Finite Group Databasebeta

← Prev (#279) Next (#281) →

C₁₁⋊C₅ #280

order	55
isoclinism	isoclinic class of C₁₁⋊C₅
automorphism	Aut(G) = AGL(1,𝔽₁₁) (order 110) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = (C₁₁⋊C₅)⋊AGL(1,𝔽₁₁)
homology	Abelianization (H₁) = C₅ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₁₁⋊C₅

Splittings

Fitting, socle, derived, Sylow

C₁₁ ⋊ C₅

Subgroups

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

4 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:

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

Polycyclic presentation:

5,11|b^a=b^4

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇
size	1	5	5	11	11	11	11
order	1	11	11	5	5	5	5
χ₁	1	1	1	1	1	1	1
χ₂	1	1	1	−ζ−ζ⁶−ζ¹⁶−ζ²¹−ζ²⁶−ζ³¹−ζ³⁶−ζ⁴¹−ζ⁴⁶−ζ⁵¹	−ζ²−ζ⁷−ζ¹²−ζ¹⁷−ζ²⁷−ζ³²−ζ³⁷−ζ⁴²−ζ⁴⁷−ζ⁵²	−ζ³−ζ⁸−ζ¹³−ζ¹⁸−ζ²³−ζ²⁸−ζ³⁸−ζ⁴³−ζ⁴⁸−ζ⁵³	−ζ⁴−ζ⁹−ζ¹⁴−ζ¹⁹−ζ²⁴−ζ²⁹−ζ³⁴−ζ³⁹−ζ⁴⁹−ζ⁵⁴
χ₃	1	1	1	−ζ²−ζ⁷−ζ¹²−ζ¹⁷−ζ²⁷−ζ³²−ζ³⁷−ζ⁴²−ζ⁴⁷−ζ⁵²	−ζ⁴−ζ⁹−ζ¹⁴−ζ¹⁹−ζ²⁴−ζ²⁹−ζ³⁴−ζ³⁹−ζ⁴⁹−ζ⁵⁴	−ζ−ζ⁶−ζ¹⁶−ζ²¹−ζ²⁶−ζ³¹−ζ³⁶−ζ⁴¹−ζ⁴⁶−ζ⁵¹	−ζ³−ζ⁸−ζ¹³−ζ¹⁸−ζ²³−ζ²⁸−ζ³⁸−ζ⁴³−ζ⁴⁸−ζ⁵³
χ₄	1	1	1	−ζ³−ζ⁸−ζ¹³−ζ¹⁸−ζ²³−ζ²⁸−ζ³⁸−ζ⁴³−ζ⁴⁸−ζ⁵³	−ζ−ζ⁶−ζ¹⁶−ζ²¹−ζ²⁶−ζ³¹−ζ³⁶−ζ⁴¹−ζ⁴⁶−ζ⁵¹	−ζ⁴−ζ⁹−ζ¹⁴−ζ¹⁹−ζ²⁴−ζ²⁹−ζ³⁴−ζ³⁹−ζ⁴⁹−ζ⁵⁴	−ζ²−ζ⁷−ζ¹²−ζ¹⁷−ζ²⁷−ζ³²−ζ³⁷−ζ⁴²−ζ⁴⁷−ζ⁵²
χ₅	1	1	1	−ζ⁴−ζ⁹−ζ¹⁴−ζ¹⁹−ζ²⁴−ζ²⁹−ζ³⁴−ζ³⁹−ζ⁴⁹−ζ⁵⁴	−ζ³−ζ⁸−ζ¹³−ζ¹⁸−ζ²³−ζ²⁸−ζ³⁸−ζ⁴³−ζ⁴⁸−ζ⁵³	−ζ²−ζ⁷−ζ¹²−ζ¹⁷−ζ²⁷−ζ³²−ζ³⁷−ζ⁴²−ζ⁴⁷−ζ⁵²	−ζ−ζ⁶−ζ¹⁶−ζ²¹−ζ²⁶−ζ³¹−ζ³⁶−ζ⁴¹−ζ⁴⁶−ζ⁵¹
χ₆	5	$\frac{- 1 + i \sqrt{11}}{2}$	$\frac{- 1 - i \sqrt{11}}{2}$	0	0	0	0
χ₇	5	$\frac{- 1 - i \sqrt{11}}{2}$	$\frac{- 1 + i \sqrt{11}}{2}$	0	0	0	0

Here ζ is a primitive root of unity of order 55 (ζ⁵⁵ = 1); i = ζ¹³.

Relationships

maximal subgroup of	AGL(1,𝔽₁₁), C₂×C₁₁⋊C₅, C₃×C₁₁⋊C₅, …
maximal normal subgroup of	AGL(1,𝔽₁₁), C₂×C₁₁⋊C₅, C₃×C₁₁⋊C₅, …
maximal quotient of	C₂×C₁₁⋊C₅, C₃×C₁₁⋊C₅, C₁₁⋊C₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(55, 1);