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C₃⋊D₃×C₂₇ #92316

order	486
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₂₇ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃⋊D₃×C₂₇
unambiguous semidirect product	C₃⋊(D₃×C₂₇)

Splittings

Fitting, Sylow	C₃²×C₂₇ ⋊ C₂
derived	C₃² ⋊ C₂×C₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂₇	C₃⋊D₃
commutator, hypocenter	C₃²	C₂×C₂₇
Frattini	C₉	C₃×C₃⋊D₃
Fitting	C₃²×C₂₇	C₂
socle	C₃³	C₂×C₉
prime radical	C₃×C₃⋊D₃	C₉
cosocle	C₃²×C₉	C₂×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂₇

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 33 points:

(2,3)(5,6)(7,8,11,9,12,17,14,19,24,10,13,18,15,20,25,22,27,30,16,21,26,23,28,31,29,32,33);(4,5,6);(1,2,3)

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

derived subgroup of	D₃²⋊D₂₇, #157⋊D₂₇, C₂₇⋊PSU(3,𝔽₂)
maximal subgroup of	C₂₇×#157, C₂₇⋊#157, D₃²×C₂₇, …
maximal normal subgroup of	C₂₇×#157, C₂₇⋊#157, D₃²×C₂₇, …
maximal quotient of	C₂₇×C₃⋊Dic₃, C₂×C₃⋊D₃×C₂₇, C₂₇×C₃⋊D₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 161);