# General linear group:GL(2,4)

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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## Definition

This group is defined in the following equivalent ways:

1. The general linear group of degree two over the field of four elements.
2. The direct product of the alternating group of degree five and cyclic group of order three.

## Arithmetic functions

Function Value Explanation
order 180 $(4^2 - 1)(4^2 - 4) = 180$
exponent 30 Elements of order $2,3,5,6,15$. Also fits general formula: exponent of $GL(2,q)$ is $p(q^2 - 1)$ where $p$ is the prime (here $p = 2, q = 4$).
derived length -- not a solvable group.
nilpotency class -- not a nilpotent group.
Frattini length 1 Frattini-free group: intersection of maximal subgroups is trivial.
max-length 5
number of subgroups 148
number of conjugacy classes 15
number of conjugacy classes of subgroups 21
composition length 2

## GAP implementation

### Group ID

This finite group has order 180 and has ID 19 among the groups of order 180 in GAP's SmallGroup library. For context, there are 37 groups of order 180. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(180,19)

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

gap> G := SmallGroup(180,19);

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

IdGroup(G) = [180,19]

or just do:

IdGroup(G)

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

### Other definitions

The group can also be defined using GAP's GeneralLinearGroup command:

`GeneralLinearGroup(2,4)`

or, the shorter way of invoking the command:

`GL(2,4)`