# 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.

## 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.