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## G = D70order 140 = 22·5·7

### Dihedral group

Aliases: D70, C2×D35, C14⋊D5, C10⋊D7, C52D14, C72D10, C701C2, C352C22, sometimes denoted D140 or Dih70 or Dih140, SmallGroup(140,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — D70
 Chief series C1 — C7 — C35 — D35 — D70
 Lower central C35 — D70
 Upper central C1 — C2

Generators and relations for D70
G = < a,b | a70=b2=1, bab=a-1 >

Smallest permutation representation of D70
On 70 points
Generators in S70
```(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 70)
(1 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)```

`G:=sub<Sym(70)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70)], [(1,70),(2,69),(3,68),(4,67),(5,66),(6,65),(7,64),(8,63),(9,62),(10,61),(11,60),(12,59),(13,58),(14,57),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,50),(22,49),(23,48),(24,47),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36)]])`

D70 is a maximal subgroup of   D70.C2  C5⋊D28  C7⋊D20  D140  C357D4  C2×D5×D7
D70 is a maximal quotient of   Dic70  D140  C357D4

38 conjugacy classes

 class 1 2A 2B 2C 5A 5B 7A 7B 7C 10A 10B 14A 14B 14C 35A ··· 35L 70A ··· 70L order 1 2 2 2 5 5 7 7 7 10 10 14 14 14 35 ··· 35 70 ··· 70 size 1 1 35 35 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

38 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 D5 D7 D10 D14 D35 D70 kernel D70 D35 C70 C14 C10 C7 C5 C2 C1 # reps 1 2 1 2 3 2 3 12 12

Matrix representation of D70 in GL2(𝔽71) generated by

 46 11 60 19
,
 46 11 53 25
`G:=sub<GL(2,GF(71))| [46,60,11,19],[46,53,11,25] >;`

D70 in GAP, Magma, Sage, TeX

`D_{70}`
`% in TeX`

`G:=Group("D70");`
`// GroupNames label`

`G:=SmallGroup(140,10);`
`// by ID`

`G=gap.SmallGroup(140,10);`
`# by ID`

`G:=PCGroup([4,-2,-2,-5,-7,194,1923]);`
`// Polycyclic`

`G:=Group<a,b|a^70=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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