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Derek Holt
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The following calculations seem to distinguish between them.

>  #LowIndexSubgroups(GA,4);
30
>  #LowIndexSubgroups(GB,4);
26

They have different numbers of homomorphisms onto $A_4$:

> #Homomorphisms(GA,Alt(4),Sym(4));
5
> #Homomorphisms(GB,Alt(4),Sym(4));
1

(The options third argument $\mathsf{Sym(4)}$ means count (surjective homomorphisms) up to conjugacy in $\mathsf{Sym(4)}$.)

Derek Holt
  • 37.4k
  • 4
  • 95
  • 149