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## G = D47order 94 = 2·47

### Dihedral group

Aliases: D47, C47⋊C2, sometimes denoted D94 or Dih47 or Dih94, SmallGroup(94,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C47 — D47
 Chief series C1 — C47 — D47
 Lower central C47 — D47
 Upper central C1

Generators and relations for D47
G = < a,b | a47=b2=1, bab=a-1 >

Character table of D47

 class 1 2 47A 47B 47C 47D 47E 47F 47G 47H 47I 47J 47K 47L 47M 47N 47O 47P 47Q 47R 47S 47T 47U 47V 47W size 1 47 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 0 ζ4725+ζ4722 ζ4733+ζ4714 ζ4744+ζ473 ζ4739+ζ478 ζ4728+ζ4719 ζ4730+ζ4717 ζ4741+ζ476 ζ4742+ζ475 ζ4731+ζ4716 ζ4727+ζ4720 ζ4738+ζ479 ζ4745+ζ472 ζ4734+ζ4713 ζ4724+ζ4723 ζ4735+ζ4712 ζ4746+ζ47 ζ4737+ζ4710 ζ4726+ζ4721 ζ4732+ζ4715 ζ4743+ζ474 ζ4740+ζ477 ζ4729+ζ4718 ζ4736+ζ4711 orthogonal faithful ρ4 2 0 ζ4732+ζ4715 ζ4746+ζ47 ζ4730+ζ4717 ζ4733+ζ4714 ζ4745+ζ472 ζ4729+ζ4718 ζ4734+ζ4713 ζ4744+ζ473 ζ4728+ζ4719 ζ4735+ζ4712 ζ4743+ζ474 ζ4727+ζ4720 ζ4736+ζ4711 ζ4742+ζ475 ζ4726+ζ4721 ζ4737+ζ4710 ζ4741+ζ476 ζ4725+ζ4722 ζ4738+ζ479 ζ4740+ζ477 ζ4724+ζ4723 ζ4739+ζ478 ζ4731+ζ4716 orthogonal faithful ρ5 2 0 ζ4734+ζ4713 ζ4743+ζ474 ζ4726+ζ4721 ζ4738+ζ479 ζ4739+ζ478 ζ4725+ζ4722 ζ4742+ζ475 ζ4735+ζ4712 ζ4729+ζ4718 ζ4746+ζ47 ζ4731+ζ4716 ζ4733+ζ4714 ζ4744+ζ473 ζ4727+ζ4720 ζ4737+ζ4710 ζ4740+ζ477 ζ4724+ζ4723 ζ4741+ζ476 ζ4736+ζ4711 ζ4728+ζ4719 ζ4745+ζ472 ζ4732+ζ4715 ζ4730+ζ4717 orthogonal faithful ρ6 2 0 ζ4745+ζ472 ζ4744+ζ473 ζ4743+ζ474 ζ4742+ζ475 ζ4741+ζ476 ζ4740+ζ477 ζ4739+ζ478 ζ4738+ζ479 ζ4737+ζ4710 ζ4736+ζ4711 ζ4735+ζ4712 ζ4734+ζ4713 ζ4733+ζ4714 ζ4732+ζ4715 ζ4731+ζ4716 ζ4730+ζ4717 ζ4729+ζ4718 ζ4728+ζ4719 ζ4727+ζ4720 ζ4726+ζ4721 ζ4725+ζ4722 ζ4724+ζ4723 ζ4746+ζ47 orthogonal faithful ρ7 2 0 ζ4735+ζ4712 ζ4729+ζ4718 ζ4724+ζ4723 ζ4730+ζ4717 ζ4736+ζ4711 ζ4742+ζ475 ζ4746+ζ47 ζ4740+ζ477 ζ4734+ζ4713 ζ4728+ζ4719 ζ4725+ζ4722 ζ4731+ζ4716 ζ4737+ζ4710 ζ4743+ζ474 ζ4745+ζ472 ζ4739+ζ478 ζ4733+ζ4714 ζ4727+ζ4720 ζ4726+ζ4721 ζ4732+ζ4715 ζ4738+ζ479 ζ4744+ζ473 ζ4741+ζ476 orthogonal faithful ρ8 2 0 ζ4741+ζ476 ζ4738+ζ479 ζ4735+ζ4712 ζ4732+ζ4715 ζ4729+ζ4718 ζ4726+ζ4721 ζ4724+ζ4723 ζ4727+ζ4720 ζ4730+ζ4717 ζ4733+ζ4714 ζ4736+ζ4711 ζ4739+ζ478 ζ4742+ζ475 ζ4745+ζ472 ζ4746+ζ47 ζ4743+ζ474 ζ4740+ζ477 ζ4737+ζ4710 ζ4734+ζ4713 ζ4731+ζ4716 ζ4728+ζ4719 ζ4725+ζ4722 ζ4744+ζ473 orthogonal faithful ρ9 2 0 ζ4739+ζ478 ζ4735+ζ4712 ζ4731+ζ4716 ζ4727+ζ4720 ζ4724+ζ4723 ζ4728+ζ4719 ζ4732+ζ4715 ζ4736+ζ4711 ζ4740+ζ477 ζ4744+ζ473 ζ4746+ζ47 ζ4742+ζ475 ζ4738+ζ479 ζ4734+ζ4713 ζ4730+ζ4717 ζ4726+ζ4721 ζ4725+ζ4722 ζ4729+ζ4718 ζ4733+ζ4714 ζ4737+ζ4710 ζ4741+ζ476 ζ4745+ζ472 ζ4743+ζ474 orthogonal faithful ρ10 2 0 ζ4746+ζ47 ζ4725+ζ4722 ζ4745+ζ472 ζ4726+ζ4721 ζ4744+ζ473 ζ4727+ζ4720 ζ4743+ζ474 ζ4728+ζ4719 ζ4742+ζ475 ζ4729+ζ4718 ζ4741+ζ476 ζ4730+ζ4717 ζ4740+ζ477 ζ4731+ζ4716 ζ4739+ζ478 ζ4732+ζ4715 ζ4738+ζ479 ζ4733+ζ4714 ζ4737+ζ4710 ζ4734+ζ4713 ζ4736+ζ4711 ζ4735+ζ4712 ζ4724+ζ4723 orthogonal faithful ρ11 2 0 ζ4736+ζ4711 ζ4740+ζ477 ζ4725+ζ4722 ζ4743+ζ474 ζ4733+ζ4714 ζ4732+ζ4715 ζ4744+ζ473 ζ4726+ζ4721 ζ4739+ζ478 ζ4737+ζ4710 ζ4728+ζ4719 ζ4746+ζ47 ζ4730+ζ4717 ζ4735+ζ4712 ζ4741+ζ476 ζ4724+ζ4723 ζ4742+ζ475 ζ4734+ζ4713 ζ4731+ζ4716 ζ4745+ζ472 ζ4727+ζ4720 ζ4738+ζ479 ζ4729+ζ4718 orthogonal faithful ρ12 2 0 ζ4729+ζ4718 ζ4727+ζ4720 ζ4736+ζ4711 ζ4745+ζ472 ζ4740+ζ477 ζ4731+ζ4716 ζ4725+ζ4722 ζ4734+ζ4713 ζ4743+ζ474 ζ4742+ζ475 ζ4733+ζ4714 ζ4724+ζ4723 ζ4732+ζ4715 ζ4741+ζ476 ζ4744+ζ473 ζ4735+ζ4712 ζ4726+ζ4721 ζ4730+ζ4717 ζ4739+ζ478 ζ4746+ζ47 ζ4737+ζ4710 ζ4728+ζ4719 ζ4738+ζ479 orthogonal faithful ρ13 2 0 ζ4724+ζ4723 ζ4736+ζ4711 ζ4746+ζ47 ζ4734+ζ4713 ζ4725+ζ4722 ζ4737+ζ4710 ζ4745+ζ472 ζ4733+ζ4714 ζ4726+ζ4721 ζ4738+ζ479 ζ4744+ζ473 ζ4732+ζ4715 ζ4727+ζ4720 ζ4739+ζ478 ζ4743+ζ474 ζ4731+ζ4716 ζ4728+ζ4719 ζ4740+ζ477 ζ4742+ζ475 ζ4730+ζ4717 ζ4729+ζ4718 ζ4741+ζ476 ζ4735+ζ4712 orthogonal faithful ρ14 2 0 ζ4737+ζ4710 ζ4732+ζ4715 ζ4727+ζ4720 ζ4725+ζ4722 ζ4730+ζ4717 ζ4735+ζ4712 ζ4740+ζ477 ζ4745+ζ472 ζ4744+ζ473 ζ4739+ζ478 ζ4734+ζ4713 ζ4729+ζ4718 ζ4724+ζ4723 ζ4728+ζ4719 ζ4733+ζ4714 ζ4738+ζ479 ζ4743+ζ474 ζ4746+ζ47 ζ4741+ζ476 ζ4736+ζ4711 ζ4731+ζ4716 ζ4726+ζ4721 ζ4742+ζ475 orthogonal faithful ρ15 2 0 ζ4742+ζ475 ζ4731+ζ4716 ζ4737+ζ4710 ζ4736+ζ4711 ζ4732+ζ4715 ζ4741+ζ476 ζ4727+ζ4720 ζ4746+ζ47 ζ4725+ζ4722 ζ4743+ζ474 ζ4730+ζ4717 ζ4738+ζ479 ζ4735+ζ4712 ζ4733+ζ4714 ζ4740+ζ477 ζ4728+ζ4719 ζ4745+ζ472 ζ4724+ζ4723 ζ4744+ζ473 ζ4729+ζ4718 ζ4739+ζ478 ζ4734+ζ4713 ζ4726+ζ4721 orthogonal faithful ρ16 2 0 ζ4727+ζ4720 ζ4730+ζ4717 ζ4740+ζ477 ζ4744+ζ473 ζ4734+ζ4713 ζ4724+ζ4723 ζ4733+ζ4714 ζ4743+ζ474 ζ4741+ζ476 ζ4731+ζ4716 ζ4726+ζ4721 ζ4736+ζ4711 ζ4746+ζ47 ζ4738+ζ479 ζ4728+ζ4719 ζ4729+ζ4718 ζ4739+ζ478 ζ4745+ζ472 ζ4735+ζ4712 ζ4725+ζ4722 ζ4732+ζ4715 ζ4742+ζ475 ζ4737+ζ4710 orthogonal faithful ρ17 2 0 ζ4726+ζ4721 ζ4739+ζ478 ζ4742+ζ475 ζ4729+ζ4718 ζ4731+ζ4716 ζ4744+ζ473 ζ4737+ζ4710 ζ4724+ζ4723 ζ4736+ζ4711 ζ4745+ζ472 ζ4732+ζ4715 ζ4728+ζ4719 ζ4741+ζ476 ζ4740+ζ477 ζ4727+ζ4720 ζ4733+ζ4714 ζ4746+ζ47 ζ4735+ζ4712 ζ4725+ζ4722 ζ4738+ζ479 ζ4743+ζ474 ζ4730+ζ4717 ζ4734+ζ4713 orthogonal faithful ρ18 2 0 ζ4730+ζ4717 ζ4745+ζ472 ζ4734+ζ4713 ζ4728+ζ4719 ζ4743+ζ474 ζ4736+ζ4711 ζ4726+ζ4721 ζ4741+ζ476 ζ4738+ζ479 ζ4724+ζ4723 ζ4739+ζ478 ζ4740+ζ477 ζ4725+ζ4722 ζ4737+ζ4710 ζ4742+ζ475 ζ4727+ζ4720 ζ4735+ζ4712 ζ4744+ζ473 ζ4729+ζ4718 ζ4733+ζ4714 ζ4746+ζ47 ζ4731+ζ4716 ζ4732+ζ4715 orthogonal faithful ρ19 2 0 ζ4738+ζ479 ζ4737+ζ4710 ζ4729+ζ4718 ζ4746+ζ47 ζ4727+ζ4720 ζ4739+ζ478 ζ4736+ζ4711 ζ4730+ζ4717 ζ4745+ζ472 ζ4726+ζ4721 ζ4740+ζ477 ζ4735+ζ4712 ζ4731+ζ4716 ζ4744+ζ473 ζ4725+ζ4722 ζ4741+ζ476 ζ4734+ζ4713 ζ4732+ζ4715 ζ4743+ζ474 ζ4724+ζ4723 ζ4742+ζ475 ζ4733+ζ4714 ζ4728+ζ4719 orthogonal faithful ρ20 2 0 ζ4728+ζ4719 ζ4742+ζ475 ζ4738+ζ479 ζ4724+ζ4723 ζ4737+ζ4710 ζ4743+ζ474 ζ4729+ζ4718 ζ4732+ζ4715 ζ4746+ζ47 ζ4734+ζ4713 ζ4727+ζ4720 ζ4741+ζ476 ζ4739+ζ478 ζ4725+ζ4722 ζ4736+ζ4711 ζ4744+ζ473 ζ4730+ζ4717 ζ4731+ζ4716 ζ4745+ζ472 ζ4735+ζ4712 ζ4726+ζ4721 ζ4740+ζ477 ζ4733+ζ4714 orthogonal faithful ρ21 2 0 ζ4744+ζ473 ζ4728+ζ4719 ζ4741+ζ476 ζ4731+ζ4716 ζ4738+ζ479 ζ4734+ζ4713 ζ4735+ζ4712 ζ4737+ζ4710 ζ4732+ζ4715 ζ4740+ζ477 ζ4729+ζ4718 ζ4743+ζ474 ζ4726+ζ4721 ζ4746+ζ47 ζ4724+ζ4723 ζ4745+ζ472 ζ4727+ζ4720 ζ4742+ζ475 ζ4730+ζ4717 ζ4739+ζ478 ζ4733+ζ4714 ζ4736+ζ4711 ζ4725+ζ4722 orthogonal faithful ρ22 2 0 ζ4743+ζ474 ζ4741+ζ476 ζ4739+ζ478 ζ4737+ζ4710 ζ4735+ζ4712 ζ4733+ζ4714 ζ4731+ζ4716 ζ4729+ζ4718 ζ4727+ζ4720 ζ4725+ζ4722 ζ4724+ζ4723 ζ4726+ζ4721 ζ4728+ζ4719 ζ4730+ζ4717 ζ4732+ζ4715 ζ4734+ζ4713 ζ4736+ζ4711 ζ4738+ζ479 ζ4740+ζ477 ζ4742+ζ475 ζ4744+ζ473 ζ4746+ζ47 ζ4745+ζ472 orthogonal faithful ρ23 2 0 ζ4731+ζ4716 ζ4724+ζ4723 ζ4732+ζ4715 ζ4740+ζ477 ζ4746+ζ47 ζ4738+ζ479 ζ4730+ζ4717 ζ4725+ζ4722 ζ4733+ζ4714 ζ4741+ζ476 ζ4745+ζ472 ζ4737+ζ4710 ζ4729+ζ4718 ζ4726+ζ4721 ζ4734+ζ4713 ζ4742+ζ475 ζ4744+ζ473 ζ4736+ζ4711 ζ4728+ζ4719 ζ4727+ζ4720 ζ4735+ζ4712 ζ4743+ζ474 ζ4739+ζ478 orthogonal faithful ρ24 2 0 ζ4733+ζ4714 ζ4726+ζ4721 ζ4728+ζ4719 ζ4735+ζ4712 ζ4742+ζ475 ζ4745+ζ472 ζ4738+ζ479 ζ4731+ζ4716 ζ4724+ζ4723 ζ4730+ζ4717 ζ4737+ζ4710 ζ4744+ζ473 ζ4743+ζ474 ζ4736+ζ4711 ζ4729+ζ4718 ζ4725+ζ4722 ζ4732+ζ4715 ζ4739+ζ478 ζ4746+ζ47 ζ4741+ζ476 ζ4734+ζ4713 ζ4727+ζ4720 ζ4740+ζ477 orthogonal faithful ρ25 2 0 ζ4740+ζ477 ζ4734+ζ4713 ζ4733+ζ4714 ζ4741+ζ476 ζ4726+ζ4721 ζ4746+ζ47 ζ4728+ζ4719 ζ4739+ζ478 ζ4735+ζ4712 ζ4732+ζ4715 ζ4742+ζ475 ζ4725+ζ4722 ζ4745+ζ472 ζ4729+ζ4718 ζ4738+ζ479 ζ4736+ζ4711 ζ4731+ζ4716 ζ4743+ζ474 ζ4724+ζ4723 ζ4744+ζ473 ζ4730+ζ4717 ζ4737+ζ4710 ζ4727+ζ4720 orthogonal faithful

Smallest permutation representation of D47
On 47 points: primitive
Generators in S47
```(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)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)```

`G:=sub<Sym(47)| (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)>;`

`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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25) );`

`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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25)]])`

D47 is a maximal subgroup of   D141  D235
D47 is a maximal quotient of   Dic47  D141  D235

Matrix representation of D47 in GL2(𝔽283) generated by

 27 282 1 0
,
 27 282 162 256
`G:=sub<GL(2,GF(283))| [27,1,282,0],[27,162,282,256] >;`

D47 in GAP, Magma, Sage, TeX

`D_{47}`
`% in TeX`

`G:=Group("D47");`
`// GroupNames label`

`G:=SmallGroup(94,1);`
`// by ID`

`G=gap.SmallGroup(94,1);`
`# by ID`

`G:=PCGroup([2,-2,-47,369]);`
`// Polycyclic`

`G:=Group<a,b|a^47=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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