Copied to
clipboard

G = F11order 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

C1C11 — F11
C1C11C11⋊C5 — F11
C11 — F11
C1

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

11C2
11C5
11C10

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
ρ41-1ζ5ζ52ζ54ζ5354535251    linear of order 10
ρ511ζ54ζ53ζ5ζ52ζ5ζ52ζ53ζ541    linear of order 5
ρ61-1ζ53ζ5ζ52ζ5452545531    linear of order 10
ρ71-1ζ54ζ53ζ5ζ5255253541    linear of order 10
ρ811ζ5ζ52ζ54ζ53ζ54ζ53ζ52ζ51    linear of order 5
ρ91-1ζ52ζ54ζ53ζ553554521    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

Export

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

׿
×
𝔽