direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C12×D13, C52⋊6C6, C156⋊4C2, D26.4C6, C6.14D26, Dic13⋊5C6, C78.14C22, C39⋊8(C2×C4), C13⋊6(C2×C12), C2.1(C6×D13), C26.10(C2×C6), (C6×D13).4C2, (C3×Dic13)⋊5C2, SmallGroup(312,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C12×D13 |
Generators and relations for C12×D13
G = < a,b,c | a12=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C12×D13
►On 156 points
Generators in S156
(1 149 46 82 55 127 21 137 35 95 66 105)(2 150 47 83 56 128 22 138 36 96 67 106)(3 151 48 84 57 129 23 139 37 97 68 107)(4 152 49 85 58 130 24 140 38 98 69 108)(5 153 50 86 59 118 25 141 39 99 70 109)(6 154 51 87 60 119 26 142 27 100 71 110)(7 155 52 88 61 120 14 143 28 101 72 111)(8 156 40 89 62 121 15 131 29 102 73 112)(9 144 41 90 63 122 16 132 30 103 74 113)(10 145 42 91 64 123 17 133 31 104 75 114)(11 146 43 79 65 124 18 134 32 92 76 115)(12 147 44 80 53 125 19 135 33 93 77 116)(13 148 45 81 54 126 20 136 34 94 78 117)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(27 29)(30 39)(31 38)(32 37)(33 36)(34 35)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(53 56)(54 55)(57 65)(58 64)(59 63)(60 62)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(79 84)(80 83)(81 82)(85 91)(86 90)(87 89)(92 97)(93 96)(94 95)(98 104)(99 103)(100 102)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(118 122)(119 121)(123 130)(124 129)(125 128)(126 127)(131 142)(132 141)(133 140)(134 139)(135 138)(136 137)(144 153)(145 152)(146 151)(147 150)(148 149)(154 156)
G:=sub<Sym(156)| (1,149,46,82,55,127,21,137,35,95,66,105)(2,150,47,83,56,128,22,138,36,96,67,106)(3,151,48,84,57,129,23,139,37,97,68,107)(4,152,49,85,58,130,24,140,38,98,69,108)(5,153,50,86,59,118,25,141,39,99,70,109)(6,154,51,87,60,119,26,142,27,100,71,110)(7,155,52,88,61,120,14,143,28,101,72,111)(8,156,40,89,62,121,15,131,29,102,73,112)(9,144,41,90,63,122,16,132,30,103,74,113)(10,145,42,91,64,123,17,133,31,104,75,114)(11,146,43,79,65,124,18,134,32,92,76,115)(12,147,44,80,53,125,19,135,33,93,77,116)(13,148,45,81,54,126,20,136,34,94,78,117), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,29)(30,39)(31,38)(32,37)(33,36)(34,35)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(53,56)(54,55)(57,65)(58,64)(59,63)(60,62)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(79,84)(80,83)(81,82)(85,91)(86,90)(87,89)(92,97)(93,96)(94,95)(98,104)(99,103)(100,102)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(118,122)(119,121)(123,130)(124,129)(125,128)(126,127)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)(144,153)(145,152)(146,151)(147,150)(148,149)(154,156)>;
G:=Group( (1,149,46,82,55,127,21,137,35,95,66,105)(2,150,47,83,56,128,22,138,36,96,67,106)(3,151,48,84,57,129,23,139,37,97,68,107)(4,152,49,85,58,130,24,140,38,98,69,108)(5,153,50,86,59,118,25,141,39,99,70,109)(6,154,51,87,60,119,26,142,27,100,71,110)(7,155,52,88,61,120,14,143,28,101,72,111)(8,156,40,89,62,121,15,131,29,102,73,112)(9,144,41,90,63,122,16,132,30,103,74,113)(10,145,42,91,64,123,17,133,31,104,75,114)(11,146,43,79,65,124,18,134,32,92,76,115)(12,147,44,80,53,125,19,135,33,93,77,116)(13,148,45,81,54,126,20,136,34,94,78,117), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,29)(30,39)(31,38)(32,37)(33,36)(34,35)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(53,56)(54,55)(57,65)(58,64)(59,63)(60,62)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(79,84)(80,83)(81,82)(85,91)(86,90)(87,89)(92,97)(93,96)(94,95)(98,104)(99,103)(100,102)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(118,122)(119,121)(123,130)(124,129)(125,128)(126,127)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)(144,153)(145,152)(146,151)(147,150)(148,149)(154,156) );
G=PermutationGroup([[(1,149,46,82,55,127,21,137,35,95,66,105),(2,150,47,83,56,128,22,138,36,96,67,106),(3,151,48,84,57,129,23,139,37,97,68,107),(4,152,49,85,58,130,24,140,38,98,69,108),(5,153,50,86,59,118,25,141,39,99,70,109),(6,154,51,87,60,119,26,142,27,100,71,110),(7,155,52,88,61,120,14,143,28,101,72,111),(8,156,40,89,62,121,15,131,29,102,73,112),(9,144,41,90,63,122,16,132,30,103,74,113),(10,145,42,91,64,123,17,133,31,104,75,114),(11,146,43,79,65,124,18,134,32,92,76,115),(12,147,44,80,53,125,19,135,33,93,77,116),(13,148,45,81,54,126,20,136,34,94,78,117)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(27,29),(30,39),(31,38),(32,37),(33,36),(34,35),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(53,56),(54,55),(57,65),(58,64),(59,63),(60,62),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(79,84),(80,83),(81,82),(85,91),(86,90),(87,89),(92,97),(93,96),(94,95),(98,104),(99,103),(100,102),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112),(118,122),(119,121),(123,130),(124,129),(125,128),(126,127),(131,142),(132,141),(133,140),(134,139),(135,138),(136,137),(144,153),(145,152),(146,151),(147,150),(148,149),(154,156)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39L | 52A | ··· | 52L | 78A | ··· | 78L | 156A | ··· | 156X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 1 | 1 | 13 | 13 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D13 | D26 | C3×D13 | C4×D13 | C6×D13 | C12×D13 |
| kernel | C12×D13 | C3×Dic13 | C156 | C6×D13 | C4×D13 | C3×D13 | Dic13 | C52 | D26 | D13 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 6 | 6 | 12 | 12 | 12 | 24 |
Matrix representation of C12×D13 ►in GL2(𝔽157) generated by
| 22 | 0 |
| 0 | 22 |
| 124 | 1 |
| 156 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(157))| [22,0,0,22],[124,156,1,0],[0,1,1,0] >;
C12×D13 in GAP, Magma, Sage, TeX
C_{12}\times D_{13} % in TeX
G:=Group("C12xD13"); // GroupNames label
G:=SmallGroup(312,28);
// by ID
G=gap.SmallGroup(312,28);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-13,66,7204]);
// Polycyclic
G:=Group<a,b,c|a^12=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export