Finite Group Databasebeta

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C₃×#825 #63611

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

Fitting	C₃²×M₁₆ ⋊ C₂
Sylow	C₃² ⋊ #104

Subgroups

	subgroup	G/H
center	C₃×C₄	#67
commutator	C₃×C₄	C₂×C₃×C₄
Frattini	C₂×C₄	C₂×C₃×D₃
Fitting	C₃²×M₁₆	C₂
socle	C₂×C₃²	C₂²⋊C₄
prime radical	C₃×#235	C₂
cosocle	C₂×C₃×C₄	C₂²×C₃
hypercenter	C₃×M₁₆	D₃
hypocenter	C₃	C₃×#104
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₃×M₁₆

98 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)(7,14,11,9,12,13,10,8);(5,6)(7,8)(9,12)(10,13)(11,14)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10590975, #10590976, #10590977, …
maximal normal subgroup of	#10590975, #10590976, #10590977, …
maximal quotient of	C₃×#4464, C₃×#4465, C₃×#4467, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 259);