C₃³×A₄ #66479

order	324
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃⁴ · Schur multiplier (H₂) = C₂×C₃⁶

Aliases

direct product	C₃³×A₄
unambiguous semidirect product	C₂²⋊C₃⁴
unambiguous semidirect product	(C₂²×C₃)⋊C₃³

Fitting, socle	C₂²×C₃³ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃⁴

	subgroup	G/H
center, hypercenter	C₃³	A₄
commutator, cosocle, hypocenter	C₂²	C₃⁴
Frattini	C₁	G
Fitting, socle	C₂²×C₃³	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃³

452 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 4 generators on 13 points:

(2,3,4);(1,4)(2,3)(11,12,13);(8,9,10);(5,6,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 171);