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C₁₁ #19

order	11
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₅ (order 10) · Inn(G) = C₁ · Out(G) = C₂×C₅
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₁₁)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

C₁₁

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₁

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 11

A minimal faithful permutation representation — 1 generator on 11 points:

(1,2,3,4,5,6,7,8,9,10,11)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁
size	1	1	1	1	1	1	1	1	1	1	1
order	1	11	11	11	11	11	11	11	11	11	11
χ₁	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	ζ	ζ²	ζ³	ζ⁴	ζ⁵	ζ⁶	ζ⁷	ζ⁸	ζ⁹	ζ¹⁰
χ₃	1	ζ²	ζ⁴	ζ⁶	ζ⁸	ζ¹⁰	ζ	ζ³	ζ⁵	ζ⁷	ζ⁹
χ₄	1	ζ³	ζ⁶	ζ⁹	ζ	ζ⁴	ζ⁷	ζ¹⁰	ζ²	ζ⁵	ζ⁸
χ₅	1	ζ⁴	ζ⁸	ζ	ζ⁵	ζ⁹	ζ²	ζ⁶	ζ¹⁰	ζ³	ζ⁷
χ₆	1	ζ⁵	ζ¹⁰	ζ⁴	ζ⁹	ζ³	ζ⁸	ζ²	ζ⁷	ζ	ζ⁶
χ₇	1	ζ⁶	ζ	ζ⁷	ζ²	ζ⁸	ζ³	ζ⁹	ζ⁴	ζ¹⁰	ζ⁵
χ₈	1	ζ⁷	ζ³	ζ¹⁰	ζ⁶	ζ²	ζ⁹	ζ⁵	ζ	ζ⁸	ζ⁴
χ₉	1	ζ⁸	ζ⁵	ζ²	ζ¹⁰	ζ⁷	ζ⁴	ζ	ζ⁹	ζ⁶	ζ³
χ₁₀	1	ζ⁹	ζ⁷	ζ⁵	ζ³	ζ	ζ¹⁰	ζ⁸	ζ⁶	ζ⁴	ζ²
χ₁₁	1	ζ¹⁰	ζ⁹	ζ⁸	ζ⁷	ζ⁶	ζ⁵	ζ⁴	ζ³	ζ²	ζ

Here ζ is a primitive root of unity of order 11 (ζ¹¹ = 1).

Relationships

center of	D₃×C₁₁, D₅×C₁₁, C₁₁×A₄, …
derived subgroup of	D₁₁, Dic₁₁, C₂×D₁₁, …
Frattini subgroup of	C₁₂₁, D₁₂₁, C₂×C₁₂₁, …
Fitting subgroup of	D₁₁, C₁₁⋊C₅, AGL(1,𝔽₁₁), …
socle of	D₁₁, C₁₁⋊C₅, AGL(1,𝔽₁₁), …
maximal subgroup of	D₁₁, C₂×C₁₁, C₃×C₁₁, …
maximal normal subgroup of	D₁₁, C₂×C₁₁, C₃×C₁₁, …
maximal quotient of	C₂×C₁₁, C₃×C₁₁, C₅×C₁₁, …
outer automorphism group of	D₂₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11, 1);