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C₃×#4325 #10587174

order	528
isoclinism	isoclinic class of #4323
automorphism	Aut(G) = C₂²×D₄×AGL(1,𝔽₁₁) (order 3520) · Inn(G) = C₂²⋊D₁₁ · Out(G) = C₂³×C₅
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₃×Q₈×C₁₁ ⋊ C₂
Hall	C₃×C₁₁ ⋊ SD₁₆
Sylow	C₁₁ ⋊ C₃×SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂²⋊D₁₁
commutator, cosocle	C₄×C₁₁	C₂²×C₃
Frattini	C₄	C₂×C₃×D₁₁
Fitting	C₃×Q₈×C₁₁	C₂
socle	C₂×C₃×C₁₁	D₄
prime radical	C₃×D₄₄	C₂
hypercenter	C₃×Q₈	D₁₁
hypocenter	C₁₁	C₃×SD₁₆
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₈

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 264

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2,3)(4,5,7,9,11,13,14,12,10,8,6)(15,16,18,20)(17,22,21,19);(5,6)(7,8)(9,10)(11,12)(13,14)(16,19)(17,21)(20,22)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#4325, #11760954, #11760955, …
maximal normal subgroup of	D₃×#4325, #11760954, #11760955, …
maximal quotient of	C₃×#66872, C₃×#66873, C₃×#66899, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(528, 44);