direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C13×D12, C39⋊6D4, C52⋊3S3, C156⋊5C2, C12⋊1C26, D6⋊1C26, C26.15D6, C78.20C22, C4⋊(S3×C13), C3⋊1(D4×C13), (S3×C26)⋊4C2, C2.4(S3×C26), C6.3(C2×C26), SmallGroup(312,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×D12
G = < a,b,c | a13=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C13×D12
►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 75 53 42 145 37 84 140 99 25 111 118)(2 76 54 43 146 38 85 141 100 26 112 119)(3 77 55 44 147 39 86 142 101 14 113 120)(4 78 56 45 148 27 87 143 102 15 114 121)(5 66 57 46 149 28 88 131 103 16 115 122)(6 67 58 47 150 29 89 132 104 17 116 123)(7 68 59 48 151 30 90 133 92 18 117 124)(8 69 60 49 152 31 91 134 93 19 105 125)(9 70 61 50 153 32 79 135 94 20 106 126)(10 71 62 51 154 33 80 136 95 21 107 127)(11 72 63 52 155 34 81 137 96 22 108 128)(12 73 64 40 156 35 82 138 97 23 109 129)(13 74 65 41 144 36 83 139 98 24 110 130)
(1 118)(2 119)(3 120)(4 121)(5 122)(6 123)(7 124)(8 125)(9 126)(10 127)(11 128)(12 129)(13 130)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 53)(26 54)(27 87)(28 88)(29 89)(30 90)(31 91)(32 79)(33 80)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 92)(49 93)(50 94)(51 95)(52 96)(66 115)(67 116)(68 117)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(131 149)(132 150)(133 151)(134 152)(135 153)(136 154)(137 155)(138 156)(139 144)(140 145)(141 146)(142 147)(143 148)
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), (1,75,53,42,145,37,84,140,99,25,111,118)(2,76,54,43,146,38,85,141,100,26,112,119)(3,77,55,44,147,39,86,142,101,14,113,120)(4,78,56,45,148,27,87,143,102,15,114,121)(5,66,57,46,149,28,88,131,103,16,115,122)(6,67,58,47,150,29,89,132,104,17,116,123)(7,68,59,48,151,30,90,133,92,18,117,124)(8,69,60,49,152,31,91,134,93,19,105,125)(9,70,61,50,153,32,79,135,94,20,106,126)(10,71,62,51,154,33,80,136,95,21,107,127)(11,72,63,52,155,34,81,137,96,22,108,128)(12,73,64,40,156,35,82,138,97,23,109,129)(13,74,65,41,144,36,83,139,98,24,110,130), (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,53)(26,54)(27,87)(28,88)(29,89)(30,90)(31,91)(32,79)(33,80)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,97)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,92)(49,93)(50,94)(51,95)(52,96)(66,115)(67,116)(68,117)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(131,149)(132,150)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,144)(140,145)(141,146)(142,147)(143,148)>;
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), (1,75,53,42,145,37,84,140,99,25,111,118)(2,76,54,43,146,38,85,141,100,26,112,119)(3,77,55,44,147,39,86,142,101,14,113,120)(4,78,56,45,148,27,87,143,102,15,114,121)(5,66,57,46,149,28,88,131,103,16,115,122)(6,67,58,47,150,29,89,132,104,17,116,123)(7,68,59,48,151,30,90,133,92,18,117,124)(8,69,60,49,152,31,91,134,93,19,105,125)(9,70,61,50,153,32,79,135,94,20,106,126)(10,71,62,51,154,33,80,136,95,21,107,127)(11,72,63,52,155,34,81,137,96,22,108,128)(12,73,64,40,156,35,82,138,97,23,109,129)(13,74,65,41,144,36,83,139,98,24,110,130), (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,53)(26,54)(27,87)(28,88)(29,89)(30,90)(31,91)(32,79)(33,80)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,97)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,92)(49,93)(50,94)(51,95)(52,96)(66,115)(67,116)(68,117)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(131,149)(132,150)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,144)(140,145)(141,146)(142,147)(143,148) );
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)], [(1,75,53,42,145,37,84,140,99,25,111,118),(2,76,54,43,146,38,85,141,100,26,112,119),(3,77,55,44,147,39,86,142,101,14,113,120),(4,78,56,45,148,27,87,143,102,15,114,121),(5,66,57,46,149,28,88,131,103,16,115,122),(6,67,58,47,150,29,89,132,104,17,116,123),(7,68,59,48,151,30,90,133,92,18,117,124),(8,69,60,49,152,31,91,134,93,19,105,125),(9,70,61,50,153,32,79,135,94,20,106,126),(10,71,62,51,154,33,80,136,95,21,107,127),(11,72,63,52,155,34,81,137,96,22,108,128),(12,73,64,40,156,35,82,138,97,23,109,129),(13,74,65,41,144,36,83,139,98,24,110,130)], [(1,118),(2,119),(3,120),(4,121),(5,122),(6,123),(7,124),(8,125),(9,126),(10,127),(11,128),(12,129),(13,130),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,65),(25,53),(26,54),(27,87),(28,88),(29,89),(30,90),(31,91),(32,79),(33,80),(34,81),(35,82),(36,83),(37,84),(38,85),(39,86),(40,97),(41,98),(42,99),(43,100),(44,101),(45,102),(46,103),(47,104),(48,92),(49,93),(50,94),(51,95),(52,96),(66,115),(67,116),(68,117),(69,105),(70,106),(71,107),(72,108),(73,109),(74,110),(75,111),(76,112),(77,113),(78,114),(131,149),(132,150),(133,151),(134,152),(135,153),(136,154),(137,155),(138,156),(139,144),(140,145),(141,146),(142,147),(143,148)]])
117 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 12A | 12B | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26AJ | 39A | ··· | 39L | 52A | ··· | 52L | 78A | ··· | 78L | 156A | ··· | 156X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | S3 | D4 | D6 | D12 | S3×C13 | D4×C13 | S3×C26 | C13×D12 |
| kernel | C13×D12 | C156 | S3×C26 | D12 | C12 | D6 | C52 | C39 | C26 | C13 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 12 | 12 | 24 | 1 | 1 | 1 | 2 | 12 | 12 | 12 | 24 |
Matrix representation of C13×D12 ►in GL4(𝔽157) generated by
| 39 | 0 | 0 | 0 |
| 0 | 39 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 156 | 156 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 24 | 133 |
| 0 | 0 | 24 | 48 |
| 156 | 156 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 24 | 133 |
| 0 | 0 | 109 | 133 |
G:=sub<GL(4,GF(157))| [39,0,0,0,0,39,0,0,0,0,1,0,0,0,0,1],[156,1,0,0,156,0,0,0,0,0,24,24,0,0,133,48],[156,0,0,0,156,1,0,0,0,0,24,109,0,0,133,133] >;
C13×D12 in GAP, Magma, Sage, TeX
C_{13}\times D_{12} % in TeX
G:=Group("C13xD12"); // GroupNames label
G:=SmallGroup(312,34);
// by ID
G=gap.SmallGroup(312,34);
# by ID
G:=PCGroup([5,-2,-2,-13,-2,-3,541,266,5204]);
// Polycyclic
G:=Group<a,b,c|a^13=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export