C₃×#3689 #88599

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	C₂×C₃²	S₄
LCS	G ▷ Q₈⋊C₉
UCS	C₁ ◁ C₂×C₃

57 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 246);