Finite Group Databasebeta

← Prev (#1119) Next (#1121) →

AGL(1,𝔽₁₁) #1120

order	110
isoclinism	isoclinic class of AGL(1,𝔽₁₁)
automorphism	Aut(G) = AGL(1,𝔽₁₁) (order 110) · Inn(G) = G · Out(G) = C₁
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₁₁)²
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

affine general linear	AGL(1,𝔽₁₁)
unambiguous semidirect product	D₁₁⋊C₅
unambiguous semidirect product	(C₁₁⋊C₅)⋊C₂

Splittings

Fitting, socle, derived, Sylow	C₁₁ ⋊ C₂×C₅
Hall	C₁₁⋊C₅ ⋊ C₂
Hall	D₁₁ ⋊ C₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁	C₂×C₅
LCS	G ▷ C₁₁
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 11 points:

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

Polycyclic presentation:

2,5,11|c^a=c^10,c^b=c^4

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁
size	1	10	11	11	11	11	11	11	11	11	11
order	1	11	2	5	5	5	5	10	10	10	10
χ₁	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	ζ	ζ²	ζ³	ζ⁴	ζ	ζ²	ζ³	ζ⁴
χ₃	1	1	1	ζ²	ζ⁴	ζ	ζ³	ζ²	ζ⁴	ζ	ζ³
χ₄	1	1	1	ζ³	ζ	ζ⁴	ζ²	ζ³	ζ	ζ⁴	ζ²
χ₅	1	1	1	ζ⁴	ζ³	ζ²	ζ	ζ⁴	ζ³	ζ²	ζ
χ₆	1	1	-1	1	1	1	1	-1	-1	-1	-1
χ₇	1	1	-1	ζ	ζ²	ζ³	ζ⁴	−ζ	−ζ²	−ζ³	−ζ⁴
χ₈	1	1	-1	ζ²	ζ⁴	ζ	ζ³	−ζ²	−ζ⁴	−ζ	−ζ³
χ₉	1	1	-1	ζ³	ζ	ζ⁴	ζ²	−ζ³	−ζ	−ζ⁴	−ζ²
χ₁₀	1	1	-1	ζ⁴	ζ³	ζ²	ζ	−ζ⁴	−ζ³	−ζ²	−ζ
χ₁₁	10	-1	0	0	0	0	0	0	0	0	0

Here ζ is a primitive root of unity of order 5 (ζ⁵ = 1).

Relationships

maximal subgroup of	C₂×AGL(1,𝔽₁₁), C₃×AGL(1,𝔽₁₁), C₃⋊AGL(1,𝔽₁₁), …
maximal normal subgroup of	C₂×AGL(1,𝔽₁₁), C₃×AGL(1,𝔽₁₁), C₅×AGL(1,𝔽₁₁), …
maximal quotient of	Dic₁₁⋊C₅, C₂×AGL(1,𝔽₁₁), C₃×AGL(1,𝔽₁₁), …
automorphism group of	D₁₁, C₁₁⋊C₅, AGL(1,𝔽₁₁), …
holomorph of	C₁₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(110, 1);