C₃²×#41⋊C₃ #89052

order	432
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃³

Aliases

direct product	C₃²×#41⋊C₃
unambiguous semidirect product	#41⋊C₃³
unambiguous semidirect product	(C₃×#41)⋊C₃²

Fitting	C₃²×#41 ⋊ C₃
Sylow	#41 ⋊ C₃³

	subgroup	G/H
center, hypercenter	C₃²×C₄	A₄
commutator, cosocle, hypocenter	Q₈	C₂×C₃³
Frattini	C₂	C₂×C₃²×A₄
Fitting	C₃²×#41	C₃
socle	C₂×C₃²	C₂×A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₃²×C₄

170 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 699);