Finite matrix group SL_2(2)

The finite matrix group $\mathrm{SL}_2(2)$ consists of determinant-one $2\times2$ matrices over the field $\mathbb F_2$ , with matrix multiplication.

Order and nonsimplicity

There are $(2^2-1)(2^2-2)=6$ invertible matrices. Every nonzero determinant equals one, so $\mathrm{GL}_2(2)=\mathrm{SL}_2(2)$ . Acting on the three nonzero vectors gives an injective map to $S_3$ : a matrix fixing all three vectors fixes a basis. Both groups have order six, so this is an isomorphism. The normal subgroup corresponding to $A_3$ shows that it is not simple.

References

Peter J. Cameron, Notes on Classical Groups, §2.1, Theorems 2.2–2.3, pp. 15–16: orders, determinant and scalar kernels; §2.4, Theorem 2.10, pp. 20–22: simplicity.