order	64
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂²×#3003 (order 512) · Inn(G) = D₄ · Out(G) = C₂³×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

C₄×Q₁₆

Subgroups

55 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);(5,6,9,12)(7,13,15,20)(8,16,18,11)(10,17,19,14);(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 120);