# Special linear group:SL(2,7)

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## Definition

This group is defined as the special linear group of degree two over field:F7, the field of seven elements.

It is denoted $SL(2,7)$ or $SL_2(7)$.

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 336 groups with same order As $\! SL(2,q), q = 7$: $\! q^3 - q = q(q-1)(q+1) = 7^3 - 7 = 7(7 - 1)(7 + 1) = 7(6)(8) = 336$
exponent of a group 168 groups with same order and exponent of a group | groups with same exponent of a group
Frattini length 2 groups with same order and Frattini length | groups with same Frattini length Follows from special linear group is quasisimple and the fact that the center is isomorphic to cyclic group:Z2.

### Arithmetic function values of a counting nature

Function Value Explanation
number of conjugacy classes 11 As $SL(2,q), q = 7$, $q$ odd: $q + 4 = 7 + 4 = 11$. See element structure of special linear group of degree two.
number of conjugacy classes of subgroups 19
number of subgroups 224

## Group properties

Function Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No
simple group No
perfect group Yes See special linear group is perfect.
quasisimple group Yes See special linear group is quasisimple. The group is perfect, and the inner automorphism group is isomorphic to projective special linear group:PSL(2,7), which is simple.
almost simple group No
one-headed group Yes The center of order two is the unique maximal normal subgroup.
monolithic group Yes The center of order two is the unique minimal normal subgroup.
T-group Yes

## GAP implementation

### Group ID

This finite group has order 336 and has ID 114 among the groups of order 336 in GAP's SmallGroup library. For context, there are 228 groups of order 336. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(336,114)

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

gap> G := SmallGroup(336,114);

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

IdGroup(G) = [336,114]

or just do:

IdGroup(G)

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

### Other descriptions

Description Functions used
SL(2,7) SL