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C₃²×Dic₉ #66398

order	324
isoclinism	isoclinic class of D₉
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

C₃²×Dic₉

Splittings

derived	C₉ ⋊ C₃²×C₄
Sylow	C₃²×C₉ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃²	D₉
commutator, hypocenter	C₉	C₃²×C₄
Frattini	C₂×C₃	C₃²×D₃
Fitting	C₂×C₃²×C₉	C₂
socle, prime radical	C₂×C₃³	D₃
cosocle	C₂×C₉	C₂×C₃²
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₃²

86 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#6189, C₃×(C₃⋊D₉)⋊C₄, C₃×D₃×Dic₉, …
maximal normal subgroup of	C₃×#6189, C₃×(C₃⋊D₉)⋊C₄, C₃×D₃×Dic₉, …
maximal quotient of	C₃²×C₉⋊C₈, C₂×C₃²×Dic₉, C₃×C₉×Dic₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 90);