Central product of SL(2,5) and Z4

Definition
This group is defined in the following equivalent ways:


 * 1) It is the unique defining ingredient::subgroup of index two inside defining ingredient::general linear group:GL(2,5). Explicitly, it is the subgroup of general linear group:GL(2,5) comprising those matrices whose determinant is a square in field:F5 (which in this case means the determinant is $$\pm 1$$.
 * 2) It is the defining ingredient::central product of defining ingredient::special linear group:SL(2,5) and defining ingredient::cyclic group:Z4 with shared central subgroup cyclic group:Z2 (this is center of special linear group:SL(2,5) in SL(2,5) and Z2 in Z4 in cyclic group:Z4).

In symbols, the group is:

$$SL(2,5) *_{\mathbb{Z}_2} \mathbb{Z}_4$$