d^2 = x^2 = e, ab = ba, ac = ca, ad = da, bc = cb, bd = db, cd = dc, xax^{-1} = ab, xb = bx, xcx^{-1} = cd, xd = dx \rangle</math>

It can also be described as the subgroup of [[upper-triangular unipotent matrix group:U(4,2)]] given by matrices with the <math>(1,2)</math>-entry equal to zero, i.e., matrices of the form:

<math>\begin{pmatrix} 1 & 0 & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & * \\ 0 & 0 & 0 & 1 \\\end{pmatrix}</math>

It can also be defined as the 2-[[Sylow subgroup]] of the [[automorphism group]] of the [[homocyclic group]] given as the [[direct product of Z4 and Z4]].

Another group that occurs as a faithful semidirect product of the elementary abelian group of order eight and the Klein four-group is [[SmallGroup(32,49)]].

==Position in classifications==

{{quotation|Get more information about groups of the same order at [[Groups of order 32#The list]]}}

{| class="sortable" border="1"
! Type of classification !! Position/number in classification
|-
| GAP ID || <math>(32,27)</math>, i.e., <math>27^{th}</math> among groups of order 32
|-
| Hall-Senior number || 33 among groups of order 32
|-
| Hall-Senior symbol || <math>32\Gamma_4a_1</math>
|}

==Arithmetic functions==

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|32}} ||
|-
| {{arithmetic function value order p-log|5}} ||
|-
| {{arithmetic function value given order|exponent of a group|4|32}} ||
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|derived length|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|4|32|5}} ||
|-
| {{arithmetic function value given order and p-log|rank of a p-group|4|32|5}} ||
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|4|32|5}} ||
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|4|32|5}} ||
|}

==Group properties==

{| class="sortable" border="1"
! Property !! Satisfied? !! Explanation
|-
| [[Dissatisfies property::Cyclic group]] || No ||
|-
| [[Dissatisfies property::Abelian group]] || No ||
|-
| [[Dissatisfies property::Metacyclic group]] || No ||
|-
| [[Satisfies property::Metabelian group]] || Yes || Has elementary abelian maximal subgroup
|-
| [[Satisfies property::Group of nilpotency class two]] || Yes || Derived subgroup is the plane of translation, which is in the center
|}

==GAP implementation==

{{GAP ID|32|27}}

SmallGroup(32,27)

2021-02-26T03:29:20Z

Anarchic Fox: /* Definition */ Clarified presentation.

{{particular group}}

==Definition==

This group is a [[semidirect product]] of [[elementary abelian group:E8]] and [[Klein four-group]] where the latter acts faithfully by transvections relative to a particular plane. It is given by the following [[presentation]]:

<math>\langle a,b,c,d,x \mid a^2 = b^2 = c^2 = d^2 = x^2 = e, ab = ba, ac = ca, ad = da, bc = cb, bd = db, cd = dc, xax^{-1} = ab, xb = bx, xcx^{-1} = cd, xd = dx \rangle</math>

It can also be described as the subgroup of [[upper-triangular unipotent matrix group:U(4,2)]] given by matrices with the <math>(1,2)</math>-entry equal to zero, i.e., matrices of the form:

<math>\begin{pmatrix} 1 & 0 & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & * \\ 0 & 0 & 0 & 1 \\\end{pmatrix}</math>

It can also be defined as the 2-[[Sylow subgroup]] of the [[automorphism group]] of the [[homocyclic group]] given as the [[direct product of Z4 and Z4]].

Another group that occurs as a faithful semidirect product of the elementary abelian group of order eight and the Klein four-group is [[SmallGroup(32,49)]].

==Position in classifications==

{{quotation|Get more information about groups of the same order at [[Groups of order 32#The list]]}}

{| class="sortable" border="1"
! Type of classification !! Position/number in classification
|-
| GAP ID || <math>(32,27)</math>, i.e., <math>27^{th}</math> among groups of order 32
|-
| Hall-Senior number || 33 among groups of order 32
|-
| Hall-Senior symbol || <math>32\Gamma_4a_1</math>
|}

==Arithmetic functions==

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|32}} ||
|-
| {{arithmetic function value order p-log|5}} ||
|-
| {{arithmetic function value given order|exponent of a group|4|32}} ||
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|derived length|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|4|32|5}} ||
|-
| {{arithmetic function value given order and p-log|rank of a p-group|4|32|5}} ||
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|4|32|5}} ||
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|4|32|5}} ||
|}

==Group properties==

{| class="sortable" border="1"
! Property !! Satisfied? !! Explanation
|-
| [[Dissatisfies property::Cyclic group]] || No ||
|-
| [[Dissatisfies property::Abelian group]] || No ||
|-
| [[Dissatisfies property::Metacyclic group]] || No ||
|-
| [[Satisfies property::Metabelian group]] || Yes || Has elementary abelian maximal subgroup
|-
| [[Satisfies property::Group of nilpotency class two]] || Yes || Derived subgroup is the plane of translation, which is in the center
|}

==GAP implementation==

{{GAP ID|32|27}}

SmallGroup(32,7)

2021-02-26T03:11:56Z

Anarchic Fox: /* Definition */

{{particular group}}

==Definition==

This group is a [[semidirect product]] <math>(Z_8 \rtimes Z_2) \rtimes Z_2</math>, and can be given by the presentation:

<math>\langle a, x, y \mid a^8 = x^2 = y^2 = e, xy = yx, xax^{-1} = a^5, yay^{-1} = ax\rangle</math>

==Position in classifications==

{{quotation|Get more information about groups of the same order at [[Groups of order 32#The list]]}}

{| class="sortable" border="1"
! Type of classification !! Position/number in classification
|-
| GAP ID || <math>(32,7)</math>, i.e., <math>7^{th}</math> among groups of order 32
|-
| Hall-Senior number || 47 among groups of order 32
|-
| Hall-Senior symbol || <math>32\Gamma_7a_2</math>
|}

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 32}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|32}} ||
|-
| {{arithmetic function value order p-log|5}} ||
|-
| {{arithmetic function value given order|exponent of a group|8|32}} ||
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|derived length|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|rank of a p-group|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|2|32|5}} ||
|}
==Group properties==

{| class="sortable" border="1"
! Property !! Satisfied? !! Explanation !! Comment
|-
| {{group properties because p-group}}
|-
| [[dissatisfies property::abelian group]] || No || ||
|}
==GAP implementation==

{{GAP ID|32|7}}

Central product of D8 and Z4

2021-02-26T02:39:10Z

Anarchic Fox: /* Definition */

{{particular group}}
[[importance rank::3| ]]
==Definition==

This group of order 16 is defined in the following equivalent ways:

# It is the [[central product]] of the [[dihedral group:D8|dihedral group of order eight]] and [[cyclic group:Z4|cyclic group of order four]] over a common cyclic central subgroup of order two.
# It is the [[central product]] of the [[quaternion group]] and [[cyclic group:Z4|cyclic group of order four]] over a common cyclic central subgroup of order two.

It is given by the presentation:

<math>G := \langle a,x,y \mid a^4 = y^4 = x^2 = e, a^2 = y^2, xax = a^{-1}, ay = ya, xy = yx \rangle</math>.

Here, <math>\langle a,x \rangle</math> is the dihedral group of order eight and <math>\langle y \rangle</math> is the cyclic group of order four.

==Arithmetic functions==

{{compare and contrast arithmetic functions|order=16}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|16}} ||
|-
| {{arithmetic function value order p-log etc|4}}
|-
| {{arithmetic function value given order|exponent of a group|4|16}} ||
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|derived length|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|rank of a p-group|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|1|16|4}} ||
|}

==Group properties==

{| class="sortable" border="1"
!Property !! Satisfied !! Explanation !! Comment
|-
|[[Dissatisfies property::Abelian group]] || No || <math>a</math> and <math>x</math> don't commute ||
|-
|[[Satisfies property::Nilpotent group]] || Yes || [[Prime power order implies nilpotent]] ||
|-
|[[Dissatisfies property::UL-equivalent group]] || No || Nilpotency class two, but center does not coincide with derived subgroup. || See [[nilpotent not implies UL-equivalent]]
|-
|[[Dissatisfies property::Metacyclic group]] || No || ||
|-
|[[Satisfies property::Supersolvable group]] || Yes || ||
|-
|[[Satisfies property::Solvable group]] || Yes || Nilpotent implies solvable ||
|-
|[[Dissatisfies property::T-group]] || No || <math>\langle a^4,x \rangle \triangleleft \langle a^2,x \rangle</math>, which is normal, but <math>\langle a^4, x \rangle</math> is not normal ||
|-
|[[Satisfies property::Monolithic group]] || Yes|| Unique minimal normal subgroup of order two ||
|-
|[[Dissatisfies property::One-headed group]] || No || Seven distinct maximal normal subgroups of order eight ||
|-
|[[Satisfies property::Directly indecomposable group]] || Yes || ||
|-
|[[Dissatisfies property::Centrally indecomposable group]] || No || ||
|-
|[[Dissatisfies property::Splitting-simple group]] || No || ||
|}

==Subgroups==

{{further|[[Subgroup structure of central product of D8 and Z4]]}}

# The trivial subgroup. Isomorphic to [[subgroup::trivial group]]. (1)
# The subgroup <math>\langle a^2 \rangle</math>. This is the unique normal subgroup of order two, and is contained in the center. Isomorphic to [[subgroup::cyclic group:Z2]]. (1)
# The subgroups <math>\langle x \rangle</math>, <math>\langle ax \rangle</math>, <math>\langle a^2x \rangle</math>, <math>\langle a^3x \rangle</math>. These come in two conjugacy classes of [[2-subnormal subgroup]]s, one comprising <math>\langle x \rangle</math> and <math>\langle a^2x \rangle</math> and the other comprising <math>\langle ax \rangle</math> and <math>\langle a^3x \rangle</math>. However, they are all [[automorphic subgroups]]. Isomorphic to [[subgroup::cyclic group:Z2]]. (4)
# The subgroups <math>\langle ay \rangle</math> and <math>\langle a^3y \rangle</math>. These form a single conjugacy class of [[2-subnormal subgroup]]s. Isomorphic to [[subgroup::cyclic group:Z2]]. (2)
# The subgroup <math>\langle y \rangle</math> of order four. This is the [[center]]. Isomorphic to [[subgroup::cyclic group:Z4]]. (1)
# The subgroups <math>\langle a \rangle</math>, <math>\langle xy \rangle</math> and <math>\langle axy \rangle</math>. These are [[normal subgroup]]s but are [[automorphic subgroups]]: they are related by outer automorphisms. Isomorphic to [[subgroup::cyclic group:Z4]]. (3)
# The subgroup <math>\langle a^2, x \rangle</math>, <math>\langle a^2, ax \rangle</math> and <math>\langle a^2, ay \rangle</math>. These are all normal subgroups but are related by outer automorphisms. Isomorphic to [[subgroup::Klein four-group]]. (3)
# The subgroup <math>\langle a, xy \rangle</math>. This is an [[isomorph-free subgroup]] of order eight, containing the three non-characteristic cyclic subgroups of order four. Isomorphic to [[subgroup::quaternion group]]. (3)
# The subgroups <math>\langle a,y \rangle</math>, <math>\langle x, y \rangle</math> and <math>\langle ax, y \rangle</math>. These are all normal and related by outer automorphisms. Isomorphic to [[subgroup::direct product of Z4 and Z2]]. (3)
# The subgroups <math>\langle a,x \rangle</math>, <math>\langle xy, ay \rangle</math> and <math>\langle axy, ay \rangle</math>. These are all normal and are related by outer automorphisms. Isomorphic to [[subgroup::dihedral group:D8]]. (3)
# The whole group. (1)

==GAP implementation==

{{GAP ID|16|13}}

Nontrivial semidirect product of Z4 and Z4

2021-02-26T01:55:46Z

Anarchic Fox: /* Definition */ Added more information.

{{particular group}}
[[Importance rank::3| ]]
==Definition==

===A presentation as a metacyclic group===

The group, a [[semidirect product]] <math>Z_4 \rtimes Z_4</math>, can be defined by:

<math>G := \langle a,x \mid a^4 = x^4 = e, xax^{-1} = a^{-1} \rangle</math>.

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 16}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|16}} ||
|-
| {{arithmetic function value order p-log etc|4}}
|-
| {{arithmetic function value given order|exponent of a group|4|16}} ||
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|derived length|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|2|16|4}}
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|rank of a p-group|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|number of subgroups|15|16|4}} ||
|-
| {{arithmetic function value given order and p-log|number of conjugacy classes of subgroups|13|16|4}} ||
|-
| {{arithmetic function value given order and p-log|number of conjugacy classes|10|16|4}} ||
|}

==Group properties==

{{compare and contrast group properties|order = 16}}

{| class="sortable" border="1"
!Property !! Satisfied? !! Explanation !! Comment
|-
|[[Dissatisfies property::Abelian group]] || No || ||
|-
|[[Satisfies property::Group of prime power order]] || Yes || ||
|-
|[[Satisfies property::Nilpotent group]] || Yes || ||
|-
| [[Satisfies property::Metabelian group]] || Yes || ||
|-
| [[Satisfies property::Metacyclic group]] || Yes || ||
|-
|[[Satisfies property::Supersolvable group]] || Yes || ||
|-
|[[Satisfies property::Group of nilpotency class two]] || Yes || ||
|-
|[[Dissatisfies property::T-group]] || No|| ||
|-
|[[Satisfies property::Directly indecomposable group]] || Yes || ||
|-
|[[Dissatisfies property::Splitting-simple group]] || No || ||
|-
|[[Dissatisfies property::UL-equivalent group]] || No || || See also [[nilpotent not implies UL-equivalent]]
|}

==Subgroups==

{{further|[[subgroup structure of nontrivial semidirect product of Z4 and Z4]]}}

{{#lst:subgroup structure of nontrivial semidirect product of Z4 and Z4|summary}}

==GAP implementation==

{{GAP ID|16|4}}

===Other descriptions===
<pre>gap> G := F/[F.1^2, F.2^4, F.1*F.2*F.1^(-1)*F.2^(-1),F.3^4,F.3*F.1*F.3^(-1)*F.1^(-1),F.3*F.2*F.3^(-1)*F.2^(-1)*F.1^(-1), F.3^2 * F.2^2];
<fp group on the generators [ f1, f2, f3 ]>
gap> IdGroup(G);
[ 16, 4 ]</pre>

SmallGroup(16,3)

2021-02-26T01:27:46Z

Anarchic Fox: /* Definition */ Added more information.

{{particular group}}

==Definition==

This group is a [[semidirect product]] <math>(Z_4 \times Z_2) \rtimes Z_2</math>, and can be defined using the following presentation:

<math>G := \langle a,b,x \mid a^4 = b^2 = x^2 = e, ab = ba, bx = xb, xax^{-1} = ab \rangle</math>

Note that <math>G</math> is generated by <math>a,c</math> alone, because the final relation allows us to write <math>b</math> in terms of <math>a</math> and <math>c</math>.

The subgroup <math>\langle a,b \rangle</math> is isomorphic to the [[direct product of Z4 and Z2]], and the element <math>c</math> is an element of order two that acts on the subgroup <math>\langle a,b \rangle</math> by conjugation by fixing <math>b</math> and sending <math>a</math> to <math>ab</math>.

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 16}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|16}} ||
|-
| {{arithmetic function value order p-log etc|4}}
|-
| {{arithmetic function value given order|exponent of a group|4|16}} ||
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|derived length|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|2|16|4}} ||
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|rank of a p-group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|3|16|4}} ||
|-
| {{arithmetic function value given order and p-log|number of conjugacy classes|10|16|4}} ||
|-
| {{arithmetic function value given order and p-log|number of conjugacy classes of subgroups|17|16|4}} ||
|}

==Group properties==

{{quotation|Want to compare with other groups of the same order? Check out [[groups of order 16#Group properties]].}}

{| class="wikitable" border="1"
!Property !! Satisfied !! Explanation
|-
|[[Dissatisfies property::abelian group]] || No ||<math>a,c</math> don't commute
|-
|[[Satisfies property::group of prime power order]] || Yes ||
|-
|[[Satisfies property::nilpotent group]] || Yes || [[prime power order implies nilpotent]]
|-
|[[Satisfies property::group of nilpotency class two]] || Yes ||
|-
|[[Satisfies property::supersolvable group]] || Yes ||
|-
|[[Dissatisfies property::T-group]] || No || The subgroup <math>\langle a \rangle</math> is [[2-subnormal subgroup|2-subnormal]], not [[normal subgroup|normal]]
|-
|[[Dissatisfies property::monolithic group]] || No ||
|-
|[[Dissatisfies property::one-headed group]] || No ||
|-
|[[Dissatisfies property::ambivalent group]] || No ||
|-
|[[Dissatisfies property::rational group]] || No ||
|-
|[[Dissatisfies property::rational-representation group]] || No ||
|}

==Subgroups==

{{further|[[subgroup structure of SmallGroup(16,3)]]}}

# The [[subgroup::trivial group]]. (1)
# The subgroup <math>\langle b \rangle</math>, which is the unique characteristic subgroup of order <math>2</math>. Isomorphic to [[subgroup::cyclic group:Z2]]. It is the [[commutator subgroup]], and can also be described as the unique group of order two containing an element that is ''not'' a square but is a product of squares. The quotient group is isomorphic to [[quotient group::direct product of Z4 and Z2]]. (1)
# The subgroups <math>\langle a^2 \rangle</math> and <math>\langle a^2b \rangle</math>, which are both normal subgroups related by an outer automorphism. Isomorphic to [[subgroup::cyclic group:Z2]]. The quotient group for each is isomorphic to [[quotient group::dihedral group:D8]]. (2)
# The subgroups <math>\langle c \rangle</math>, <math>\langle bc \rangle</math>, <math>\langle a^2c \rangle</math> and <math>\langle a^2bc \rangle</math>. Neither is normal, and they come in two conjugacy classes of size two each. Isomorphic to [[subgroup::cyclic group:Z2]]. (4)
# The subgroup <math>\langle a^2,b \rangle</math>, which is the [[center]], [[agemo subgroups of group of prime power order|first agemo subgroup]], and [[Frattini subgroup]]. Isomorphic to [[subgroup::Klein four-group]]. The quotient group is isomorphic to [[quotient group::Klein four-group]]. (1)
# The subgroups <math>\langle b,c \rangle</math> and <math>\langle b, a^2c\rangle</math>. Both are [[normal subgroup]]s and are related by an [[outer automorphism]]. Isomorphic to [[subgroup::Klein four-group]]. The quotient group is isomorphic to [[quotient group::cyclic group:Z4]]. (2)
# The subgroups <math>\langle a^2,c \rangle</math>, <math>\langle a^2,bc \rangle</math>, <math>\langle a^2b,c\rangle</math>, and <math>\langle a^2b, bc \rangle</math>. Two conjugacy classes of size two each. Isomorphic to [[subgroup::Klein four-group]]. The quotient group is also isomorphic to a [[quotient group::Klein four-group]]. (4)
# The subgroups <math>\langle a \rangle</math>, <math>\langle ab \rangle</math>, <math>\langle ac \rangle</math>, and <math>\langle abc \rangle</math>. All of them are related by outer automorphisms, and they form two conjugacy classes of subgroups of size two each: <math>\langle a \rangle</math> is conjugate to <math>\langle ab \rangle</math>, while <math>\langle ac \rangle</math> is conjugate to <math>\langle abc \rangle</math>. Isomorphic to [[subgroup::cyclic group:Z4]]. (4)
# The subgroups <math>\langle a,b \rangle</math> and <math>\langle ac, b \rangle</math>. These are both [[normal subgroup]]s related by an outer automorphism. Isomorphic to [[subgroup::direct product of Z4 and Z2]]. The quotient group is isomorphic to [[quotient group::cyclic group:Z2]]. (2)
# The subgroup <math>\langle a^2,b,c \rangle</math>. Isomorphic to [[subgroup::elementary abelian group:E8]]. The quotient group is isomorphic to [[quotient group::cyclic group:Z2]]. (1)
# The whole group. (1)

==Subgroup-defining functions==

{| class="wikitable" border="1"
! Subgroup-defining function !! Subgroup type in list !! Page on subgroup embedding !! Isomorphism class !! Comment
|-
| [[Center]] || (4) || [[subgroup-defining function value as embedding::center;center of SmallGroup(16,3)| ]][[center of SmallGroup(16,3)]] || [[subgroup-defining function value::center;Klein four-group| ]][[Klein four-group]] ||
|-
| [[Commutator subgroup]] || (2) || [[subgroup-defining function value as embedding::commutator subgroup;commutator subgroup of SmallGroup(16,3)| ]][[commutator subgroup of SmallGroup(16,3)]] || [[subgroup-defining function value::commutator subgroup;cyclic group:Z2| ]][[cyclic group:Z2]] ||
|-
| [[Frattini subgroup]] || (4) || [[subgroup-defining function value as embedding::center;center of SmallGroup(16,3)| ]][[center of SmallGroup(16,3)]] || [[subgroup-defining function value::center;Klein four-group| ]][[Klein four-group]] ||
|-
| [[Socle]] || (4) || [[subgroup-defining function value as embedding::center;center of SmallGroup(16,3)| ]][[center of SmallGroup(16,3)]] || [[subgroup-defining function value::center;Klein four-group| ]][[Klein four-group]] ||
|-
| [[omega subgroups of group of prime power order|first omega subgroup]] || (7) || || [[elementary abelian group:E8]] ||
|}

==GAP implementation==

{{GAP ID|16|3}}

===Other descriptions===

The group can be constructed using the following GAP commands:

<pre>gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^4, F.2^2, F.1*F.2*F.1^(-1)*F.2^(-1),F.3^2,F.3*F.2*F.3^(-1)*F.2^(-1),F.3*F.1*F.3^(-1)*F.2^(-1)*F.1^(-1)];
<fp group on the generators [ f1, f2, f3 ]></pre>

SmallGroup(48,51)

2021-02-22T02:22:02Z

Anarchic Fox: Sorry, wrong page.

{{particular group}}

This page was created in error. It will be corrected at a later date.

SmallGroup(48,50)

2021-02-22T02:20:56Z

Anarchic Fox: Created group page.

{{particular group}}

==Definition==

This [[group]] is a [[semidirect product]] <math>(Z_{2})^4 \rtimes Z_3</math>. It is given by:

<math>\langle a,b,c,d,x \mid a^2 = b^2 = c^2 = d^2 = x^3 = e, ab = ba, ac = ca, ad = da, bc = cb, bd = db, cd = dc, xax^{-1} = d, xbx^{-1} = bc, xcx^{-1} = b, xdx^{-1} = ad \rangle</math>

where <math>e</math> is the identity element.

The element <math>x</math> acts as the following matrix if we use <math>\{a,b,c,d \}</math> as the basis for [[elementary abelian group:E16]] viewed as a four-dimensional vector space over [[field:F2]]:

<math>\begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 \\\end{pmatrix}</math>

Note that this matrix has order three, explaining the group structure.

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 48}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups
|-
| {{arithmetic function value order|48}} ||
|-
| {{arithmetic function value given order|exponent of a group|6|48}} ||
|}

==GAP implementation==

{{GAP ID|48|50}}