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G = D11order 22 = 2·11

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D11, C11⋊C2, sometimes denoted D22 or Dih11 or Dih22, SmallGroup(22,1)

Series: Derived Chief Lower central Upper central

C1C11 — D11
C1C11 — D11
C11 — D11
C1

Generators and relations for D11
 G = < a,b | a11=b2=1, bab=a-1 >

11C2

Character table of D11

 class 1211A11B11C11D11E
 size 11122222
ρ11111111    trivial
ρ21-111111    linear of order 2
ρ320ζ111011ζ117114ζ119112ζ118113ζ116115    orthogonal faithful
ρ420ζ118113ζ111011ζ116115ζ119112ζ117114    orthogonal faithful
ρ520ζ116115ζ119112ζ111011ζ117114ζ118113    orthogonal faithful
ρ620ζ119112ζ118113ζ117114ζ116115ζ111011    orthogonal faithful
ρ720ζ117114ζ116115ζ118113ζ111011ζ119112    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);

Polynomial with Galois group D11 over ℚ
actionf(x)Disc(f)
11T2x11-5x10-4x9+54x8-53x7-127x6+208x5+69x4-222x3+29x2+56x-554·12975

Matrix representation of D11 in GL2(𝔽23) generated by

1022
10
,
1022
713
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

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