order	432
isoclinism	isoclinic class of C₃⋊(#41⋊C₃)
automorphism	Aut(G) = C₂×D₃²×S₄ (order 1728) · Inn(G) = D₃×A₄ · Out(G) = C₂²×D₃
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×C₃⋊(#41⋊C₃)
unambiguous semidirect product	C₃⋊(C₃×#41⋊C₃)
unambiguous semidirect product	#239⋊C₃²
unambiguous semidirect product	(C₃×#239)⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃×A₄
commutator, cosocle, hypocenter	C₃×Q₈	C₂×C₃²
Frattini	C₂	C₃×D₃×A₄
Fitting	C₃²×Q₈	C₂×C₃
socle	C₂×C₃²	C₂×A₄
LCS	G ▷ C₃×Q₈
UCS	C₁ ◁ C₂×C₃

105 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 22 points:

(1,8,5,2)(3,15,12,6,9,16,10,13,4,14,11,7)(21,22);(1,11,5,9)(2,15,8,13)(3,4,10,12)(6,7,14,16)(17,18,19)(20,21,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 622);