Copied to
clipboard

## G = D93order 186 = 2·3·31

### Dihedral group

Aliases: D93, C31⋊S3, C3⋊D31, C931C2, sometimes denoted D186 or Dih93 or Dih186, SmallGroup(186,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C93 — D93
 Chief series C1 — C31 — C93 — D93
 Lower central C93 — D93
 Upper central C1

Generators and relations for D93
G = < a,b | a93=b2=1, bab=a-1 >

93C2
31S3
3D31

Smallest permutation representation of D93
On 93 points
Generators in S93
```(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)
(1 93)(2 92)(3 91)(4 90)(5 89)(6 88)(7 87)(8 86)(9 85)(10 84)(11 83)(12 82)(13 81)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 73)(22 72)(23 71)(24 70)(25 69)(26 68)(27 67)(28 66)(29 65)(30 64)(31 63)(32 62)(33 61)(34 60)(35 59)(36 58)(37 57)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)```

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

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

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

D93 is a maximal subgroup of   S3×D31
D93 is a maximal quotient of   Dic93

48 conjugacy classes

 class 1 2 3 31A ··· 31O 93A ··· 93AD order 1 2 3 31 ··· 31 93 ··· 93 size 1 93 2 2 ··· 2 2 ··· 2

48 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 S3 D31 D93 kernel D93 C93 C31 C3 C1 # reps 1 1 1 15 30

Matrix representation of D93 in GL2(𝔽373) generated by

 248 132 241 211
,
 248 132 119 125
`G:=sub<GL(2,GF(373))| [248,241,132,211],[248,119,132,125] >;`

D93 in GAP, Magma, Sage, TeX

`D_{93}`
`% in TeX`

`G:=Group("D93");`
`// GroupNames label`

`G:=SmallGroup(186,5);`
`// by ID`

`G=gap.SmallGroup(186,5);`
`# by ID`

`G:=PCGroup([3,-2,-3,-31,25,1622]);`
`// Polycyclic`

`G:=Group<a,b|a^93=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

Export

׿
×
𝔽