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 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