C₃×#3688 #88597

order	432
isoclinism	isoclinic class of #3688
automorphism	Aut(G) = C₂²×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 864) · Inn(G) = (C₂²⋊C₉)⋊C₂ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, hypercenter	C₂×C₃	(C₂²⋊C₉)⋊C₂
commutator, cosocle, hypocenter	Q₈⋊C₉	C₂×C₃
Frattini	C₂×C₃	C₃×S₄
Fitting	C₃²×Q₈	D₃
socle, prime radical	C₂×C₃²	S₄
LCS	G ▷ Q₈⋊C₉
UCS	C₁ ◁ C₂×C₃

49 conjugacy classes of subgroups.

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 28 points:

(1,15,10,7,5,6,12,13)(2,3,14,9,8,11,16,4)(17,18,19)(21,23)(22,26)(24,25)(27,28);(3,9,12)(4,10,11)(6,13,16)(7,14,15)(20,28,23,26,25,24,22,21,27)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 244);