Finite Group Databasebeta

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C₃×#837 #63623

order	288
isoclinism	isoclinic class of #213
automorphism	Aut(G) = C₂³×D₃×D₄ (order 384) · Inn(G) = #67 · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

Splittings

C₃² ⋊ #104

Subgroups

	subgroup	G/H
center	C₃×C₄	#67
commutator	C₃×C₄	C₂×C₃×C₄
Frattini	C₂×C₄	C₂×C₃×D₃
Fitting, prime radical	C₃²×#41	C₂
socle	C₂×C₃²	C₂²⋊C₄
cosocle	C₂×C₃×C₄	C₂²×C₃
hypercenter	C₃×#41	D₃
hypocenter	C₃	C₃×#104
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₃×#41

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10591662, #10591663, #10591666, …
maximal normal subgroup of	#10591662, #10591663, #10591666, …
maximal quotient of	C₃×#4532, C₃×D₄⋊(C₃⋊C₈), C₃×#4535, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 271);