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## G = D121order 242 = 2·112

### Dihedral group

Aliases: D121, C121⋊C2, C11.D11, sometimes denoted D242 or Dih121 or Dih242, SmallGroup(242,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C121 — D121
 Chief series C1 — C11 — C121 — D121
 Lower central C121 — D121
 Upper central C1

Generators and relations for D121
G = < a,b | a121=b2=1, bab=a-1 >

121C2
11D11

Smallest permutation representation of D121
On 121 points
Generators in S121
```(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)
(1 121)(2 120)(3 119)(4 118)(5 117)(6 116)(7 115)(8 114)(9 113)(10 112)(11 111)(12 110)(13 109)(14 108)(15 107)(16 106)(17 105)(18 104)(19 103)(20 102)(21 101)(22 100)(23 99)(24 98)(25 97)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 85)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)```

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

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

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

D121 is a maximal quotient of   Dic121

62 conjugacy classes

 class 1 2 11A ··· 11E 121A ··· 121BC order 1 2 11 ··· 11 121 ··· 121 size 1 121 2 ··· 2 2 ··· 2

62 irreducible representations

 dim 1 1 2 2 type + + + + image C1 C2 D11 D121 kernel D121 C121 C11 C1 # reps 1 1 5 55

Matrix representation of D121 in GL2(𝔽727) generated by

 251 315 412 239
,
 306 79 462 421
`G:=sub<GL(2,GF(727))| [251,412,315,239],[306,462,79,421] >;`

D121 in GAP, Magma, Sage, TeX

`D_{121}`
`% in TeX`

`G:=Group("D121");`
`// GroupNames label`

`G:=SmallGroup(242,1);`
`// by ID`

`G=gap.SmallGroup(242,1);`
`# by ID`

`G:=PCGroup([3,-2,-11,-11,1441,70,1982]);`
`// Polycyclic`

`G:=Group<a,b|a^121=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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