SL(2,3) in SL(2,5)

Definition
The group $$G$$ is taken as special linear group:SL(2,5): the special linear group of degree two over field:F5.

The subgroup $$H$$ is taken as a representative of the unique conjugacy class of subgroups of order 24. (The conjugacy class contains 5 subgroups).

This subgroup can be obtained explicitly using the quaternionic representation of special linear group:SL(2,3) over field:F5 (see also linear representation theory of special linear group:SL(2,3)).