There is a useful convention to decorate some of the groups with an index which is the smallest $n$ for which the group embeds in $S_n$ (edit:) as a transitive subgroup. Your $S_n, D_n, Q_8, ...$ suggest that you are using it. Of course, there is also another convention to use the size of the group instead...

There is a useful convention to decorate some of the groups with an index which is the smallest $n$ for which the group can act transitively on $n$ points, i.e. embeds in $S_n$ as a transitive subgroup. The notation for $S_n, A_n, D_n, C_n$, your $Q_8$ and for example Mathieu groups $M_{11}, M_{12}, M_{22}$ (although not other sporadic simple groups) follow this pattern. 

Of course, there is also another convention to use the size of the group instead...

There is a useful convention to decorate some of the groups with an index which is the smallest $n$ for which the group embeds in $S_n$. Your $S_n, D_n, Q_8, ...$ suggest that you are using it. Of course, there is also another convention to use the size of the group instead...

There is a useful convention to decorate some of the groups with an index which is the smallest $n$ for which the group embeds in $S_n$ (edit:) as a transitive subgroup. Your $S_n, D_n, Q_8, ...$ suggest that you are using it. Of course, there is also another convention to use the size of the group instead...