SmallGroup(16,3)

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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Definition

This group can be defined using the following presentation:

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

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

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

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions
Function Value Similar groups Explanation for function value
underlying prime of p-group 2
order (number of elements, equivalently, cardinality or size of underlying set) 16 groups with same order
prime-base logarithm of order 4 groups with same prime-base logarithm of order
max-length of a group 4 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 4 chief length equals prime-base logarithm of order for group of prime power order
composition length 4 composition length equals prime-base logarithm of order for group of prime power order
exponent of a group 4 groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent 2 groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class 2 groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
derived length 2 groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length
Frattini length 2 groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group 3 groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group
rank of a p-group 3 groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group
normal rank of a p-group 3 groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group
characteristic rank of a p-group 3 groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group
number of conjugacy classes 10 groups with same order and number of conjugacy classes | groups with same prime-base logarithm of order and number of conjugacy classes | groups with same number of conjugacy classes
number of conjugacy classes of subgroups 17 groups with same order and number of conjugacy classes of subgroups | groups with same prime-base logarithm of order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups

Group properties

Want to compare with other groups of the same order? Check out groups of order 16#Group properties.
Property Satisfied Explanation
abelian group No a,c don't commute
group of prime power order Yes
nilpotent group Yes prime power order implies nilpotent
group of nilpotency class two Yes
supersolvable group Yes
T-group No The subgroup \langle a \rangle is 2-subnormal, not normal
monolithic group No
one-headed group No
ambivalent group No
rational group No
rational-representation group No

Subgroups

Further information: subgroup structure of SmallGroup(16,3)

  1. The trivial group. (1)
  2. The subgroup \langle b \rangle, which is the unique characteristic subgroup of order 2. Isomorphic to 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 direct product of Z4 and Z2. (1)
  3. The subgroups \langle a^2 \rangle and \langle a^2b \rangle, which are both normal subgroups related by an outer automorphism. Isomorphic to cyclic group:Z2. The quotient group for each is isomorphic to dihedral group:D8. (2)
  4. The subgroups \langle c \rangle, \langle bc \rangle, \langle a^2c \rangle and \langle a^2bc \rangle. Neither is normal, and they come in two conjugacy classes of size two each. Isomorphic to cyclic group:Z2. (4)
  5. The subgroup \langle a^2,b \rangle, which is the center, first agemo subgroup, and Frattini subgroup. Isomorphic to Klein four-group. The quotient group is isomorphic to Klein four-group. (1)
  6. The subgroups \langle b,c \rangle and \langle b, a^2c\rangle. Both are normal subgroups and are related by an outer automorphism. Isomorphic to Klein four-group. The quotient group is isomorphic to cyclic group:Z4. (2)
  7. The subgroups \langle a^2,c \rangle, \langle a^2,bc \rangle, \langle a^2b,c\rangle, and \langle a^2b, bc \rangle. Two conjugacy classes of size two each. Isomorphic to Klein four-group. The quotient group is also isomorphic to a Klein four-group. (4)
  8. The subgroups \langle a \rangle, \langle ab \rangle, \langle ac \rangle, and \langle abc \rangle. All of them are related by outer automorphisms, and they form two conjugacy classes of subgroups of size two each: \langle a \rangle is conjugate to \langle ab \rangle, while \langle ac \rangle is conjugate to \langle abc \rangle. Isomorphic to cyclic group:Z4. (4)
  9. The subgroups \langle a,b \rangle and \langle ac, b \rangle. These are both normal subgroups related by an outer automorphism. Isomorphic to direct product of Z4 and Z2. The quotient group is isomorphic to cyclic group:Z2. (2)
  10. The subgroup \langle a^2,b,c \rangle. Isomorphic to elementary abelian group:E8. The quotient group is isomorphic to cyclic group:Z2. (1)
  11. The whole group. (1)

Subgroup-defining functions

Subgroup-defining function Subgroup type in list Page on subgroup embedding Isomorphism class Comment
Center (4) center of SmallGroup(16,3) Klein four-group
Commutator subgroup (2) commutator subgroup of SmallGroup(16,3) cyclic group:Z2
Frattini subgroup (4) center of SmallGroup(16,3) Klein four-group
Socle (4) center of SmallGroup(16,3) Klein four-group
first omega subgroup (7) elementary abelian group:E8

GAP implementation

Group ID

This finite group has order 16 and has ID 3 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(16,3)

For instance, we can use the following assignment in GAP to create the group and name it G:

gap> G := SmallGroup(16,3);

Conversely, to check whether a given group G is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [16,3]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.


Other descriptions

The group can be constructed using the following GAP commands:

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 ]>