order	432
isoclinism	isoclinic class of ((C₂²⋊C₉)⋊C₂)⋊C₃
automorphism	Aut(G) = C₂×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 432) · Inn(G) = ((C₂²⋊C₉)⋊C₂)⋊C₃ · Out(G) = C₂
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×((C₂²⋊C₉)⋊C₂)⋊C₃
unambiguous semidirect product	((C₂²⋊C₉)⋊C₂)⋊(C₂×C₃)
unambiguous semidirect product	#1094⋊C₂²
unambiguous semidirect product	(C₂×(C₂²⋊C₉)⋊C₂)⋊C₃
unambiguous semidirect product	(C₂×#1094)⋊C₂

Splittings

	subgroup	G/H
center, hypercenter	C₂	((C₂²⋊C₉)⋊C₂)⋊C₃
commutator, cosocle, hypocenter	C₂²⋊C₉	C₂²×C₃
Frattini	C₃	C₂×C₃×S₄
Fitting	C₂³×C₃²	D₃
socle	C₂³×C₃	C₃×D₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₂

120 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 15 points:

(3,4,6,5)(8,12,15,10,11,14)(9,13);(1,2)(4,5,6)(7,15,10,13,11,12,9,8,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 533);