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C₉×#216 #88512

order	432
isoclinism	isoclinic class of #213
automorphism	Aut(G) = C₂²×C₃×D₃×D₄ (order 576) · Inn(G) = #67 · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₁

Aliases

Splittings

C₃×C₉ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂×C₉	#67
commutator	C₃×C₄	C₂²×C₉
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	C₃×Q₈×C₉	C₂
socle, prime radical	C₂×C₃²	C₃×D₄
cosocle	C₃²×C₄	C₂²×C₃
hypercenter	Q₈×C₉	D₃
hypocenter	C₃	C₉×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ Q₈×C₉

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×#216, #11721219, #11721220, …
maximal normal subgroup of	D₉×#216, #11721219, #11721220, …
maximal quotient of	C₉×#807, C₉×#810, C₉×#835, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 159);