Element structure of special linear group:SL(2,3)

This article gives detailed information about the element structure of special linear group:SL(2,3).

See also element structure of special linear group of degree two over a finite field.

Order computation
The group $$SL(2,3)$$ has order 24. with prime factorization $$24 = 2^3 \cdot 3^1 = 8 \cdot 3$$. Below are listed various methods that can be used to compute the order, all of which should give the answer 24:

Conjugacy and automorphism class structure


Conjugacy classes
Note that since we are over field:F3, $$-1 = 2$$, so all the $$-1$$s below can be rewritten as $$2$$s.

Automorphism classes
Below are the orbits under the action of the automorphism group, i.e., the automorphism classes of elements of the group.



Interpretation as double cover of alternating group
$$SL(2,3)$$ is isomorphic to $$2 \cdot A_n,n = 4$$. Recall that we have the following rules to determine splitting and orders. The rules listed below are only for partitions that already correspond to even permutations, i.e., partitions that have an even number of even parts: