metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic31, C31⋊C4, C62.C2, C2.D31, SmallGroup(124,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — Dic31 |
Generators and relations for Dic31
G = < a,b | a62=1, b2=a31, bab-1=a-1 >
Smallest permutation representation of Dic31
►Regular action 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 98 32 67)(2 97 33 66)(3 96 34 65)(4 95 35 64)(5 94 36 63)(6 93 37 124)(7 92 38 123)(8 91 39 122)(9 90 40 121)(10 89 41 120)(11 88 42 119)(12 87 43 118)(13 86 44 117)(14 85 45 116)(15 84 46 115)(16 83 47 114)(17 82 48 113)(18 81 49 112)(19 80 50 111)(20 79 51 110)(21 78 52 109)(22 77 53 108)(23 76 54 107)(24 75 55 106)(25 74 56 105)(26 73 57 104)(27 72 58 103)(28 71 59 102)(29 70 60 101)(30 69 61 100)(31 68 62 99)
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), (1,98,32,67)(2,97,33,66)(3,96,34,65)(4,95,35,64)(5,94,36,63)(6,93,37,124)(7,92,38,123)(8,91,39,122)(9,90,40,121)(10,89,41,120)(11,88,42,119)(12,87,43,118)(13,86,44,117)(14,85,45,116)(15,84,46,115)(16,83,47,114)(17,82,48,113)(18,81,49,112)(19,80,50,111)(20,79,51,110)(21,78,52,109)(22,77,53,108)(23,76,54,107)(24,75,55,106)(25,74,56,105)(26,73,57,104)(27,72,58,103)(28,71,59,102)(29,70,60,101)(30,69,61,100)(31,68,62,99)>;
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), (1,98,32,67)(2,97,33,66)(3,96,34,65)(4,95,35,64)(5,94,36,63)(6,93,37,124)(7,92,38,123)(8,91,39,122)(9,90,40,121)(10,89,41,120)(11,88,42,119)(12,87,43,118)(13,86,44,117)(14,85,45,116)(15,84,46,115)(16,83,47,114)(17,82,48,113)(18,81,49,112)(19,80,50,111)(20,79,51,110)(21,78,52,109)(22,77,53,108)(23,76,54,107)(24,75,55,106)(25,74,56,105)(26,73,57,104)(27,72,58,103)(28,71,59,102)(29,70,60,101)(30,69,61,100)(31,68,62,99) );
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)], [(1,98,32,67),(2,97,33,66),(3,96,34,65),(4,95,35,64),(5,94,36,63),(6,93,37,124),(7,92,38,123),(8,91,39,122),(9,90,40,121),(10,89,41,120),(11,88,42,119),(12,87,43,118),(13,86,44,117),(14,85,45,116),(15,84,46,115),(16,83,47,114),(17,82,48,113),(18,81,49,112),(19,80,50,111),(20,79,51,110),(21,78,52,109),(22,77,53,108),(23,76,54,107),(24,75,55,106),(25,74,56,105),(26,73,57,104),(27,72,58,103),(28,71,59,102),(29,70,60,101),(30,69,61,100),(31,68,62,99)]])
Dic31 is a maximal subgroup of
Dic62 C4×D31 C31⋊D4 C31⋊C12 Dic93
Dic31 is a maximal quotient of C31⋊C8 Dic93
34 conjugacy classes
| class | 1 | 2 | 4A | 4B | 31A | ··· | 31O | 62A | ··· | 62O |
| order | 1 | 2 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 |
| size | 1 | 1 | 31 | 31 | 2 | ··· | 2 | 2 | ··· | 2 |
34 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D31 | Dic31 |
| kernel | Dic31 | C62 | C31 | C2 | C1 |
| # reps | 1 | 1 | 2 | 15 | 15 |
Matrix representation of Dic31 ►in GL3(𝔽373) generated by
| 372 | 0 | 0 |
| 0 | 316 | 372 |
| 0 | 1 | 0 |
| 104 | 0 | 0 |
| 0 | 274 | 241 |
| 0 | 289 | 99 |
G:=sub<GL(3,GF(373))| [372,0,0,0,316,1,0,372,0],[104,0,0,0,274,289,0,241,99] >;
Dic31 in GAP, Magma, Sage, TeX
{\rm Dic}_{31} % in TeX
G:=Group("Dic31"); // GroupNames label
G:=SmallGroup(124,1);
// by ID
G=gap.SmallGroup(124,1);
# by ID
G:=PCGroup([3,-2,-2,-31,6,1082]);
// Polycyclic
G:=Group<a,b|a^62=1,b^2=a^31,b*a*b^-1=a^-1>;
// generators/relations
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