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## G = D22order 44 = 22·11

### Dihedral group

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

 Derived series C1 — C11 — D22
 Chief series C1 — C11 — D11 — D22
 Lower central C11 — D22
 Upper central C1 — C2

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

Character table of D22

 class 1 2A 2B 2C 11A 11B 11C 11D 11E 22A 22B 22C 22D 22E size 1 1 11 11 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 2 0 0 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 orthogonal lifted from D11 ρ6 2 2 0 0 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 orthogonal lifted from D11 ρ7 2 2 0 0 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 orthogonal lifted from D11 ρ8 2 2 0 0 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 orthogonal lifted from D11 ρ9 2 -2 0 0 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 -ζ116-ζ115 -ζ1110-ζ11 orthogonal faithful ρ10 2 -2 0 0 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 -ζ117-ζ114 -ζ118-ζ113 orthogonal faithful ρ11 2 -2 0 0 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 -ζ118-ζ113 -ζ116-ζ115 orthogonal faithful ρ12 2 2 0 0 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 orthogonal lifted from D11 ρ13 2 -2 0 0 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 -ζ1110-ζ11 -ζ119-ζ112 orthogonal faithful ρ14 2 -2 0 0 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 -ζ119-ζ112 -ζ117-ζ114 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);`

D22 is a maximal subgroup of   D44  C11⋊D4
D22 is a maximal quotient of   Dic22  D44  C11⋊D4

Matrix representation of D22 in GL2(𝔽23) generated by

 20 7 16 16
,
 20 7 12 3
`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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