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C₉×C₃⋊Dic₃ #66405

order	324
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₉ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₉×C₃⋊Dic₃
unambiguous semidirect product	C₃⋊(C₉×Dic₃)

Splittings

derived	C₃² ⋊ C₄×C₉
Sylow	C₃²×C₉ ⋊ C₄

Subgroups

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

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 — 3 generators on 19 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13048248, #13048249, #13048250, …
maximal subgroup of	C₉×#622, C₉⋊#622, D₃×C₉×Dic₃, …
maximal normal subgroup of	C₉×#622, C₉⋊#622, D₃×C₉×Dic₃, …
maximal quotient of	C₉×C₃⋊(C₃⋊C₈), C₂×C₉×C₃⋊Dic₃, C₉×C₃⋊Dic₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 97);