C₄×#41 #517

order	64
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = #72216167 (order 3072) · Inn(G) = C₂² · Out(G) = S₄×#120
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

155 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 12 points:

(1,2,3,4);(5,6)(7,10)(8,9)(11,12);(6,9)(10,12);(5,7,8,11)(6,10,9,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 198);