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C₃²×#216 #88831

order	432
isoclinism	isoclinic class of #213
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₃

Aliases

C₃²×#216

Splittings

C₃³ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃²	#67
commutator, cosocle	C₃×C₄	C₂²×C₃²
Frattini	C₄	C₂×C₃²×D₃
Fitting	C₃³×Q₈	C₂
socle, prime radical	C₂×C₃³	D₄
hypercenter	C₃²×Q₈	D₃
hypocenter	C₃	C₃²×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×C₄ ◁ C₃²×Q₈

156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 25 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×#216, C₃×#63943, C₃×#63944, …
maximal normal subgroup of	C₃×D₃×#216, C₃×#63943, C₃×#63944, …
maximal quotient of	C₃²×#807, C₃²×#810, C₃²×#835, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 478);