Special linear group:SL(2,3)

Definition
The special linear group $$SL(2,3)$$ is defined in the following equivalent ways:


 * It is the of degree two over a field of three elements. In other words, it is the group of $$2 \times 2$$ matrices with determinant $$1$$ over the field of three elements.
 * It is the double cover of alternating group $$2 \cdot A_4$$, i.e., it is the double cover of alternating group:A4.
 * It is a binary von Dyck group with the parameters $$(2,3,3)$$, i.e., it has the presentation:

$$\langle a,b,c \mid a^3 = b^3 = c^2 = abc \rangle$$.

It is a group of order $$24$$.

Automorphisms
The automorphism group of $$SL(2,3)$$ is isomorphic to the symmetric group of degree four. The inner automorphism group, which is the quotient of $$SL(2,3)$$ by its center, and is also the projective special linear group $$PSL(2,3)$$, is isomorphic to the alternating group of degree four.

To see how the outer automorphisms act, we can view $$SL(2,3)$$ as a subgroup of index two in $$GL(2,3)$$, the general linear group of order two over the field of three elements. The inner automorphism group, which is $$PGL(2,3)$$, is isomorphic to the symmetric group of degree four. Since $$SL(2,3)$$ is a normal subgroup, and since its center equals the center of $$GL(2,3)$$, the automorphisms of $$GL(2,3)$$ all restrict to automorphisms of $$SL(2,3)$$. $$SL(2,3)$$ has no other automorphisms.

Supergroups
The special linear group $$SL(2,3)$$ is a subgroup of the general linear group $$GL(2,3)$$. It is also a subgroup of $$SL(2,q)$$ for any prime power $$q \ge 3$$ (note that $$q$$ need not be a power of $$3$$).

Description by presentation
F := FreeGroup(3); G := F/[F.1^3*F.2^(-3), F.2^3*F.3^(-2), F.1^3*(F.1*F.2*F.3)^(-1)];