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G = F11  order 110 = 2·5·11

Frobenius group

metacyclic, supersoluble, monomial, Z-group

Aliases: F11, AGL1(𝔽11), C11⋊C10, D11⋊C5, C11⋊C5⋊C2, Aut(D11), Hol(C11), SmallGroup(110,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C11 — F11
Chief series
C1 — C11 — C11⋊C5 — F11
Lower central
C11 — F11
Upper central
C1

Generators and relations for F11
 G = < a,b | a11=b10=1, bab-1=a6 >

C1
11C2
11C5
C11
11C10
D11
C11⋊C5
F11

Character table of F11

 class 125A5B5C5D10A10B10C10D11
 size 111111111111111111110
ρ111111111111    trivial
ρ21-11111-1-1-1-11    linear of order 2
ρ311ζ52ζ54ζ53ζ5ζ53ζ5ζ54ζ521    linear of order 5
ρ41-1ζ5ζ52ζ54ζ53-ζ54-ζ53-ζ52-ζ51    linear of order 10
ρ511ζ54ζ53ζ5ζ52ζ5ζ52ζ53ζ541    linear of order 5
ρ61-1ζ53ζ5ζ52ζ54-ζ52-ζ54-ζ5-ζ531    linear of order 10
ρ71-1ζ54ζ53ζ5ζ52-ζ5-ζ52-ζ53-ζ541    linear of order 10
ρ811ζ5ζ52ζ54ζ53ζ54ζ53ζ52ζ51    linear of order 5
ρ91-1ζ52ζ54ζ53ζ5-ζ53-ζ5-ζ54-ζ521    linear of order 10
ρ1011ζ53ζ5ζ52ζ54ζ52ζ54ζ5ζ531    linear of order 5
ρ1110000000000-1    orthogonal faithful

Permutation representations of F11
►On 11 points: primitive, sharply doubly transitive - transitive group 11T4
Generators in S11
(1 2 3 4 5 6 7 8 9 10 11)

(2 3 5 9 6 11 10 8 4 7)
 
G:=sub<Sym(11)| (1,2,3,4,5,6,7,8,9,10,11), (2,3,5,9,6,11,10,8,4,7)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11), (2,3,5,9,6,11,10,8,4,7) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11)], [(2,3,5,9,6,11,10,8,4,7)]])
 
G:=TransitiveGroup(11,4);
 

►On 22 points - transitive group 22T4
Generators in S22
(1 2 3 4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19 20 21 22)

(1 12)(2 14 5 20 6 22 10 19 4 18)(3 16 9 17 11 21 8 15 7 13)
 
G:=sub<Sym(22)| (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,12)(2,14,5,20,6,22,10,19,4,18)(3,16,9,17,11,21,8,15,7,13)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,12)(2,14,5,20,6,22,10,19,4,18)(3,16,9,17,11,21,8,15,7,13) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22)], [(1,12),(2,14,5,20,6,22,10,19,4,18),(3,16,9,17,11,21,8,15,7,13)]])
 
G:=TransitiveGroup(22,4);
 

F11 is a maximal subgroup of   C3⋊F11
F11 is a maximal quotient of   C11⋊C20  C3⋊F11

Polynomial with Galois group F11 over ℚ
actionf(x)Disc(f)
11T4x11+x10-53x9-31x8+996x7+28x6-7812x5+5784x4+20673x3-36219x2+19909x-3279210·55·710·118·535032·2029492

Matrix representation of F11 ►in GL10(ℤ)

0100000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
-1-1-1-1-1-1-1-1-1-1
,
1000000000
0000001000
0100000000
0000000100
0010000000
0000000010
0001000000
0000000001
0000100000
-1-1-1-1-1-1-1-1-1-1

G:=sub<GL(10,Integers())| [0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,1,0,-1] >;
 

F11 in GAP, Magma, Sage, TeX 

F_{11}
 
% in TeX
 
G:=Group("F11");
 
// GroupNames label
 
G:=SmallGroup(110,1);
 
// by ID
 
G=gap.SmallGroup(110,1);
 
# by ID
 
G:=PCGroup([3,-2,-5,-11,902,185]);
 
// Polycyclic
 
G:=Group<a,b|a^11=b^10=1,b*a*b^-1=a^6>;
 
// generators/relations
 

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Subgroup lattice of F11 in TeX
Character table of F11 in TeX

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