direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D122, C2×D61, C122⋊C2, C61⋊C22, sometimes denoted D244 or Dih122 or Dih244, SmallGroup(244,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C61 — D122 |
Generators and relations for D122
G = < a,b | a122=b2=1, bab=a-1 >
Smallest permutation representation of D122
►On 122 points
Generators in S122
(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)
(1 122)(2 121)(3 120)(4 119)(5 118)(6 117)(7 116)(8 115)(9 114)(10 113)(11 112)(12 111)(13 110)(14 109)(15 108)(16 107)(17 106)(18 105)(19 104)(20 103)(21 102)(22 101)(23 100)(24 99)(25 98)(26 97)(27 96)(28 95)(29 94)(30 93)(31 92)(32 91)(33 90)(34 89)(35 88)(36 87)(37 86)(38 85)(39 84)(40 83)(41 82)(42 81)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)
G:=sub<Sym(122)| (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), (1,122)(2,121)(3,120)(4,119)(5,118)(6,117)(7,116)(8,115)(9,114)(10,113)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122), (1,122)(2,121)(3,120)(4,119)(5,118)(6,117)(7,116)(8,115)(9,114)(10,113)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122)], [(1,122),(2,121),(3,120),(4,119),(5,118),(6,117),(7,116),(8,115),(9,114),(10,113),(11,112),(12,111),(13,110),(14,109),(15,108),(16,107),(17,106),(18,105),(19,104),(20,103),(21,102),(22,101),(23,100),(24,99),(25,98),(26,97),(27,96),(28,95),(29,94),(30,93),(31,92),(32,91),(33,90),(34,89),(35,88),(36,87),(37,86),(38,85),(39,84),(40,83),(41,82),(42,81),(43,80),(44,79),(45,78),(46,77),(47,76),(48,75),(49,74),(50,73),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(61,62)]])
D122 is a maximal subgroup of
D244 C61⋊D4
D122 is a maximal quotient of Dic122 D244 C61⋊D4
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 61A | ··· | 61AD | 122A | ··· | 122AD |
| order | 1 | 2 | 2 | 2 | 61 | ··· | 61 | 122 | ··· | 122 |
| size | 1 | 1 | 61 | 61 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D61 | D122 |
| kernel | D122 | D61 | C122 | C2 | C1 |
| # reps | 1 | 2 | 1 | 30 | 30 |
Matrix representation of D122 ►in GL2(𝔽367) generated by
| 274 | 361 |
| 68 | 32 |
| 81 | 184 |
| 92 | 286 |
G:=sub<GL(2,GF(367))| [274,68,361,32],[81,92,184,286] >;
D122 in GAP, Magma, Sage, TeX
D_{122} % in TeX
G:=Group("D122"); // GroupNames label
G:=SmallGroup(244,4);
// by ID
G=gap.SmallGroup(244,4);
# by ID
G:=PCGroup([3,-2,-2,-61,2162]);
// Polycyclic
G:=Group<a,b|a^122=b^2=1,b*a*b=a^-1>;
// generators/relations
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