Direct product of A5 and Z7

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Definition

This group is defined as the external direct product of alternating group:A5 (a simple non-abelian group of order 60) and cyclic group:Z7 (a group of order 7).

It is the smallest example of a non-solvable group that is a p-solvable group for some prime number $p$ dividing its order (in this case, $p = 7$). In fact, the group is a p-nilpotent group for $p = 7$.

GAP implementation

Group ID

This finite group has order 420 and has ID 13 among the groups of order 420 in GAP's SmallGroup library. For context, there are 41 groups of order 420. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(420,13)

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

gap> G := SmallGroup(420,13);

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

IdGroup(G) = [420,13]

or just do:

IdGroup(G)

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