Groups of order 8

This article gives basic information comparing and contrasting the groups of order $8$.

The list

Common name for group	Second part of GAP ID (GAP ID is (8,second part))	Hall-Senior number	Hall-Senior symbol
cyclic group:Z8	1	3	$(3)$
direct product of Z4 and Z2	2	2	$(21)$
dihedral group:D8	3	4	$8\Gamma _{2}a_{1}$
quaternion group	4	5	$8\Gamma _{2}a_{2}$
elementary abelian group:E8	5	1	$(1^{3})$

Subgroup/quotient relationships

Subgroup relationships

Quotient relationships

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

Here now is the same table along with various measures of averages and deviations: [SHOW MORE]

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			2	1.4	1.4	2	2	2	1.8	1.8	1.6	0.4	0.8
Median			2	1	1	2	2	2	2	2	1	0	0
Mode			2	1	1	2	2	2	1	1	1	0	0
Standard deviation			0.71	0.55	0.55	0.71	0.71	0.84	0.84	0.89	0.55	1.1
Standard deviation to mean ratio		0.35	0.39	0.39	0.35	0.35	0.35	0.46	0.46	0.56	1.37	1.37

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
number of conjugacy classes	8	8	5	5	8
number of subgroups	4	8	10	6	16
number of normal subgroups	4	8	6	6	16
number of conjugacy classes of subgroups	4	8	8	6	16

Here are the correlations between these various arithmetic functions across the groups of order 8: [SHOW MORE]

Arithmetic function	prime-base logarithm of exponent	nilpotency class	derived length	Frattini length	minimum size of generating set	subgroup rank	rank	normal rank	characteristic rank	prime-base logarithm of order of derived subgroup	prime-base logarithm of order of inner automorphism group
prime-base logarithm of exponent	1	0	0	1	-1	-1	-0.85	-0.85	-0.79	0	0
nilpotency class	0	1	1	0	0	0	-0.33	-0.33	-0.61	1	1
derived length	0	1	1	0	0	0	-0.33	-0.33	-0.61	1	1
Frattini length	1	0	0	1	-1	-1	-0.85	-0.85	-0.79	0	0
minimum size of generating set	-1	0	0	-1	1	1	0.85	0.85	0.79	0	0
subgroup rank	-1	0	0	-1	1	1	0.85	0.85	0.79	0	0
rank as p-group	-0.85	-0.33	-0.33	-0.85	0.85	0.85	1	1	0.87	-0.33	-0.33
normal rank as p-group	-0.85	-0.33	-0.33	-0.85	0.85	0.85	1	1	0.87	-0.33	-0.33
characteristic rank as p-group	-0.79	-0.61	-0.61	-0.79	0.79	0.79	0.87	0.87	1	-0.61	-0.61
prime-base logarithm of order of derived subgroup	0	1	1	0	0	0	-0.33	-0.33	-0.61	1	1
prime-base logarithm of order of inner automorphism group	0	1	1	0	0	0	-0.33	-0.33	-0.61	1	1

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

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

Classification and families

Up to isoclinism

Up to the relation of being isoclinic, there are two equivalence classes:

Description of equivalence class	Members	Hall-Senior name	Second parts of IDs of members
abelian groups of order eight	cyclic group:Z8, direct product of Z4 and Z2, elementary abelian group:E8	$8\Gamma _{1}$	1,2,5
non-abelian groups of order eight	dihedral group:D8, quaternion group	$8\Gamma _{2}$	3,4

Up to Hall-Senior genus

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

Description of equivalence class	Members	Hall-Senior name	Hall-Senior numbers	Second parts of GAP ID of members
Cyclic group	cyclic group:Z8	$(3)$	3	1
Abelian group for partition $3=2+1$	direct product of Z4 and Z2	$(21)$	2	2
Non-abelian groups	dihedral group:D8, quaternion group	$8\Gamma _{2}a$ (the dihedral group is $8\Gamma _{2}a_{1}$ and the quaternion group is $8\Gamma _{2}a_{2}$)	4,5	3,4
Elementary abelian group	elementary abelian group:E8	$(1^{3})$	1	5

Element structure

Order statistics

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

Subgroup structure

Number of subgroups per isomorphism type

The number in each column is the number of subgroups in the given group of that isomorphism type:

Group	Second part of GAP ID	Hall-Senior number	cyclic group:Z2	cyclic group:Z4	Klein four-group	Total (row sum + 2, for trivial group and whole group)
cyclic group:Z8	1	3	1	1	0	4
direct product of Z4 and Z2	2	2	3	2	1	8
dihedral group:D8	3	4	5	1	2	10
quaternion group	4	5	1	3	0	6
elementary abelian group:E8	5	1	7	0	7	16

Number of normal subgroups per isomorphism type

Group	Second part of GAP ID	Hall-Senior number	cyclic group:Z2	cyclic group:Z4	Klein four-group	Total (row sum + 2, for trivial group and whole group)
cyclic group:Z8	1	3	1	1	0	4
direct product of Z4 and Z2	2	2	3	2	1	8
dihedral group:D8	3	4	1	1	2	6
quaternion group	4	5	1	3	0	6
elementary abelian group:E8	5	1	7	0	7	16

Number of subgroups of various kinds per order

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

Possibilities for maximal subgroups

Collection of isomorphism classes of maximal subgroups	Groups
cyclic group:Z4 only	cyclic group:Z8, quaternion group
Klein four-group only	elementary abelian group:E8
cyclic group:Z4 and Klein four-group	direct product of Z4 and Z2, dihedral group:D8