Copied to
clipboard

## G = D69order 138 = 2·3·23

### Dihedral group

Aliases: D69, C23⋊S3, C3⋊D23, C691C2, sometimes denoted D138 or Dih69 or Dih138, SmallGroup(138,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C69 — D69
 Chief series C1 — C23 — C69 — D69
 Lower central C69 — D69
 Upper central C1

Generators and relations for D69
G = < a,b | a69=b2=1, bab=a-1 >

69C2
23S3
3D23

Smallest permutation representation of D69
On 69 points
Generators in S69
```(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 69)
(1 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)```

`G:=sub<Sym(69)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,60),(11,59),(12,58),(13,57),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,48),(23,47),(24,46),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36)]])`

D69 is a maximal subgroup of   S3×D23  D207  C3⋊D69
D69 is a maximal quotient of   Dic69  D207  C3⋊D69

36 conjugacy classes

 class 1 2 3 23A ··· 23K 69A ··· 69V order 1 2 3 23 ··· 23 69 ··· 69 size 1 69 2 2 ··· 2 2 ··· 2

36 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 S3 D23 D69 kernel D69 C69 C23 C3 C1 # reps 1 1 1 11 22

Matrix representation of D69 in GL2(𝔽139) generated by

 78 36 103 3
,
 78 36 82 61
`G:=sub<GL(2,GF(139))| [78,103,36,3],[78,82,36,61] >;`

D69 in GAP, Magma, Sage, TeX

`D_{69}`
`% in TeX`

`G:=Group("D69");`
`// GroupNames label`

`G:=SmallGroup(138,3);`
`// by ID`

`G=gap.SmallGroup(138,3);`
`# by ID`

`G:=PCGroup([3,-2,-3,-23,25,1190]);`
`// Polycyclic`

`G:=Group<a,b|a^69=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

Export

׿
×
𝔽