# M32

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## Definition

The group is the semidirect product of the cyclic group of order sixteen and cyclic group of order two, where the latter acts by the $9^{th}$ power map: $G := \langle a,x \mid a^{16} = x^2 = e, xax = a^9 \rangle$.

## Position in classifications

Type of classification Position/number in classification
GAP ID $(32,17)$, i.e., $17^{th}$ among groups of order 32
Hall-Senior number 22 among groups of order 32
Hall-Senior symbol $32\Gamma_2k$

## 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
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 16 groups with same order and exponent of a group | groups with same exponent of a group cyclic subgroup of order 16.
prime-base logarithm of exponent 4 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 the derived subgroup is contained in the cyclic subgroup and is hence abelian
Frattini length 4 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 2 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 All proper subgroups are cyclic, dihedral, or Klein four-groups.
rank of a p-group 2 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 there exist Klein four-subgroups.
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 all abelian normal subgroups are cyclic.
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 There is a unique (hence characteristic) Klein four-subgroup.

## GAP implementation

### Group ID

This finite group has order 32 and has ID 17 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(32,17)

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

gap> G := SmallGroup(32,17);

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

IdGroup(G) = [32,17]

or just do:

IdGroup(G)

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

### Other descriptions

This group can be constructed using its presentation:

gap> F := FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> G := F/[F.1^(16), F.2^2, F.2 * F.1 * F.2^(-1) * F.1^7];
<fp group on the generators [ f1, f2 ]>