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C₁₁×#214 #10587189

order	528
isoclinism	isoclinic class of #213
automorphism	Aut(G) = C₂²×C₅×D₃×D₄ (order 960) · Inn(G) = #67 · Out(G) = C₂³×C₅
homology	Abelianization (H₁) = C₂²×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Hall	C₃×C₁₁ ⋊ SD₁₆
Sylow	C₃ ⋊ C₁₁×SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₁₁	#67
commutator, cosocle	C₃×C₄	C₂²×C₁₁
Frattini	C₄	C₂×D₃×C₁₁
Fitting, prime radical	C₃×D₄×C₁₁	C₂
socle	C₂×C₃×C₁₁	D₄
hypercenter	D₄×C₁₁	D₃
hypocenter	C₃	C₁₁×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₁ ◁ C₄×C₁₁ ◁ D₄×C₁₁

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:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₁×#214, #11760934, #11760937, …
maximal normal subgroup of	D₁₁×#214, #11760934, #11760937, …
maximal quotient of	C₁₁×#808, C₁₁×#810, C₁₁×#832, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(528, 59);