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## G = D105order 210 = 2·3·5·7

### Dihedral group

Aliases: D105, C3⋊D35, C5⋊D21, C7⋊D15, C351S3, C211D5, C151D7, C1051C2, sometimes denoted D210 or Dih105 or Dih210, SmallGroup(210,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C105 — D105
 Chief series C1 — C7 — C35 — C105 — D105
 Lower central C105 — D105
 Upper central C1

Generators and relations for D105
G = < a,b | a105=b2=1, bab=a-1 >

105C2
35S3
21D5
15D7
7D15
5D21
3D35

Smallest permutation representation of D105
On 105 points
Generators in S105
```(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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)
(2 105)(3 104)(4 103)(5 102)(6 101)(7 100)(8 99)(9 98)(10 97)(11 96)(12 95)(13 94)(14 93)(15 92)(16 91)(17 90)(18 89)(19 88)(20 87)(21 86)(22 85)(23 84)(24 83)(25 82)(26 81)(27 80)(28 79)(29 78)(30 77)(31 76)(32 75)(33 74)(34 73)(35 72)(36 71)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)```

`G:=sub<Sym(105)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105), (2,105)(3,104)(4,103)(5,102)(6,101)(7,100)(8,99)(9,98)(10,97)(11,96)(12,95)(13,94)(14,93)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105), (2,105)(3,104)(4,103)(5,102)(6,101)(7,100)(8,99)(9,98)(10,97)(11,96)(12,95)(13,94)(14,93)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)], [(2,105),(3,104),(4,103),(5,102),(6,101),(7,100),(8,99),(9,98),(10,97),(11,96),(12,95),(13,94),(14,93),(15,92),(16,91),(17,90),(18,89),(19,88),(20,87),(21,86),(22,85),(23,84),(24,83),(25,82),(26,81),(27,80),(28,79),(29,78),(30,77),(31,76),(32,75),(33,74),(34,73),(35,72),(36,71),(37,70),(38,69),(39,68),(40,67),(41,66),(42,65),(43,64),(44,63),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54)])`

D105 is a maximal subgroup of   D7×D15  D5×D21  S3×D35
D105 is a maximal quotient of   Dic105

54 conjugacy classes

 class 1 2 3 5A 5B 7A 7B 7C 15A 15B 15C 15D 21A ··· 21F 35A ··· 35L 105A ··· 105X order 1 2 3 5 5 7 7 7 15 15 15 15 21 ··· 21 35 ··· 35 105 ··· 105 size 1 105 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

54 irreducible representations

 dim 1 1 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 S3 D5 D7 D15 D21 D35 D105 kernel D105 C105 C35 C21 C15 C7 C5 C3 C1 # reps 1 1 1 2 3 4 6 12 24

Matrix representation of D105 in GL2(𝔽211) generated by

 181 42 121 133
,
 0 193 82 0
`G:=sub<GL(2,GF(211))| [181,121,42,133],[0,82,193,0] >;`

D105 in GAP, Magma, Sage, TeX

`D_{105}`
`% in TeX`

`G:=Group("D105");`
`// GroupNames label`

`G:=SmallGroup(210,11);`
`// by ID`

`G=gap.SmallGroup(210,11);`
`# by ID`

`G:=PCGroup([4,-2,-3,-5,-7,33,290,2883]);`
`// Polycyclic`

`G:=Group<a,b|a^105=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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