order	128
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) = C₂⁴×#442 (order 1024) · Inn(G) = D₈ · Out(G) = C₂⁴×C₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

C₄×SD₃₂

Subgroups

81 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2,3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20);(6,12)(7,19)(8,10)(9,17)(11,15)(14,20)(16,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 905);