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## G = D68order 136 = 23·17

### Dihedral group

Aliases: D68, C4⋊D17, C171D4, C681C2, D341C2, C2.4D34, C34.3C22, sometimes denoted D136 or Dih68 or Dih136, SmallGroup(136,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C34 — D68
 Chief series C1 — C17 — C34 — D34 — D68
 Lower central C17 — C34 — D68
 Upper central C1 — C2 — C4

Generators and relations for D68
G = < a,b | a68=b2=1, bab=a-1 >

34C2
34C2
17C22
17C22
2D17
2D17
17D4

Smallest permutation representation of D68
On 68 points
Generators in S68
```(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 68)
(1 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)```

`G:=sub<Sym(68)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)], [(1,68),(2,67),(3,66),(4,65),(5,64),(6,63),(7,62),(8,61),(9,60),(10,59),(11,58),(12,57),(13,56),(14,55),(15,54),(16,53),(17,52),(18,51),(19,50),(20,49),(21,48),(22,47),(23,46),(24,45),(25,44),(26,43),(27,42),(28,41),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35)]])`

D68 is a maximal subgroup of
C136⋊C2  D136  D4⋊D17  Q8⋊D17  D685C2  D4×D17  D68⋊C2  C3⋊D68  D204
D68 is a maximal quotient of
C136⋊C2  D136  Dic68  C683C4  D34⋊C4  C3⋊D68  D204

37 conjugacy classes

 class 1 2A 2B 2C 4 17A ··· 17H 34A ··· 34H 68A ··· 68P order 1 2 2 2 4 17 ··· 17 34 ··· 34 68 ··· 68 size 1 1 34 34 2 2 ··· 2 2 ··· 2 2 ··· 2

37 irreducible representations

 dim 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 D4 D17 D34 D68 kernel D68 C68 D34 C17 C4 C2 C1 # reps 1 1 2 1 8 8 16

Matrix representation of D68 in GL2(𝔽137) generated by

 37 7 130 132
,
 37 7 59 100
`G:=sub<GL(2,GF(137))| [37,130,7,132],[37,59,7,100] >;`

D68 in GAP, Magma, Sage, TeX

`D_{68}`
`% in TeX`

`G:=Group("D68");`
`// GroupNames label`

`G:=SmallGroup(136,6);`
`// by ID`

`G=gap.SmallGroup(136,6);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-17,49,21,2051]);`
`// Polycyclic`

`G:=Group<a,b|a^68=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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