order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×#72216294 (order 2048) · Inn(G) = C₂² · Out(G) = C₂×#35680
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

C₄×M₃₂

Subgroups

98 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);(6,14)(8,16)(10,18)(12,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 839);