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C₉×#213 #88503

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) = C₂

Aliases

Splittings

Fitting	C₃×D₄×C₉ ⋊ C₂
Sylow	C₃×C₉ ⋊ D₈

Subgroups

	subgroup	G/H
center	C₂×C₉	#67
commutator	C₃×C₄	C₂²×C₉
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	C₃×D₄×C₉	C₂
socle	C₂×C₃²	C₃×D₄
prime radical	C₃×#213	C₃
cosocle	C₃²×C₄	C₂²×C₃
hypercenter	D₄×C₉	D₃
hypocenter	C₃	C₉×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ D₄×C₉

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 72

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

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×#213, #11721173, #11721174, …
maximal normal subgroup of	D₉×#213, #11721173, #11721174, …
maximal quotient of	C₉×#807, C₉×#809, C₉×#826, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 150);