metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C39⋊3D4, C13⋊2D12, D6⋊2D13, D78⋊4C2, Dic13⋊S3, C6.6D26, C26.6D6, C78.6C22, (S3×C26)⋊2C2, C3⋊1(C13⋊D4), C2.6(S3×D13), (C3×Dic13)⋊3C2, SmallGroup(312,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13⋊D12
G = < a,b,c | a13=b12=c2=1, bab-1=cac=a-1, 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)
(1 117 70 126 20 102 51 144 28 89 56 134)(2 116 71 125 21 101 52 156 29 88 57 133)(3 115 72 124 22 100 40 155 30 87 58 132)(4 114 73 123 23 99 41 154 31 86 59 131)(5 113 74 122 24 98 42 153 32 85 60 143)(6 112 75 121 25 97 43 152 33 84 61 142)(7 111 76 120 26 96 44 151 34 83 62 141)(8 110 77 119 14 95 45 150 35 82 63 140)(9 109 78 118 15 94 46 149 36 81 64 139)(10 108 66 130 16 93 47 148 37 80 65 138)(11 107 67 129 17 92 48 147 38 79 53 137)(12 106 68 128 18 104 49 146 39 91 54 136)(13 105 69 127 19 103 50 145 27 90 55 135)
(1 28)(2 27)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(40 68)(41 67)(42 66)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 59)(54 58)(55 57)(60 65)(61 64)(62 63)(79 131)(80 143)(81 142)(82 141)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 123)(93 122)(94 121)(95 120)(96 119)(97 118)(98 130)(99 129)(100 128)(101 127)(102 126)(103 125)(104 124)(105 156)(106 155)(107 154)(108 153)(109 152)(110 151)(111 150)(112 149)(113 148)(114 147)(115 146)(116 145)(117 144)
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), (1,117,70,126,20,102,51,144,28,89,56,134)(2,116,71,125,21,101,52,156,29,88,57,133)(3,115,72,124,22,100,40,155,30,87,58,132)(4,114,73,123,23,99,41,154,31,86,59,131)(5,113,74,122,24,98,42,153,32,85,60,143)(6,112,75,121,25,97,43,152,33,84,61,142)(7,111,76,120,26,96,44,151,34,83,62,141)(8,110,77,119,14,95,45,150,35,82,63,140)(9,109,78,118,15,94,46,149,36,81,64,139)(10,108,66,130,16,93,47,148,37,80,65,138)(11,107,67,129,17,92,48,147,38,79,53,137)(12,106,68,128,18,104,49,146,39,91,54,136)(13,105,69,127,19,103,50,145,27,90,55,135), (1,28)(2,27)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,68)(41,67)(42,66)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,59)(54,58)(55,57)(60,65)(61,64)(62,63)(79,131)(80,143)(81,142)(82,141)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,123)(93,122)(94,121)(95,120)(96,119)(97,118)(98,130)(99,129)(100,128)(101,127)(102,126)(103,125)(104,124)(105,156)(106,155)(107,154)(108,153)(109,152)(110,151)(111,150)(112,149)(113,148)(114,147)(115,146)(116,145)(117,144)>;
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), (1,117,70,126,20,102,51,144,28,89,56,134)(2,116,71,125,21,101,52,156,29,88,57,133)(3,115,72,124,22,100,40,155,30,87,58,132)(4,114,73,123,23,99,41,154,31,86,59,131)(5,113,74,122,24,98,42,153,32,85,60,143)(6,112,75,121,25,97,43,152,33,84,61,142)(7,111,76,120,26,96,44,151,34,83,62,141)(8,110,77,119,14,95,45,150,35,82,63,140)(9,109,78,118,15,94,46,149,36,81,64,139)(10,108,66,130,16,93,47,148,37,80,65,138)(11,107,67,129,17,92,48,147,38,79,53,137)(12,106,68,128,18,104,49,146,39,91,54,136)(13,105,69,127,19,103,50,145,27,90,55,135), (1,28)(2,27)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,68)(41,67)(42,66)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,59)(54,58)(55,57)(60,65)(61,64)(62,63)(79,131)(80,143)(81,142)(82,141)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,123)(93,122)(94,121)(95,120)(96,119)(97,118)(98,130)(99,129)(100,128)(101,127)(102,126)(103,125)(104,124)(105,156)(106,155)(107,154)(108,153)(109,152)(110,151)(111,150)(112,149)(113,148)(114,147)(115,146)(116,145)(117,144) );
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)], [(1,117,70,126,20,102,51,144,28,89,56,134),(2,116,71,125,21,101,52,156,29,88,57,133),(3,115,72,124,22,100,40,155,30,87,58,132),(4,114,73,123,23,99,41,154,31,86,59,131),(5,113,74,122,24,98,42,153,32,85,60,143),(6,112,75,121,25,97,43,152,33,84,61,142),(7,111,76,120,26,96,44,151,34,83,62,141),(8,110,77,119,14,95,45,150,35,82,63,140),(9,109,78,118,15,94,46,149,36,81,64,139),(10,108,66,130,16,93,47,148,37,80,65,138),(11,107,67,129,17,92,48,147,38,79,53,137),(12,106,68,128,18,104,49,146,39,91,54,136),(13,105,69,127,19,103,50,145,27,90,55,135)], [(1,28),(2,27),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21),(40,68),(41,67),(42,66),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,59),(54,58),(55,57),(60,65),(61,64),(62,63),(79,131),(80,143),(81,142),(82,141),(83,140),(84,139),(85,138),(86,137),(87,136),(88,135),(89,134),(90,133),(91,132),(92,123),(93,122),(94,121),(95,120),(96,119),(97,118),(98,130),(99,129),(100,128),(101,127),(102,126),(103,125),(104,124),(105,156),(106,155),(107,154),(108,153),(109,152),(110,151),(111,150),(112,149),(113,148),(114,147),(115,146),(116,145),(117,144)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 12A | 12B | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26R | 39A | ··· | 39F | 78A | ··· | 78F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 6 | 78 | 2 | 26 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D12 | D13 | D26 | C13⋊D4 | S3×D13 | C13⋊D12 |
| kernel | C13⋊D12 | C3×Dic13 | S3×C26 | D78 | Dic13 | C39 | C26 | C13 | D6 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 6 | 6 |
Matrix representation of C13⋊D12 ►in GL4(𝔽157) generated by
| 0 | 1 | 0 | 0 |
| 156 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 114 | 146 | 0 | 0 |
| 54 | 43 | 0 | 0 |
| 0 | 0 | 2 | 67 |
| 0 | 0 | 89 | 156 |
| 1 | 0 | 0 | 0 |
| 35 | 156 | 0 | 0 |
| 0 | 0 | 155 | 90 |
| 0 | 0 | 68 | 2 |
G:=sub<GL(4,GF(157))| [0,156,0,0,1,35,0,0,0,0,1,0,0,0,0,1],[114,54,0,0,146,43,0,0,0,0,2,89,0,0,67,156],[1,35,0,0,0,156,0,0,0,0,155,68,0,0,90,2] >;
C13⋊D12 in GAP, Magma, Sage, TeX
C_{13}\rtimes D_{12} % in TeX
G:=Group("C13:D12"); // GroupNames label
G:=SmallGroup(312,20);
// by ID
G=gap.SmallGroup(312,20);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,20,61,168,7204]);
// Polycyclic
G:=Group<a,b,c|a^13=b^12=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export