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G = D106order 212 = 22·53

Dihedral group

Aliases: D106, C2×D53, C106⋊C2, C53⋊C22, sometimes denoted D212 or Dih106 or Dih212, SmallGroup(212,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C53 — D106
 Chief series C1 — C53 — D53 — D106
 Lower central C53 — D106
 Upper central C1 — C2

Generators and relations for D106
G = < a,b | a106=b2=1, bab=a-1 >

Smallest permutation representation of D106
On 106 points
Generators in S106
```(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 99 100 101 102 103 104 105 106)
(1 106)(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(106)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106), (1,106)(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( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106), (1,106)(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([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106)], [(1,106),(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)]])`

D106 is a maximal subgroup of   D212  C53⋊D4
D106 is a maximal quotient of   Dic106  D212  C53⋊D4

56 conjugacy classes

 class 1 2A 2B 2C 53A ··· 53Z 106A ··· 106Z order 1 2 2 2 53 ··· 53 106 ··· 106 size 1 1 53 53 2 ··· 2 2 ··· 2

56 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D53 D106 kernel D106 D53 C106 C2 C1 # reps 1 2 1 26 26

Matrix representation of D106 in GL2(𝔽107) generated by

 6 34 50 16
,
 56 21 80 51
`G:=sub<GL(2,GF(107))| [6,50,34,16],[56,80,21,51] >;`

D106 in GAP, Magma, Sage, TeX

`D_{106}`
`% in TeX`

`G:=Group("D106");`
`// GroupNames label`

`G:=SmallGroup(212,4);`
`// by ID`

`G=gap.SmallGroup(212,4);`
`# by ID`

`G:=PCGroup([3,-2,-2,-53,1874]);`
`// Polycyclic`

`G:=Group<a,b|a^106=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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