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## G = D88order 176 = 24·11

### Dihedral group

Aliases: D88, C111D8, C881C2, C81D11, D441C2, C4.9D22, C2.4D44, C22.2D4, C44.9C22, sometimes denoted D176 or Dih88 or Dih176, SmallGroup(176,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — D88
 Chief series C1 — C11 — C22 — C44 — D44 — D88
 Lower central C11 — C22 — C44 — D88
 Upper central C1 — C2 — C4 — C8

Generators and relations for D88
G = < a,b | a88=b2=1, bab=a-1 >

44C2
44C2
22C22
22C22
4D11
4D11
11D4
11D4
2D22
2D22
11D8

Smallest permutation representation of D88
On 88 points
Generators in S88
```(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)
(1 88)(2 87)(3 86)(4 85)(5 84)(6 83)(7 82)(8 81)(9 80)(10 79)(11 78)(12 77)(13 76)(14 75)(15 74)(16 73)(17 72)(18 71)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)```

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

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

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

D88 is a maximal subgroup of
D176  C176⋊C2  C11⋊D16  C8.6D22  D887C2  C8⋊D22  D8×D11  D88⋊C2  D885C2
D88 is a maximal quotient of
D176  C176⋊C2  Dic88  C44.5Q8  C2.D88

47 conjugacy classes

 class 1 2A 2B 2C 4 8A 8B 11A ··· 11E 22A ··· 22E 44A ··· 44J 88A ··· 88T order 1 2 2 2 4 8 8 11 ··· 11 22 ··· 22 44 ··· 44 88 ··· 88 size 1 1 44 44 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

47 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 D4 D8 D11 D22 D44 D88 kernel D88 C88 D44 C22 C11 C8 C4 C2 C1 # reps 1 1 2 1 2 5 5 10 20

Matrix representation of D88 in GL2(𝔽89) generated by

 17 35 85 18
,
 79 3 56 10
`G:=sub<GL(2,GF(89))| [17,85,35,18],[79,56,3,10] >;`

D88 in GAP, Magma, Sage, TeX

`D_{88}`
`% in TeX`

`G:=Group("D88");`
`// GroupNames label`

`G:=SmallGroup(176,6);`
`// by ID`

`G=gap.SmallGroup(176,6);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-11,61,66,182,42,4004]);`
`// Polycyclic`

`G:=Group<a,b|a^88=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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