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## G = D112order 224 = 25·7

### Dihedral group

Aliases: D112, C71D16, C161D7, C1121C2, D561C2, C4.1D28, C2.3D56, C14.1D8, C28.24D4, C8.13D14, C56.14C22, sometimes denoted D224 or Dih112 or Dih224, SmallGroup(224,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — D112
 Chief series C1 — C7 — C14 — C28 — C56 — D56 — D112
 Lower central C7 — C14 — C28 — C56 — D112
 Upper central C1 — C2 — C4 — C8 — C16

Generators and relations for D112
G = < a,b | a112=b2=1, bab=a-1 >

56C2
56C2
28C22
28C22
8D7
8D7
14D4
14D4
4D14
4D14
7D8
7D8
2D28
2D28
7D16

Smallest permutation representation of D112
On 112 points
Generators in S112
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 112)(2 111)(3 110)(4 109)(5 108)(6 107)(7 106)(8 105)(9 104)(10 103)(11 102)(12 101)(13 100)(14 99)(15 98)(16 97)(17 96)(18 95)(19 94)(20 93)(21 92)(22 91)(23 90)(24 89)(25 88)(26 87)(27 86)(28 85)(29 84)(30 83)(31 82)(32 81)(33 80)(34 79)(35 78)(36 77)(37 76)(38 75)(39 74)(40 73)(41 72)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)```

`G:=sub<Sym(112)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,112)(2,111)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,112)(2,111)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,112),(2,111),(3,110),(4,109),(5,108),(6,107),(7,106),(8,105),(9,104),(10,103),(11,102),(12,101),(13,100),(14,99),(15,98),(16,97),(17,96),(18,95),(19,94),(20,93),(21,92),(22,91),(23,90),(24,89),(25,88),(26,87),(27,86),(28,85),(29,84),(30,83),(31,82),(32,81),(33,80),(34,79),(35,78),(36,77),(37,76),(38,75),(39,74),(40,73),(41,72),(42,71),(43,70),(44,69),(45,68),(46,67),(47,66),(48,65),(49,64),(50,63),(51,62),(52,61),(53,60),(54,59),(55,58),(56,57)])`

D112 is a maximal subgroup of
D224  C224⋊C2  C7⋊D32  C7⋊SD64  D1127C2  C16⋊D14  D7×D16  D112⋊C2  Q323D7
D112 is a maximal quotient of
D224  C224⋊C2  Dic112  C1125C4  C2.D112

59 conjugacy classes

 class 1 2A 2B 2C 4 7A 7B 7C 8A 8B 14A 14B 14C 16A 16B 16C 16D 28A ··· 28F 56A ··· 56L 112A ··· 112X order 1 2 2 2 4 7 7 7 8 8 14 14 14 16 16 16 16 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 56 56 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

59 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 D4 D7 D8 D14 D16 D28 D56 D112 kernel D112 C112 D56 C28 C16 C14 C8 C7 C4 C2 C1 # reps 1 1 2 1 3 2 3 4 6 12 24

Matrix representation of D112 in GL2(𝔽113) generated by

 74 35 78 14
,
 74 35 5 39
`G:=sub<GL(2,GF(113))| [74,78,35,14],[74,5,35,39] >;`

D112 in GAP, Magma, Sage, TeX

`D_{112}`
`% in TeX`

`G:=Group("D112");`
`// GroupNames label`

`G:=SmallGroup(224,5);`
`// by ID`

`G=gap.SmallGroup(224,5);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,73,79,218,122,579,69,6917]);`
`// Polycyclic`

`G:=Group<a,b|a^112=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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