C₄⋊S₄ #980

order	96
automorphism	Aut(G) = D₄×S₄ (order 192) · Inn(G) = C₂×S₄ · Out(G) = C₂²
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

C₄⋊S₄

C₂²×C₄ ⋊ D₃

56 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 2 generators on 8 points:

(1,2,3,4)(5,6,8);(2,4)(6,7)

Polycyclic presentation:

2^3,3,2^2|b^2=c,b^a=bc,d^a=d^2,e^a=f,f^a=e,e^d=f,f^d=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 187);