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C₃²×Dic₁₂ #88821

order	432
isoclinism	isoclinic class of #204
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₃

Aliases

C₃²×Dic₁₂

Splittings

C₃³ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃²	D₁₂
commutator, cosocle	C₃×C₄	C₂²×C₃²
Frattini	C₄	C₂×C₃²×D₃
Fitting	C₃³×C₈	C₂
socle, prime radical	C₂×C₃³	D₄
hypercenter	C₃²×C₈	D₃
hypocenter	C₃	C₃²×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×C₄ ◁ C₃²×C₈

140 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)(10,23,13,12,14,20,19,21)(11,25,16,15,17,18,22,24);(1,2,3)(8,9)(10,17,14,11)(12,25,21,18)(13,16,19,22)(15,20,24,23)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#63547, C₃×#63549, C₃×#63550, …
maximal normal subgroup of	C₃×#63547, C₃×#63549, C₃×#63550, …
maximal quotient of	C₃²×#816, C₃²×#818, C₂×C₃²×Dic₁₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 468);