direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic3×C13, C3⋊C52, C39⋊5C4, C6.C26, C78.3C2, C26.2S3, C2.(S3×C13), SmallGroup(156,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C13 |
Generators and relations for Dic3×C13
G = < a,b,c | a13=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C13
►Regular action on 156 points
Generators in S156
(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)(118 119 120 121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140 141 142 143)(144 145 146 147 148 149 150 151 152 153 154 155 156)
(1 101 105 133 83 123)(2 102 106 134 84 124)(3 103 107 135 85 125)(4 104 108 136 86 126)(5 92 109 137 87 127)(6 93 110 138 88 128)(7 94 111 139 89 129)(8 95 112 140 90 130)(9 96 113 141 91 118)(10 97 114 142 79 119)(11 98 115 143 80 120)(12 99 116 131 81 121)(13 100 117 132 82 122)(14 59 49 33 77 150)(15 60 50 34 78 151)(16 61 51 35 66 152)(17 62 52 36 67 153)(18 63 40 37 68 154)(19 64 41 38 69 155)(20 65 42 39 70 156)(21 53 43 27 71 144)(22 54 44 28 72 145)(23 55 45 29 73 146)(24 56 46 30 74 147)(25 57 47 31 75 148)(26 58 48 32 76 149)
(1 44 133 145)(2 45 134 146)(3 46 135 147)(4 47 136 148)(5 48 137 149)(6 49 138 150)(7 50 139 151)(8 51 140 152)(9 52 141 153)(10 40 142 154)(11 41 143 155)(12 42 131 156)(13 43 132 144)(14 128 33 110)(15 129 34 111)(16 130 35 112)(17 118 36 113)(18 119 37 114)(19 120 38 115)(20 121 39 116)(21 122 27 117)(22 123 28 105)(23 124 29 106)(24 125 30 107)(25 126 31 108)(26 127 32 109)(53 82 71 100)(54 83 72 101)(55 84 73 102)(56 85 74 103)(57 86 75 104)(58 87 76 92)(59 88 77 93)(60 89 78 94)(61 90 66 95)(62 91 67 96)(63 79 68 97)(64 80 69 98)(65 81 70 99)
G:=sub<Sym(156)| (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)(118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143)(144,145,146,147,148,149,150,151,152,153,154,155,156), (1,101,105,133,83,123)(2,102,106,134,84,124)(3,103,107,135,85,125)(4,104,108,136,86,126)(5,92,109,137,87,127)(6,93,110,138,88,128)(7,94,111,139,89,129)(8,95,112,140,90,130)(9,96,113,141,91,118)(10,97,114,142,79,119)(11,98,115,143,80,120)(12,99,116,131,81,121)(13,100,117,132,82,122)(14,59,49,33,77,150)(15,60,50,34,78,151)(16,61,51,35,66,152)(17,62,52,36,67,153)(18,63,40,37,68,154)(19,64,41,38,69,155)(20,65,42,39,70,156)(21,53,43,27,71,144)(22,54,44,28,72,145)(23,55,45,29,73,146)(24,56,46,30,74,147)(25,57,47,31,75,148)(26,58,48,32,76,149), (1,44,133,145)(2,45,134,146)(3,46,135,147)(4,47,136,148)(5,48,137,149)(6,49,138,150)(7,50,139,151)(8,51,140,152)(9,52,141,153)(10,40,142,154)(11,41,143,155)(12,42,131,156)(13,43,132,144)(14,128,33,110)(15,129,34,111)(16,130,35,112)(17,118,36,113)(18,119,37,114)(19,120,38,115)(20,121,39,116)(21,122,27,117)(22,123,28,105)(23,124,29,106)(24,125,30,107)(25,126,31,108)(26,127,32,109)(53,82,71,100)(54,83,72,101)(55,84,73,102)(56,85,74,103)(57,86,75,104)(58,87,76,92)(59,88,77,93)(60,89,78,94)(61,90,66,95)(62,91,67,96)(63,79,68,97)(64,80,69,98)(65,81,70,99)>;
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)(118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143)(144,145,146,147,148,149,150,151,152,153,154,155,156), (1,101,105,133,83,123)(2,102,106,134,84,124)(3,103,107,135,85,125)(4,104,108,136,86,126)(5,92,109,137,87,127)(6,93,110,138,88,128)(7,94,111,139,89,129)(8,95,112,140,90,130)(9,96,113,141,91,118)(10,97,114,142,79,119)(11,98,115,143,80,120)(12,99,116,131,81,121)(13,100,117,132,82,122)(14,59,49,33,77,150)(15,60,50,34,78,151)(16,61,51,35,66,152)(17,62,52,36,67,153)(18,63,40,37,68,154)(19,64,41,38,69,155)(20,65,42,39,70,156)(21,53,43,27,71,144)(22,54,44,28,72,145)(23,55,45,29,73,146)(24,56,46,30,74,147)(25,57,47,31,75,148)(26,58,48,32,76,149), (1,44,133,145)(2,45,134,146)(3,46,135,147)(4,47,136,148)(5,48,137,149)(6,49,138,150)(7,50,139,151)(8,51,140,152)(9,52,141,153)(10,40,142,154)(11,41,143,155)(12,42,131,156)(13,43,132,144)(14,128,33,110)(15,129,34,111)(16,130,35,112)(17,118,36,113)(18,119,37,114)(19,120,38,115)(20,121,39,116)(21,122,27,117)(22,123,28,105)(23,124,29,106)(24,125,30,107)(25,126,31,108)(26,127,32,109)(53,82,71,100)(54,83,72,101)(55,84,73,102)(56,85,74,103)(57,86,75,104)(58,87,76,92)(59,88,77,93)(60,89,78,94)(61,90,66,95)(62,91,67,96)(63,79,68,97)(64,80,69,98)(65,81,70,99) );
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),(118,119,120,121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140,141,142,143),(144,145,146,147,148,149,150,151,152,153,154,155,156)], [(1,101,105,133,83,123),(2,102,106,134,84,124),(3,103,107,135,85,125),(4,104,108,136,86,126),(5,92,109,137,87,127),(6,93,110,138,88,128),(7,94,111,139,89,129),(8,95,112,140,90,130),(9,96,113,141,91,118),(10,97,114,142,79,119),(11,98,115,143,80,120),(12,99,116,131,81,121),(13,100,117,132,82,122),(14,59,49,33,77,150),(15,60,50,34,78,151),(16,61,51,35,66,152),(17,62,52,36,67,153),(18,63,40,37,68,154),(19,64,41,38,69,155),(20,65,42,39,70,156),(21,53,43,27,71,144),(22,54,44,28,72,145),(23,55,45,29,73,146),(24,56,46,30,74,147),(25,57,47,31,75,148),(26,58,48,32,76,149)], [(1,44,133,145),(2,45,134,146),(3,46,135,147),(4,47,136,148),(5,48,137,149),(6,49,138,150),(7,50,139,151),(8,51,140,152),(9,52,141,153),(10,40,142,154),(11,41,143,155),(12,42,131,156),(13,43,132,144),(14,128,33,110),(15,129,34,111),(16,130,35,112),(17,118,36,113),(18,119,37,114),(19,120,38,115),(20,121,39,116),(21,122,27,117),(22,123,28,105),(23,124,29,106),(24,125,30,107),(25,126,31,108),(26,127,32,109),(53,82,71,100),(54,83,72,101),(55,84,73,102),(56,85,74,103),(57,86,75,104),(58,87,76,92),(59,88,77,93),(60,89,78,94),(61,90,66,95),(62,91,67,96),(63,79,68,97),(64,80,69,98),(65,81,70,99)]])
Dic3×C13 is a maximal subgroup of
D78.C2 C3⋊D52 C39⋊Q8 S3×C52
78 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39L | 52A | ··· | 52X | 78A | ··· | 78L |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 |
| size | 1 | 1 | 2 | 3 | 3 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C13 | C26 | C52 | S3 | Dic3 | S3×C13 | Dic3×C13 |
| kernel | Dic3×C13 | C78 | C39 | Dic3 | C6 | C3 | C26 | C13 | C2 | C1 |
| # reps | 1 | 1 | 2 | 12 | 12 | 24 | 1 | 1 | 12 | 12 |
Matrix representation of Dic3×C13 ►in GL2(𝔽157) generated by
| 101 | 0 |
| 0 | 101 |
| 1 | 156 |
| 1 | 0 |
| 149 | 122 |
| 114 | 8 |
G:=sub<GL(2,GF(157))| [101,0,0,101],[1,1,156,0],[149,114,122,8] >;
Dic3×C13 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{13} % in TeX
G:=Group("Dic3xC13"); // GroupNames label
G:=SmallGroup(156,3);
// by ID
G=gap.SmallGroup(156,3);
# by ID
G:=PCGroup([4,-2,-13,-2,-3,104,1667]);
// Polycyclic
G:=Group<a,b,c|a^13=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export