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C₃×C₉⋊A₄ #66435

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

Aliases

C₃×C₉⋊A₄

Splittings

Fitting	C₂²×C₃×C₉ ⋊ C₃
Sylow	C₂² ⋊ C₃×3₋¹⁺²

Subgroups

	subgroup	G/H
center	C₃²	C₃×A₄
commutator, cosocle	C₂²×C₃	C₃³
Frattini	C₃	C₃²×A₄
Fitting	C₂²×C₃×C₉	C₃
socle	C₂²×C₃²	C₃²
prime radical	C₃²×A₄	C₃
hypercenter	C₃×C₉	A₄
hypocenter	C₂²	C₃×3₋¹⁺²
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃×C₉

86 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₉⋊A₄, D₃×C₉⋊A₄, C₃⋊(D₉⋊A₄), …
maximal normal subgroup of	C₃×D₉⋊A₄, D₃×C₉⋊A₄, C₃⋊(D₉⋊A₄), …
maximal quotient of	C₃×C₉⋊SL(2,𝔽₃), C₂×C₃×C₉⋊A₄, C₉×C₉⋊A₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 127);