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## G = D92order 184 = 23·23

### Dihedral group

Aliases: D92, C4⋊D23, C231D4, C921C2, D461C2, C2.4D46, C46.3C22, sometimes denoted D184 or Dih92 or Dih184, SmallGroup(184,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C46 — D92
 Chief series C1 — C23 — C46 — D46 — D92
 Lower central C23 — C46 — D92
 Upper central C1 — C2 — C4

Generators and relations for D92
G = < a,b | a92=b2=1, bab=a-1 >

46C2
46C2
23C22
23C22
2D23
2D23
23D4

Smallest permutation representation of D92
On 92 points
Generators in S92
```(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)
(1 92)(2 91)(3 90)(4 89)(5 88)(6 87)(7 86)(8 85)(9 84)(10 83)(11 82)(12 81)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)```

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

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

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

D92 is a maximal subgroup of   C184⋊C2  D184  D4⋊D23  Q8⋊D23  D925C2  D4×D23  D92⋊C2
D92 is a maximal quotient of   C184⋊C2  D184  Dic92  C92⋊C4  D46⋊C4

49 conjugacy classes

 class 1 2A 2B 2C 4 23A ··· 23K 46A ··· 46K 92A ··· 92V order 1 2 2 2 4 23 ··· 23 46 ··· 46 92 ··· 92 size 1 1 46 46 2 2 ··· 2 2 ··· 2 2 ··· 2

49 irreducible representations

 dim 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 D4 D23 D46 D92 kernel D92 C92 D46 C23 C4 C2 C1 # reps 1 1 2 1 11 11 22

Matrix representation of D92 in GL2(𝔽277) generated by

 248 197 127 16
,
 145 139 56 132
`G:=sub<GL(2,GF(277))| [248,127,197,16],[145,56,139,132] >;`

D92 in GAP, Magma, Sage, TeX

`D_{92}`
`% in TeX`

`G:=Group("D92");`
`// GroupNames label`

`G:=SmallGroup(184,5);`
`// by ID`

`G=gap.SmallGroup(184,5);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-23,49,21,2819]);`
`// Polycyclic`

`G:=Group<a,b|a^92=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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