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C₃×#1094 #66442

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

Aliases

Splittings

C₂² ⋊ C₃×3₋¹⁺²

Subgroups

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

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 54

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

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72276200
maximal subgroup of	C₃×((C₂²⋊C₉)⋊C₂)⋊C₃, C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃), D₃×#1094, …
maximal normal subgroup of	C₃×((C₂²⋊C₉)⋊C₂)⋊C₃, C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃), D₃×#1094, …
maximal quotient of	C₃×#6204, C₂×C₃×#1094, C₃×(C₂²⋊C₉)⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 134);