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## G = D108order 216 = 23·33

### Dihedral group

Aliases: D108, C4⋊D27, C271D4, C3.D36, C9.D12, C1081C2, D541C2, C36.2S3, C12.2D9, C2.4D54, C6.10D18, C18.10D6, C54.3C22, sometimes denoted D216 or Dih108 or Dih216, SmallGroup(216,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C54 — D108
 Chief series C1 — C3 — C9 — C27 — C54 — D54 — D108
 Lower central C27 — C54 — D108
 Upper central C1 — C2 — C4

Generators and relations for D108
G = < a,b | a108=b2=1, bab=a-1 >

54C2
54C2
27C22
27C22
18S3
18S3
27D4
9D6
9D6
6D9
6D9
9D12
3D18
3D18
2D27
2D27
3D36

Smallest permutation representation of D108
On 108 points
Generators in S108
```(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)
(1 108)(2 107)(3 106)(4 105)(5 104)(6 103)(7 102)(8 101)(9 100)(10 99)(11 98)(12 97)(13 96)(14 95)(15 94)(16 93)(17 92)(18 91)(19 90)(20 89)(21 88)(22 87)(23 86)(24 85)(25 84)(26 83)(27 82)(28 81)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 72)(38 71)(39 70)(40 69)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)```

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

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

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

D108 is a maximal subgroup of   C216⋊C2  D216  D4⋊D27  Q82D27  D1085C2  D4×D27  Q83D27
D108 is a maximal quotient of   Dic108  C216⋊C2  D216  C4⋊Dic27  D54⋊C4

57 conjugacy classes

 class 1 2A 2B 2C 3 4 6 9A 9B 9C 12A 12B 18A 18B 18C 27A ··· 27I 36A ··· 36F 54A ··· 54I 108A ··· 108R order 1 2 2 2 3 4 6 9 9 9 12 12 18 18 18 27 ··· 27 36 ··· 36 54 ··· 54 108 ··· 108 size 1 1 54 54 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

57 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 S3 D4 D6 D9 D12 D18 D27 D36 D54 D108 kernel D108 C108 D54 C36 C27 C18 C12 C9 C6 C4 C3 C2 C1 # reps 1 1 2 1 1 1 3 2 3 9 6 9 18

Matrix representation of D108 in GL2(𝔽109) generated by

 78 66 43 35
,
 43 35 78 66
`G:=sub<GL(2,GF(109))| [78,43,66,35],[43,78,35,66] >;`

D108 in GAP, Magma, Sage, TeX

`D_{108}`
`% in TeX`

`G:=Group("D108");`
`// GroupNames label`

`G:=SmallGroup(216,6);`
`// by ID`

`G=gap.SmallGroup(216,6);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,963,381,3604,208,5189]);`
`// Polycyclic`

`G:=Group<a,b|a^108=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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