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## G = D110order 220 = 22·5·11

### Dihedral group

Aliases: D110, C2×D55, C22⋊D5, C10⋊D11, C52D22, C112D10, C1101C2, C552C22, sometimes denoted D220 or Dih110 or Dih220, SmallGroup(220,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C55 — D110
 Chief series C1 — C11 — C55 — D55 — D110
 Lower central C55 — D110
 Upper central C1 — C2

Generators and relations for D110
G = < a,b | a110=b2=1, bab=a-1 >

55C2
55C2
55C22
11D5
11D5
5D11
5D11
11D10
5D22

Smallest permutation representation of D110
On 110 points
Generators in S110
```(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 99 100 101 102 103 104 105 106 107 108 109 110)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 100)(12 99)(13 98)(14 97)(15 96)(16 95)(17 94)(18 93)(19 92)(20 91)(21 90)(22 89)(23 88)(24 87)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)```

`G:=sub<Sym(110)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,100)(12,99)(13,98)(14,97)(15,96)(16,95)(17,94)(18,93)(19,92)(20,91)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,100)(12,99)(13,98)(14,97)(15,96)(16,95)(17,94)(18,93)(19,92)(20,91)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110)], [(1,110),(2,109),(3,108),(4,107),(5,106),(6,105),(7,104),(8,103),(9,102),(10,101),(11,100),(12,99),(13,98),(14,97),(15,96),(16,95),(17,94),(18,93),(19,92),(20,91),(21,90),(22,89),(23,88),(24,87),(25,86),(26,85),(27,84),(28,83),(29,82),(30,81),(31,80),(32,79),(33,78),(34,77),(35,76),(36,75),(37,74),(38,73),(39,72),(40,71),(41,70),(42,69),(43,68),(44,67),(45,66),(46,65),(47,64),(48,63),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,56)])`

D110 is a maximal subgroup of   D552C4  C5⋊D44  C11⋊D20  D220  C557D4  C2×D5×D11
D110 is a maximal quotient of   Dic110  D220  C557D4

58 conjugacy classes

 class 1 2A 2B 2C 5A 5B 10A 10B 11A ··· 11E 22A ··· 22E 55A ··· 55T 110A ··· 110T order 1 2 2 2 5 5 10 10 11 ··· 11 22 ··· 22 55 ··· 55 110 ··· 110 size 1 1 55 55 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

58 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 D5 D10 D11 D22 D55 D110 kernel D110 D55 C110 C22 C11 C10 C5 C2 C1 # reps 1 2 1 2 2 5 5 20 20

Matrix representation of D110 in GL2(𝔽331) generated by

 133 250 81 100
,
 133 250 59 198
`G:=sub<GL(2,GF(331))| [133,81,250,100],[133,59,250,198] >;`

D110 in GAP, Magma, Sage, TeX

`D_{110}`
`% in TeX`

`G:=Group("D110");`
`// GroupNames label`

`G:=SmallGroup(220,14);`
`// by ID`

`G=gap.SmallGroup(220,14);`
`# by ID`

`G:=PCGroup([4,-2,-2,-5,-11,194,3203]);`
`// Polycyclic`

`G:=Group<a,b|a^110=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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