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C₃×C₃⋊(C₃⋊Dic₃) #66465

order	324
isoclinism	isoclinic class of C₃⋊(C₃⋊D₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₃³

Aliases

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

Splittings

derived	C₃³ ⋊ C₃×C₄
Sylow	C₃⁴ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	C₃⋊(C₃⋊D₃)
commutator, hypocenter	C₃³	C₃×C₄
Frattini	C₂	C₃×C₃⋊(C₃⋊D₃)
Fitting, socle, prime radical	C₂×C₃⁴	C₂
cosocle	C₂×C₃³	C₂×C₃
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂×C₃

324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×C₃⋊Dic₃, C₃×Dic₃×C₃⋊D₃, C₃×#6290, …
maximal normal subgroup of	C₃×D₃×C₃⋊Dic₃, C₃×Dic₃×C₃⋊D₃, C₃×#6290, …
maximal quotient of	C₃×C₃⋊(C₃⋊(C₃⋊C₈)), C₂×C₃×C₃⋊(C₃⋊Dic₃), C₃×C₃⋊(C₃⋊Dic₉), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 157);