M27

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Definition

This group, sometimes denoted $M_{27}$ or $M_3(3)$, is defined as the semidirect product of the cyclic group of order nine and a cyclic group of order three, acting on it by nontrivial automorphisms.

It is given by the presentation (with $e$ denoting the identity element):

$\langle a,b \mid a^9 = b^3 = e, bab^{-1} = a^4 \rangle$

It is part of the family of groups: semidirect product of cyclic group of prime-square order and cyclic group of prime order.

Arithmetic functions

Function	Value	Similar groups	Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set)	27	groups with same order
underlying prime of p-group	3
prime-base logarithm of order	3	groups with same prime-base logarithm of order
exponent of a group	9	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	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
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
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

Group properties

Property	Satisfied	Explanation
abelian group	No
group of prime power order	Yes
nilpotent group	Yes
extraspecial group	Yes
Frattini-in-center group	Yes

GAP implementation

Group ID

This finite group has order 27 and has ID 4 among the groups of order 27 in GAP's SmallGroup library. For context, there are groups of order 27. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(27,4)

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

gap> G := SmallGroup(27,4);

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

IdGroup(G) = [27,4]

or just do:

IdGroup(G)

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