C₄³ #374

order	64
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 86016
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₄³

Aliases

C₄³

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, socle, prime radical, cosocle	C₂³	C₂³
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₂³ ▷ C₁

129 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 55);