Holomorph of Z8

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Definition

This group (which we shall call $G$ 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 direct product of the cyclic group on eight elements, with its automorphism group.
• It is the holomorph of the ring $\Z/8\Z$. In other words, it is the general affine group $GA(1,\Z/8\Z)$.

The group has the following presentation (with $e$ denoting the identity element):

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

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#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)	32	groups with same order	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)
prime-base logarithm of order	5	groups with same prime-base logarithm of order
max-length of a group	5		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	5		chief length equals prime-base logarithm of order for group of prime power order
composition length	5		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	8	groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent	3	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	3	groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class	The derived subgroup is $\langle a^2 \rangle$, the next member of the lower central series is $\langle a^4 \rangle$
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	3	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	3	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	The subgroup $\langle a^4,x,y\rangle$ is an elementary abelian subgroup of maximum rank
normal rank of a p-group	2	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	2	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

GAP implementation

Group ID

This finite group has order 32 and has ID 43 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(32,43)

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

gap> G := SmallGroup(32,43);

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

IdGroup(G) = [32,43]

or just do:

IdGroup(G)

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

Other definitions

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

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

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

Holomorph(CyclicGroup(8))