Unitriangular matrix group:UT(4,2)

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Definition

This group is defined in the following equivalent ways:

• It is the group of upper-triangular $4\times 4$ matrices with $1$s on the diagonal over the field of two elements.
• It is a $2$-Sylow subgroup of general linear group:GL(4,2).
• It is the $2$-Sylow subgroup of the holomorph of the elementary abelian group of order eight, which is the general affine group.
• It is the $2$-Sylow subgroup of the holomorph of the direct product of Z4 and Z2.

GAP implementation

Group ID

This finite group has order 64 and has ID 138 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(64,138)

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

gap> G := SmallGroup(64,138);

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

IdGroup(G) = [64,138]

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 be constructed using the SylowSubgroup and GL functions:

SylowSubgroup(GL(4,2),2)

By first defining the Holomorph function, we can also write this as:

SylowSubgroup(Holomorph(ElementaryAbelianGroup(8)),2)

or as:

SylowSubgroup(Holomorph(DirectProduct(CyclicGroup(4),CyclicGroup(2))),2)