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## G = D56order 112 = 24·7

### Dihedral group

Aliases: D56, C71D8, C81D7, C561C2, D281C2, C2.4D28, C4.9D14, C14.2D4, C28.9C22, sometimes denoted D112 or Dih56 or Dih112, SmallGroup(112,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D56
 Chief series C1 — C7 — C14 — C28 — D28 — D56
 Lower central C7 — C14 — C28 — D56
 Upper central C1 — C2 — C4 — C8

Generators and relations for D56
G = < a,b | a56=b2=1, bab=a-1 >

28C2
28C2
14C22
14C22
4D7
4D7
7D4
7D4
2D14
2D14
7D8

Smallest permutation representation of D56
On 56 points
Generators in S56
```(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)
(1 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)```

`G:=sub<Sym(56)| (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)>;`

`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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29) );`

`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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29)]])`

D56 is a maximal subgroup of
D112  C112⋊C2  C7⋊D16  C7⋊SD32  D567C2  C8⋊D14  D7×D8  D56⋊C2  Q8.D14  D56⋊C3  C3⋊D56  D168
D56 is a maximal quotient of
D112  C112⋊C2  Dic56  C561C4  C2.D56  C3⋊D56  D168

31 conjugacy classes

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

31 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 D4 D7 D8 D14 D28 D56 kernel D56 C56 D28 C14 C8 C7 C4 C2 C1 # reps 1 1 2 1 3 2 3 6 12

Matrix representation of D56 in GL2(𝔽113) generated by

 49 44 25 34
,
 21 106 79 92
`G:=sub<GL(2,GF(113))| [49,25,44,34],[21,79,106,92] >;`

D56 in GAP, Magma, Sage, TeX

`D_{56}`
`% in TeX`

`G:=Group("D56");`
`// GroupNames label`

`G:=SmallGroup(112,6);`
`// by ID`

`G=gap.SmallGroup(112,6);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-7,61,66,182,42,2404]);`
`// Polycyclic`

`G:=Group<a,b|a^56=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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