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G = D22order 44 = 22·11

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D22, C2×D11, C22⋊C2, C11⋊C22, sometimes denoted D44 or Dih22 or Dih44, SmallGroup(44,3)

Series: Derived Chief Lower central Upper central

C1C11 — D22
C1C11D11 — D22
C11 — D22
C1C2

Generators and relations for D22
 G = < a,b | a22=b2=1, bab=a-1 >

11C2
11C2
11C22

Character table of D22

 class 12A2B2C11A11B11C11D11E22A22B22C22D22E
 size 1111112222222222
ρ111111111111111    trivial
ρ21-1-1111111-1-1-1-1-1    linear of order 2
ρ31-11-111111-1-1-1-1-1    linear of order 2
ρ411-1-11111111111    linear of order 2
ρ52200ζ116115ζ119112ζ117114ζ118113ζ111011ζ111011ζ116115ζ119112ζ117114ζ118113    orthogonal lifted from D11
ρ62200ζ117114ζ116115ζ111011ζ119112ζ118113ζ118113ζ117114ζ116115ζ111011ζ119112    orthogonal lifted from D11
ρ72200ζ118113ζ111011ζ119112ζ117114ζ116115ζ116115ζ118113ζ111011ζ119112ζ117114    orthogonal lifted from D11
ρ82200ζ119112ζ118113ζ116115ζ111011ζ117114ζ117114ζ119112ζ118113ζ116115ζ111011    orthogonal lifted from D11
ρ92-200ζ119112ζ118113ζ116115ζ111011ζ117114117114119112118113116115111011    orthogonal faithful
ρ102-200ζ116115ζ119112ζ117114ζ118113ζ111011111011116115119112117114118113    orthogonal faithful
ρ112-200ζ111011ζ117114ζ118113ζ116115ζ119112119112111011117114118113116115    orthogonal faithful
ρ122200ζ111011ζ117114ζ118113ζ116115ζ119112ζ119112ζ111011ζ117114ζ118113ζ116115    orthogonal lifted from D11
ρ132-200ζ117114ζ116115ζ111011ζ119112ζ118113118113117114116115111011119112    orthogonal faithful
ρ142-200ζ118113ζ111011ζ119112ζ117114ζ116115116115118113111011119112117114    orthogonal faithful

Permutation representations of D22
On 22 points - transitive group 22T3
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 11)(2 10)(3 9)(4 8)(5 7)(12 22)(13 21)(14 20)(15 19)(16 18)

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

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

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

G:=TransitiveGroup(22,3);

Matrix representation of D22 in GL2(𝔽23) generated by

207
1616
,
207
123
G:=sub<GL(2,GF(23))| [20,16,7,16],[20,12,7,3] >;

D22 in GAP, Magma, Sage, TeX

D_{22}
% in TeX

G:=Group("D22");
// GroupNames label

G:=SmallGroup(44,3);
// by ID

G=gap.SmallGroup(44,3);
# by ID

G:=PCGroup([3,-2,-2,-11,362]);
// Polycyclic

G:=Group<a,b|a^22=b^2=1,b*a*b=a^-1>;
// generators/relations

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