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C₉×C₂²⋊C₉ #66354

order	324
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₉² · Schur multiplier (H₂) = C₂×C₉

Aliases

direct product	C₉×C₂²⋊C₉
unambiguous semidirect product	C₂²⋊C₉²

Splittings

C₂² ⋊ C₉²

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₉	A₄
commutator, hypocenter	C₂²	C₉²
Frattini	C₃²	C₃×A₄
Fitting	C₂²×C₃×C₉	C₃
socle, prime radical, cosocle	C₂²×C₃²	C₃²
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₉

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₉⋊((C₂²⋊C₉)⋊C₂), C₉⋊((C₂²⋊C₉)⋊C₄), D₉×(C₂²⋊C₉)⋊C₂, …
maximal subgroup of	C₉×(C₂²⋊C₉)⋊C₂, C₉⋊((C₂²⋊C₉)⋊C₂), D₉×C₂²⋊C₉, …
maximal normal subgroup of	C₉×(C₂²⋊C₉)⋊C₂, C₉⋊((C₂²⋊C₉)⋊C₂), D₉×C₂²⋊C₉, …
maximal quotient of	C₉×Q₈⋊C₉, C₂×C₉×C₂²⋊C₉, C₉×C₂²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 46);