C₃²×A₄ #1114

order	108
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = S₄×AGL(2,𝔽₃) (order 10368) · Inn(G) = A₄ · Out(G) = C₂×AGL(2,𝔽₃)
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product	C₃²×A₄
unambiguous semidirect product	C₂²⋊C₃³
unambiguous semidirect product	(C₂²×C₃)⋊C₃²

Fitting, socle	C₂²×C₃² ⋊ C₃
derived, Sylow	C₂² ⋊ C₃³

	subgroup	G/H
center, hypercenter	C₃²	A₄
commutator, cosocle, hypocenter	C₂²	C₃³
Frattini	C₁	G
Fitting, socle	C₂²×C₃²	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃²

62 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 10 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 41);