Groups of order 8

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This article gives information about, and links to more details on, groups of order 8
See pages on algebraic structures of order 8| See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 8. See also more detailed information on specific subtopics through the links:

Information type Page summarizing information for groups of order 8
element structure (element orders, conjugacy classes, etc.) element structure of groups of order 8
subgroup structure subgroup structure of groups of order 8
linear representation theory linear representation theory of groups of order 8
projective representation theory of groups of order 8
modular representation theory of groups of order 8
endomorphism structure, automorphism structure endomorphism structure of groups of order 8
group cohomology group cohomology of groups of order 8

Statistics at a glance

To understand these in a broader context, see: groups of order 2^n|groups of prime-cube order

Since 8 = 2^3 is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.

Quantity Value Explanation
Total number of groups 5 See groups of prime-cube order, classification of groups of prime-cube order
Number of abelian groups 3 equals the number of unordered integer partitions of 3, the exponent part in 2^3. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of groups of class exactly two 2

The list

To learn more about how to come up with the list and prove that it is exhaustive (i.e., that these are precisely the isomorphism classes of groups of order 8), see classification of groups of prime-cube order


Common name for group Second part of GAP ID (GAP ID is (8,second part)) Hall-Senior number Hall-Senior symbol Nilpotency class Minimum size of generating set Probability in cohomology tree probability distribution
cyclic group:Z8 1 3 (3) 1 1 1/4
direct product of Z4 and Z2 2 2 (21) 1 2 7/16
dihedral group:D8 3 4 8\Gamma_2a_1 2 2 3/16
quaternion group 4 5 8\Gamma_2a_2 2 2 1/16
elementary abelian group:E8 5 1 (1^3) 1 3 1/16

Presentations

Further information: presentations for groups of order 8

Below are the power-commutator presentations for groups of order 8.


Group Second part of GAP ID (GAP ID is (p^3,2nd part) Nilpotency class Minimum size of generating set Prime-base logarithm of exponent \beta(1,2) \beta(1,3) \beta(2,3) \beta(1,2,3) full power-commutator presentation
cyclic group:Z8 1 1 1 3 1 0 1 0 [SHOW MORE]
direct product of Z4 and Z2 2 1 2 2 0 1 0 0 [SHOW MORE]
dihedral group:D8 3 2 2 2 0 0 0 1 [SHOW MORE]
quaternion group 4 2 2 2 0 1 1 1 [SHOW MORE]
elementary abelian group:E8 5 1 3 1 0 0 0 0 [SHOW MORE]


Subgroup/quotient relationships

Subgroup relationships

Orderuptoeightbysubgroupinclusion.png

Quotient relationships

Orderuptoeightbyquotients.png

Arithmetic functions

Functions taking values between 0 and 3

Group GAP ID (second part) Hall-senior number prime-base logarithm of exponent nilpotency class derived length Frattini length minimum size of generating set subgroup rank rank as p-group normal rank characteristic rank prime-base logarithm of order of derived subgroup prime-base logarithm of order of inner automorphism group
Cyclic group:Z8 1 3 3 1 1 3 1 1 1 1 1 0 0
Direct product of Z4 and Z2 2 2 2 1 1 2 2 2 2 2 2 0 0
Dihedral group:D8 3 4 2 2 2 2 2 2 2 2 1 1 2
Quaternion group 4 5 2 2 2 2 2 2 1 1 1 1 2
Elementary abelian group:E8 5 1 1 1 1 1 3 3 3 3 3 0 0
Mean (with equal weight on all groups) 2 1.4 1.4 2 2 2 1.8 1.8 1.6 0.4 0.8
Mean (using cohomology tree probability distribution) 2.1875 1.25 1.25 2.1875 1.8125 1.8125 1.75 1.75 1.5625 0.25 0.5
Here are some measures of central tendency (averages) and deviation measures: [SHOW MORE]

Same, with rows and columns interchanged:

Function Cyclic group:Z8 Direct product of Z4 and Z2 Dihedral group:D8 Quaternion group Elementary abelian group:E8
prime-base logarithm of exponent 3 2 2 2 1
nilpotency class 1 1 2 2 1
derived length 1 1 2 2 1
Frattini length 3 2 2 2 1
minimum size of generating set 1 2 2 2 3
subgroup rank 1 2 2 2 3
rank as p-group 1 2 2 1 3
normal rank as p-group 1 2 2 1 3
characteristic rank as p-group 1 2 1 1 3
Here are the correlations between these various arithmetic functions across the groups of order 8: [SHOW MORE]

Arithmetic function values of a counting nature

Group GAP ID (second part) Hall-senior number number of conjugacy classes number of subgroups number of conjugacy classes of subgroups number of normal subgroups number of automorphism classes of subgroups number of characteristic subgroups
Cyclic group:Z8 1 3 8 4 4 4 4 4
Direct product of Z4 and Z2 2 2 8 8 8 8 6 4
Dihedral group:D8 3 4 5 10 8 6 6 4
Quaternion group 4 5 5 6 6 6 4 3
Elementary abelian group:E8 5 1 8 16 16 16 4 2

Here is the same table, with rows and columns interchanged:

Function Cyclic group:Z8 Direct product of Z4 and Z2 Dihedral group:D8 Quaternion group Elementary abelian group:E8
number of conjugacy classes 8 8 5 5 8
number of subgroups 4 8 10 6 16
number of conjugacy classes of subgroups 4 8 8 6 16
number of normal subgroups 4 8 6 6 16
number of automorphism classes of subgroups 4 6 6 4 4
number of characteristic subgroups 4 4 4 3 2

Arithmetic function values of a representational nature

Group GAP ID (second part) Hall-senior number minimum degree of faithful permutation representation minimum degree of faithful transitive permutation representation minimum degree of faithful linear representation over \mathbb{C} symmetric genus
cyclic group:Z8 1 3 8 8 1  ?
direct product of Z4 and Z2 2 2 6 8 (at most) 2  ?
dihedral group:D8 3 4 4 4 2  ?
quaternion group 4 5 8 8 2  ?
elementary abelian group:E8 5 1 6 8 (at most) 3  ?

Numerical invariants

Group Conjugacy class sizes Degrees of irreducible representations
cyclic group:Z8 1 (8 times) 1 (8 times)
direct product of Z4 and Z2 1 (8 times) 1 (8 times)
dihedral group:D8 1,1,2,2,2 1,1,1,1,2
quaternion group 1,1,2,2,2 1,1,1,1,2
elementary abelian group:E8 1 (8 times) 1 (8 times)

Group properties

Property Cyclic group:Z8 Direct product of Z4 and Z2 Dihedral group:D8 Quaternion group Elementary abelian group:E8
cyclic group Yes No No No No
elementary abelian group No No No No Yes
abelian group Yes Yes No No Yes
homocyclic group Yes No No No Yes
metacyclic group Yes Yes Yes Yes No
metabelian group Yes Yes Yes Yes Yes
group of nilpotency class two Yes Yes Yes Yes Yes
maximal class group No No Yes Yes No
ambivalent group No No Yes Yes Yes
rational group No No Yes Yes Yes
rational-representation group No No Yes No Yes
group in which every element is automorphic to its inverse Yes Yes Yes Yes Yes
group in which any two elements generating the same cyclic subgroup are automorphic Yes Yes Yes Yes Yes
T-group Yes Yes No Yes Yes
C-group No No No No Yes
SC-group No No No No Yes
UL-equivalent group Yes Yes Yes Yes Yes
algebra group No Yes Yes No Yes

Families and classification

Further information: Classification of groups of order 8

Up to isoclinism

FACTS TO CHECK AGAINST for isoclinic groups (groups with an isoclinism between them):
by definition, isoclinic groups have isomorphic inner automorphism groups and isomorphic derived subgroups, with the isomorphisms compatible.
isoclinic groups have same nilpotency class|isoclinic groups have same derived length | isoclinic groups have same proportions of conjugacy class sizes | isoclinic groups have same proportions of degrees of irreducible representations
FACTS TO CHECK AGAINST for isoclinic groups (groups with an isoclinism between them):
by definition, isoclinic groups have isomorphic inner automorphism groups and isomorphic derived subgroups, with the isomorphisms compatible.
isoclinic groups have same nilpotency class|isoclinic groups have same derived length | isoclinic groups have same proportions of conjugacy class sizes | isoclinic groups have same proportions of degrees of irreducible representations

The equivalence classes up to being isoclinic were classified by Hall and Senior, and we call them Hall-Senior families.

Family name Isomorphism class of inner automorphism group Isomorphism class of derived subgroup Number of groups Nilpotency class Members Second part of GAP ID of members (sorted ascending) Hall-Senior numbers of members (sorted ascending) Smallest order of group isoclinic to these groups Stem groups (groups of smallest order)
\Gamma_1 (abelian groups) trivial group trivial group 3 1 cyclic group:Z8, direct product of Z4 and Z2, elementary abelian group:E8 1,2,5 1-3 1 trivial group
\Gamma_2 Klein four-group cyclic group:Z2 2 2 dihedral group:D8, quaternion group 3,4 4,5 8 dihedral group:D8, quaternion group
Total (2 rows) -- -- 5 -- -- -- -- -- --

Up to Hall-Senior genus

Up to the relation of groups having the same Hall-Senior genus, there are four equivalence classes:

Genus name Description of genus Members Hall-Senior numbers Second parts of GAP ID of members
(3) cyclic group cyclic group:Z8 3 1
(21) abelian group for partition 3 = 2 + 1 direct product of Z4 and Z2 2 2
8\Gamma_2a (the dihedral group is 8\Gamma_2a_1 and the quaternion group is 8\Gamma_2a_2) non-abelian groups dihedral group:D8, quaternion group 4,5 3,4
(1^3) elementary abelian group elementary abelian group:E8 1 5

Up to isologism for higher class

Since all the groups of order 8 has class at most two, we have a unique equivalence class under isologism for any class equal to or more than two.

Up to isologism for elementary abelian groups

Each of the abelian groups is in a different equivalence class under the equivalence relation of being isologic with respect to elementary abelian 2-groups. The two non-abelian groups are isologic to each other with respect to the variety of elementary abelian 2-groups.

Cohomology tree

Cohomologytreeorder8groups.png

Element structure

Further information: element structure of groups of order 8

Order statistics

FACTS TO CHECK AGAINST:

ORDER STATISTICS (cf. order statistics, order statistics-equivalent finite groups): number of nth roots is a multiple of n | Finite abelian groups with the same order statistics are isomorphic | Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring | Frobenius conjecture on nth roots

1-ISOMORPHISM (cf. 1-isomorphic groups): Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring | order statistics-equivalent not implies 1-isomorphic

Here are the statistics for a particular order.

Group Second part of GAP ID Hall-Senior number Number of elements of order 1 Number of elements of order 2 Number of elements of order 4 Number of elements of order 8
cyclic group:Z8 1 3 1 1 2 4
direct product of Z4 and Z2 2 2 1 3 4 0
dihedral group:D8 3 4 1 5 2 0
quaternion group 4 5 1 1 6 0
elementary abelian group:E8 5 1 1 7 0 0

Here are the number of n^{th} root statistics. The number of n^{th} roots equals the number of elements whose order divides n.

Group Second part of GAP ID Hall-Senior number Number of first roots Number of 2^{nd} roots Number of 4^{th} roots Number of 8^{th} roots
cyclic group:Z8 1 3 1 2 4 8
direct product of Z4 and Z2 2 2 1 4 8 8
dihedral group:D8 3 4 1 6 8 8
quaternion group 4 5 1 2 8 8
elementary abelian group:E8 5 1 1 8 8 8

Equivalence classes

No two of the groups of order 8 are order statistics-equivalent, and hence no two of them are 1-isomorphic.

Subgroup structure

Further information: subgroup structure of groups of order 8

Group Second part of GAP ID Subgroup structure page
Cyclic group:Z8 1 subgroup structure of cyclic group:Z8
Direct product of Z4 and Z2 2 subgroup structure of direct product of Z4 and Z2
Dihedral group:D8 3 subgroup structure of dihedral group:D8
Quaternion group 4 subgroup structure of quaternion group
Elementary abelian group:E8 5 subgroup structure of elementary abelian group:E8

Linear representation theory

Further information: linear representation theory of groups of order 8

Group GAP ID second part Hall-Senior number Nilpotency class Degrees as list Number of irreps of degree 1 (= order of abelianization) Number of irreps of degree 2 Total number of irreps (= number of conjugacy classes)
cyclic group:Z8 1 3 1 1,1,1,1,1,1,1,1 8 0 8
direct product of Z4 and Z2 2 2 1 1,1,1,1,1,1,1,1 8 0 8
dihedral group:D8 3 4 2 1,1,1,1,2 4 1 5
quaternion group 4 5 2 1,1,1,1,2 4 1 5
elementary abelian group:E8 5 1 1 1,1,1,1,1,1,1,1 8 0 8


Subgroup-defining functions

Values up to isomorphism type

Subgroup-defining function Cyclic group:Z8 Direct product of Z4 and Z2 Dihedral group:D8 Quaternion group Elementary abelian group:E8
center cyclic group:Z8 direct product of Z4 and Z2 cyclic group:Z2 cyclic group:Z2 elementary abelian group:E8
derived subgroup trivial group trivial group cyclic group:Z2 cyclic group:Z2 trivial group
Frattini subgroup cyclic group:Z4 cyclic group:Z2 cyclic group:Z2 cyclic group:Z2 trivial group

Automorphism groups

Group GAP ID (second part) Order of automorphism group Iso. class of automorphism group Log_2 of largest power of 2 dividing automorphism group Iso. class of 2-Sylow subgroup of automorphism group Log_2 of order of 2-core Iso. class of 2-core Log_2 of order of inner automorphism group Iso. class of inner automorphism group
cyclic group:Z8 1 4 Klein four-group 2 Klein four-group 2 Klein four-group 0 trivial group
direct product of Z4 and Z2 2 8 dihedral group:D8 3 dihedral group:D8 3 dihedral group:D8 0 trivial group
dihedral group:D8 3 8 dihedral group:D8 3 dihedral group:D8 3 dihedral group:D8 2 Klein four-group
quaternion group 4 24 symmetric group:S4 3 dihedral group:D8 2 Klein four-group 2 Klein four-group
elementary abelian group:E8 5 168 general linear group:GL(3,2) 3 dihedral group:D8 0 trivial group 0 trivial group

Associated constructs

Associated construct Cyclic group:Z8 Direct product of Z4 and Z2 Dihedral group:D8 Quaternion group Elementary abelian group:E8
automorphism group Klein four-group dihedral group:D8 dihedral group:D8 symmetric group:S4 general linear group:GL(3,2)
inner automorphism group trivial group trivial group Klein four-group Klein four-group trivial group
holomorph holomorph of Z8 unitriangular matrix group:UT(4,2) holomorph of D8 holomorph of Q8 general affine group:GA(3,2)