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C₂×D₃×C₂₇ #66374

order	324
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂²×D₃×C₉ (order 216) · Inn(G) = D₃ · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₂²×C₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₃×C₂₇
unambiguous semidirect product	C₃⋊(C₂²×C₂₇)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₂₇)

Splittings

Fitting	C₂×C₃×C₂₇ ⋊ C₂
derived	C₃ ⋊ C₂²×C₂₇
Sylow	C₃×C₂₇ ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₂₇	D₃
commutator, hypocenter	C₃	C₂²×C₂₇
Frattini	C₉	C₂×C₃×D₃
Fitting	C₂×C₃×C₂₇	C₂
socle	C₂×C₃²	C₂×C₉
prime radical	C₂×C₃×D₃	C₉
cosocle	C₃×C₉	C₂²×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₂₇

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×Dic₂₇, #10619544, #10619545, …
maximal normal subgroup of	D₃×Dic₂₇, #10619544, #10619545, …
maximal quotient of	Dic₆×C₂₇, C₄×D₃×C₂₇, D₁₂×C₂₇, …
automorphism group of	C₂²×C₈₁, D₃×C₈₁
outer automorphism group of	C₂²×C₈₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 66);