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## G = D124order 248 = 23·31

### Dihedral group

Aliases: D124, C4⋊D31, C311D4, C1241C2, D621C2, C2.4D62, C62.3C22, sometimes denoted D248 or Dih124 or Dih248, SmallGroup(248,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — D124
 Chief series C1 — C31 — C62 — D62 — D124
 Lower central C31 — C62 — D124
 Upper central C1 — C2 — C4

Generators and relations for D124
G = < a,b | a124=b2=1, bab=a-1 >

62C2
62C2
31C22
31C22
2D31
2D31
31D4

Smallest permutation representation of D124
On 124 points
Generators in S124
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124)
(1 124)(2 123)(3 122)(4 121)(5 120)(6 119)(7 118)(8 117)(9 116)(10 115)(11 114)(12 113)(13 112)(14 111)(15 110)(16 109)(17 108)(18 107)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 100)(26 99)(27 98)(28 97)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 88)(38 87)(39 86)(40 85)(41 84)(42 83)(43 82)(44 81)(45 80)(46 79)(47 78)(48 77)(49 76)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)```

`G:=sub<Sym(124)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,124)(2,123)(3,122)(4,121)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,124)(2,123)(3,122)(4,121)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124)], [(1,124),(2,123),(3,122),(4,121),(5,120),(6,119),(7,118),(8,117),(9,116),(10,115),(11,114),(12,113),(13,112),(14,111),(15,110),(16,109),(17,108),(18,107),(19,106),(20,105),(21,104),(22,103),(23,102),(24,101),(25,100),(26,99),(27,98),(28,97),(29,96),(30,95),(31,94),(32,93),(33,92),(34,91),(35,90),(36,89),(37,88),(38,87),(39,86),(40,85),(41,84),(42,83),(43,82),(44,81),(45,80),(46,79),(47,78),(48,77),(49,76),(50,75),(51,74),(52,73),(53,72),(54,71),(55,70),(56,69),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63)]])`

D124 is a maximal subgroup of   C248⋊C2  D248  D4⋊D31  Q8⋊D31  D1245C2  D4×D31  Q82D31
D124 is a maximal quotient of   C248⋊C2  D248  Dic124  C4⋊Dic31  D62⋊C4

65 conjugacy classes

 class 1 2A 2B 2C 4 31A ··· 31O 62A ··· 62O 124A ··· 124AD order 1 2 2 2 4 31 ··· 31 62 ··· 62 124 ··· 124 size 1 1 62 62 2 2 ··· 2 2 ··· 2 2 ··· 2

65 irreducible representations

 dim 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 D4 D31 D62 D124 kernel D124 C124 D62 C31 C4 C2 C1 # reps 1 1 2 1 15 15 30

Matrix representation of D124 in GL2(𝔽373) generated by

 236 299 74 285
,
 137 74 321 236
`G:=sub<GL(2,GF(373))| [236,74,299,285],[137,321,74,236] >;`

D124 in GAP, Magma, Sage, TeX

`D_{124}`
`% in TeX`

`G:=Group("D124");`
`// GroupNames label`

`G:=SmallGroup(248,5);`
`// by ID`

`G=gap.SmallGroup(248,5);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-31,49,21,3843]);`
`// Polycyclic`

`G:=Group<a,b|a^124=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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