= Composition chains in products with S5
{title2=$\ell(C_2\times S_5)=3,\quad\ell(S_5\times S_5)=4$}
The sign quotient of $S_5$ controls its cyclic composition factors, while its unique nontrivial proper normal subgroup $A_5$ supplies the nonabelian factor. There are four composition chains in $C_2\times S_5$: the three maximal normal subgroups are the kernels of the nonzero maps to $C_2$, and only $C_2\times A_5$ has two choices for its next term. In $S_5\times S_5$ there are eight chains: six pass through $A_5\times A_5$ and one of its two simple direct factors; two instead pass through a single $S_5$ direct factor. The equal-sign subgroup gives two of the six chains. It has no quotient $A_5$, since its odd diagonal element acts by an outer automorphism on either alternating factor.
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