D11 - GroupNames

class	1	2	11A	11B	11C	11D	11E
size	1	11	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	linear of order 2
ρ₃	2	0	ζ₁₁¹⁰+ζ₁₁	ζ₁₁⁷+ζ₁₁⁴	ζ₁₁⁹+ζ₁₁²	ζ₁₁⁸+ζ₁₁³	ζ₁₁⁶+ζ₁₁⁵	orthogonal faithful
ρ₄	2	0	ζ₁₁⁸+ζ₁₁³	ζ₁₁¹⁰+ζ₁₁	ζ₁₁⁶+ζ₁₁⁵	ζ₁₁⁹+ζ₁₁²	ζ₁₁⁷+ζ₁₁⁴	orthogonal faithful
ρ₅	2	0	ζ₁₁⁶+ζ₁₁⁵	ζ₁₁⁹+ζ₁₁²	ζ₁₁¹⁰+ζ₁₁	ζ₁₁⁷+ζ₁₁⁴	ζ₁₁⁸+ζ₁₁³	orthogonal faithful
ρ₆	2	0	ζ₁₁⁹+ζ₁₁²	ζ₁₁⁸+ζ₁₁³	ζ₁₁⁷+ζ₁₁⁴	ζ₁₁⁶+ζ₁₁⁵	ζ₁₁¹⁰+ζ₁₁	orthogonal faithful
ρ₇	2	0	ζ₁₁⁷+ζ₁₁⁴	ζ₁₁⁶+ζ₁₁⁵	ζ₁₁⁸+ζ₁₁³	ζ₁₁¹⁰+ζ₁₁	ζ₁₁⁹+ζ₁₁²	orthogonal faithful

Permutation representations of D₁₁
►On 11 points: primitive - transitive group 11T2

Generators in S₁₁

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

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

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

G:=Group( (1,2,3,4,5,6,7,8,9,10,11), (1,11)(2,10)(3,9)(4,8)(5,7) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11)], [(1,11),(2,10),(3,9),(4,8),(5,7)]])

G:=TransitiveGroup(11,2);

►Regular action on 22 points - transitive group 22T2

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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 22)(11 21)

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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,22),(11,21)]])

G:=TransitiveGroup(22,2);

D₁₁ is a maximal subgroup of
F₁₁ C₁₁⋊D₁₁
D_11p: D₃₃ D₅₅ D₇₇ D₁₂₁ D₁₄₃ D₁₈₇ D₂₀₉ ...
D₁₁ is a maximal quotient of
Dic₁₁ C₁₁⋊D₁₁
D_11p: D₃₃ D₅₅ D₇₇ D₁₂₁ D₁₄₃ D₁₈₇ D₂₀₉ ...

Polynomial with Galois group D₁₁ over ℚ

action	f(x)	Disc(f)
11T2	x¹¹-5x¹⁰-4x⁹+54x⁸-53x⁷-127x⁶+208x⁵+69x⁴-222x³+29x²+56x-5	5⁴·1297⁵

Matrix representation of D₁₁ ►in GL₂(𝔽₂₃) generated by

10	22
1	0

10	22
7	13

G:=sub<GL(2,GF(23))| [10,1,22,0],[10,7,22,13] >;

D₁₁ in GAP, Magma, Sage, TeX

D_{11}

% in TeX

G:=Group("D11");

// GroupNames label

G:=SmallGroup(22,1);

// by ID

G=gap.SmallGroup(22,1);

# by ID

G:=PCGroup([2,-2,-11,81]);

// Polycyclic

G:=Group<a,b|a^11=b^2=1,b*a*b=a^-1>;

// generators/relations

Export

Subgroup lattice of D₁₁ in TeX
Character table of D₁₁ in TeX

G = D11 order 22 = 2·11

Dihedral group

G = D₁₁ order 22 = 2·11