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## G = D67order 134 = 2·67

### Dihedral group

Aliases: D67, C67⋊C2, sometimes denoted D134 or Dih67 or Dih134, SmallGroup(134,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C67 — D67
 Chief series C1 — C67 — D67
 Lower central C67 — D67
 Upper central C1

Generators and relations for D67
G = < a,b | a67=b2=1, bab=a-1 >

Smallest permutation representation of D67
On 67 points: primitive
Generators in S67
```(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 67)
(1 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 60)(9 59)(10 58)(11 57)(12 56)(13 55)(14 54)(15 53)(16 52)(17 51)(18 50)(19 49)(20 48)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)```

`G:=sub<Sym(67)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67)], [(1,67),(2,66),(3,65),(4,64),(5,63),(6,62),(7,61),(8,60),(9,59),(10,58),(11,57),(12,56),(13,55),(14,54),(15,53),(16,52),(17,51),(18,50),(19,49),(20,48),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,40),(29,39),(30,38),(31,37),(32,36),(33,35)]])`

D67 is a maximal subgroup of   C67⋊C6  D201
D67 is a maximal quotient of   Dic67  D201

35 conjugacy classes

 class 1 2 67A ··· 67AG order 1 2 67 ··· 67 size 1 67 2 ··· 2

35 irreducible representations

 dim 1 1 2 type + + + image C1 C2 D67 kernel D67 C67 C1 # reps 1 1 33

Matrix representation of D67 in GL2(𝔽269) generated by

 53 268 1 0
,
 53 268 118 216
`G:=sub<GL(2,GF(269))| [53,1,268,0],[53,118,268,216] >;`

D67 in GAP, Magma, Sage, TeX

`D_{67}`
`% in TeX`

`G:=Group("D67");`
`// GroupNames label`

`G:=SmallGroup(134,1);`
`// by ID`

`G=gap.SmallGroup(134,1);`
`# by ID`

`G:=PCGroup([2,-2,-67,529]);`
`// Polycyclic`

`G:=Group<a,b|a^67=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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