What are the groups 2^6 : 3 . S_6 or 2^4 : A_8 ? Are they some subgroups of S_6 or A_8?
I believe that 2 . A_n is the double cover of A_n, and "multiplying" with a number gives a covering group. But what does : stand for?
I read Pinter "A Book of Abstract Algebra" (that's my level, the notation appears in more advanced texts). Any suggestions where this notation is explained?
Thank you
This is ATLAS notation.
The : means a split extension, and $2^4$ means an elementary abelian group of that order. So $2^4:A_8$ denotes a group $G$ having an elementary abelian normal subgroup $N$ of order $16$ and a subgroup $H \cong A_8$ such that $G = NH$ and $N \cap H = 1$. So $G \cong N \rtimes A_8$. We can also deduce that the action of $A_8$ on $N$ is nontrivial, since otherwise it would be a direct product written as $2^4 \times A_8$.
In principal the bracketing of $2^6:3\cdot S_6$ is ambiguous but the only sensible interpretation is $2^6:(3 \cdot S_6)$, which denotes a group $G \cong N \rtimes H$ with $N$ elementary abelian of order $64$ and $H \cong 3 \cdot S_6$, a 3-fold cover of $S_6$. The $\cdot$ actually denotes a nonsplit extension.