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C₃×C₉×A₄ #66434

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

Aliases

C₃×C₉×A₄

Splittings

Fitting	C₂²×C₃×C₉ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃²×C₉

Subgroups

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

110 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:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₃⋊(C₉⋊S₄), C₃⋊(C₉⋊SL(2,ℤ/4)), S₄×C₃⋊D₉, …
maximal subgroup of	C₃×C₉×S₄, C₉×C₃⋊S₄, C₃×C₉⋊S₄, …
maximal normal subgroup of	C₃×C₉×S₄, C₉×C₃⋊S₄, C₃×C₉⋊S₄, …
maximal quotient of	C₃×C₉×SL(2,𝔽₃), C₂×C₃×C₉×A₄, C₉²×A₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 126);