General linear group:GL(2,Z4)

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Definition

This group can be defined in the following equivalent ways:

1. It is the group $GL(2,\mathbb {Z} _{4})$ or $GL(2,\mathbb {Z} /4\mathbb {Z} )$, i.e., the general linear group of degree two over the ring of integers modulo $4$.
2. It is the group $GL(2,\mathbb {F} _{2}[t]/(t^{2}))$.

Note that although the rings in question are different, the corresponding general linear groups are isomorphic.

GAP implementation

Group ID

This finite group has order 96 and has ID 195 among the groups of order 96 in GAP's SmallGroup library. For context, there are groups of order 96. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(96,195)

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

gap> G := SmallGroup(96,195);

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

IdGroup(G) = [96,195]

or just do:

IdGroup(G)

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

Other descriptions

The group can also be defined using the GeneralLinearGroup and ZmodnZ functions:

GL(2,ZmodnZ(4))