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## G = D66order 132 = 22·3·11

### Dihedral group

Aliases: D66, C2×D33, C22⋊S3, C6⋊D11, C32D22, C112D6, C661C2, C332C22, sometimes denoted D132 or Dih66 or Dih132, SmallGroup(132,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — D66
 Chief series C1 — C11 — C33 — D33 — D66
 Lower central C33 — D66
 Upper central C1 — C2

Generators and relations for D66
G = < a,b | a66=b2=1, bab=a-1 >

33C2
33C2
33C22
11S3
11S3
3D11
3D11
11D6
3D22

Smallest permutation representation of D66
On 66 points
Generators in S66
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66)
(1 66)(2 65)(3 64)(4 63)(5 62)(6 61)(7 60)(8 59)(9 58)(10 57)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)```

`G:=sub<Sym(66)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66), (1,66)(2,65)(3,64)(4,63)(5,62)(6,61)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66), (1,66)(2,65)(3,64)(4,63)(5,62)(6,61)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66)], [(1,66),(2,65),(3,64),(4,63),(5,62),(6,61),(7,60),(8,59),(9,58),(10,57),(11,56),(12,55),(13,54),(14,53),(15,52),(16,51),(17,50),(18,49),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,42),(26,41),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34)]])`

D66 is a maximal subgroup of   D33⋊C4  C3⋊D44  C11⋊D12  D132  C337D4  C2×S3×D11
D66 is a maximal quotient of   Dic66  D132  C337D4

36 conjugacy classes

 class 1 2A 2B 2C 3 6 11A ··· 11E 22A ··· 22E 33A ··· 33J 66A ··· 66J order 1 2 2 2 3 6 11 ··· 11 22 ··· 22 33 ··· 33 66 ··· 66 size 1 1 33 33 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

36 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D6 D11 D22 D33 D66 kernel D66 D33 C66 C22 C11 C6 C3 C2 C1 # reps 1 2 1 1 1 5 5 10 10

Matrix representation of D66 in GL2(𝔽67) generated by

 44 15 35 15
,
 54 14 55 13
`G:=sub<GL(2,GF(67))| [44,35,15,15],[54,55,14,13] >;`

D66 in GAP, Magma, Sage, TeX

`D_{66}`
`% in TeX`

`G:=Group("D66");`
`// GroupNames label`

`G:=SmallGroup(132,9);`
`// by ID`

`G=gap.SmallGroup(132,9);`
`# by ID`

`G:=PCGroup([4,-2,-2,-3,-11,98,1923]);`
`// Polycyclic`

`G:=Group<a,b|a^66=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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