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C₃×C₂²⋊C₂₇ #66352

order	324
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₂₇ · Schur multiplier (H₂) = C₂×C₃

Aliases

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

Splittings

C₂² ⋊ C₃×C₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₉	A₄
commutator, hypocenter	C₂²	C₃×C₂₇
Frattini	C₉	C₃×A₄
Fitting	C₂²×C₃×C₉	C₃
socle, prime radical	C₂²×C₃²	C₉
cosocle	C₂²×C₉	C₃²
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₉

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 2 generators on 34 points:

(2,3,4)(8,9,12,10,13,18,15,20,25,11,14,19,16,21,26,23,28,31,17,22,27,24,29,32,30,33,34);(1,4)(2,3)(5,6,7)

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₃⋊((C₂²⋊C₂₇)⋊C₂), C₃⋊((C₂²⋊C₂₇)⋊C₄), D₃×(C₂²⋊C₂₇)⋊C₂, …
maximal subgroup of	C₃×(C₂²⋊C₂₇)⋊C₂, C₃⋊((C₂²⋊C₂₇)⋊C₂), D₃×C₂²⋊C₂₇, …
maximal normal subgroup of	C₃×(C₂²⋊C₂₇)⋊C₂, C₃⋊((C₂²⋊C₂₇)⋊C₂), D₃×C₂²⋊C₂₇, …
maximal quotient of	C₃×Q₈⋊C₂₇, C₂×C₃×C₂²⋊C₂₇, C₉×C₂²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 44);