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C₃²×C₂²⋊C₉ #66441

order	324
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉ · Schur multiplier (H₂) = C₂×C₃³

Aliases

C₃²×C₂²⋊C₉

Splittings

C₂² ⋊ C₃²×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃³	A₄
commutator, hypocenter	C₂²	C₃²×C₉
Frattini	C₃	C₃²×A₄
Fitting, socle, prime radical	C₂²×C₃³	C₃
cosocle	C₂²×C₃	C₃³
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃³

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₃⋊(C₃⋊((C₂²⋊C₉)⋊C₂)), C₃⋊(C₃⋊((C₂²⋊C₉)⋊C₄)), (C₂²⋊C₉)⋊#157, …
maximal subgroup of	C₃²×(C₂²⋊C₉)⋊C₂, C₃×C₃⋊((C₂²⋊C₉)⋊C₂), C₃⋊(C₃⋊((C₂²⋊C₉)⋊C₂)), …
maximal normal subgroup of	C₃²×(C₂²⋊C₉)⋊C₂, C₃×C₃⋊((C₂²⋊C₉)⋊C₂), C₃⋊(C₃⋊((C₂²⋊C₉)⋊C₂)), …
maximal quotient of	C₃²×Q₈⋊C₉, C₂×C₃²×C₂²⋊C₉, C₃²×C₂²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 133);