Element structure of special linear group:SL(2,5)

This article gives detailed information about the element structure of special linear group:SL(2,5), which is a group of order 120.

Order computation
The group $$SL(2,5)$$ has order 120. with prime factorization $$120 = 2^3 \cdot 3^1 \cdot 5^1 = 8 \cdot 3 \cdot 5$$. Below are listed various methods that can be used to compute the order, all of which should give the answer 120:

Interpretation as special linear group of degree two
In the table below, $$q = 5$$. Note that the information is presented for generic odd $$q$$ and then computed numerically for $$q = 5$$.

Interpretation as double cover of alternating group
$$SL(2,5)$$ is isomorphic to $$2 \cdot A_n,n = 5$$. 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:

Number of conjugacy classes
The group has 9 conjugacy classes. This number can be computed in a variety of ways: