direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×Dic13, D6.D13, C6.2D26, C26.2D6, Dic39⋊3C2, C78.2C22, C13⋊4(C4×S3), C39⋊5(C2×C4), (S3×C26).C2, (S3×C13)⋊2C4, C2.2(S3×D13), C3⋊1(C2×Dic13), (C3×Dic13)⋊1C2, SmallGroup(312,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — S3×Dic13 |
Generators and relations for S3×Dic13
G = < a,b,c,d | a3=b2=c26=1, d2=c13, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of S3×Dic13
►On 156 points
Generators in S156
(1 123 76)(2 124 77)(3 125 78)(4 126 53)(5 127 54)(6 128 55)(7 129 56)(8 130 57)(9 105 58)(10 106 59)(11 107 60)(12 108 61)(13 109 62)(14 110 63)(15 111 64)(16 112 65)(17 113 66)(18 114 67)(19 115 68)(20 116 69)(21 117 70)(22 118 71)(23 119 72)(24 120 73)(25 121 74)(26 122 75)(27 102 131)(28 103 132)(29 104 133)(30 79 134)(31 80 135)(32 81 136)(33 82 137)(34 83 138)(35 84 139)(36 85 140)(37 86 141)(38 87 142)(39 88 143)(40 89 144)(41 90 145)(42 91 146)(43 92 147)(44 93 148)(45 94 149)(46 95 150)(47 96 151)(48 97 152)(49 98 153)(50 99 154)(51 100 155)(52 101 156)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 144)(28 145)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 155)(39 156)(40 131)(41 132)(42 133)(43 134)(44 135)(45 136)(46 137)(47 138)(48 139)(49 140)(50 141)(51 142)(52 143)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)
(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 89 14 102)(2 88 15 101)(3 87 16 100)(4 86 17 99)(5 85 18 98)(6 84 19 97)(7 83 20 96)(8 82 21 95)(9 81 22 94)(10 80 23 93)(11 79 24 92)(12 104 25 91)(13 103 26 90)(27 76 40 63)(28 75 41 62)(29 74 42 61)(30 73 43 60)(31 72 44 59)(32 71 45 58)(33 70 46 57)(34 69 47 56)(35 68 48 55)(36 67 49 54)(37 66 50 53)(38 65 51 78)(39 64 52 77)(105 136 118 149)(106 135 119 148)(107 134 120 147)(108 133 121 146)(109 132 122 145)(110 131 123 144)(111 156 124 143)(112 155 125 142)(113 154 126 141)(114 153 127 140)(115 152 128 139)(116 151 129 138)(117 150 130 137)
G:=sub<Sym(156)| (1,123,76)(2,124,77)(3,125,78)(4,126,53)(5,127,54)(6,128,55)(7,129,56)(8,130,57)(9,105,58)(10,106,59)(11,107,60)(12,108,61)(13,109,62)(14,110,63)(15,111,64)(16,112,65)(17,113,66)(18,114,67)(19,115,68)(20,116,69)(21,117,70)(22,118,71)(23,119,72)(24,120,73)(25,121,74)(26,122,75)(27,102,131)(28,103,132)(29,104,133)(30,79,134)(31,80,135)(32,81,136)(33,82,137)(34,83,138)(35,84,139)(36,85,140)(37,86,141)(38,87,142)(39,88,143)(40,89,144)(41,90,145)(42,91,146)(43,92,147)(44,93,148)(45,94,149)(46,95,150)(47,96,151)(48,97,152)(49,98,153)(50,99,154)(51,100,155)(52,101,156), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,144)(28,145)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,131)(41,132)(42,133)(43,134)(44,135)(45,136)(46,137)(47,138)(48,139)(49,140)(50,141)(51,142)(52,143)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104), (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), (1,89,14,102)(2,88,15,101)(3,87,16,100)(4,86,17,99)(5,85,18,98)(6,84,19,97)(7,83,20,96)(8,82,21,95)(9,81,22,94)(10,80,23,93)(11,79,24,92)(12,104,25,91)(13,103,26,90)(27,76,40,63)(28,75,41,62)(29,74,42,61)(30,73,43,60)(31,72,44,59)(32,71,45,58)(33,70,46,57)(34,69,47,56)(35,68,48,55)(36,67,49,54)(37,66,50,53)(38,65,51,78)(39,64,52,77)(105,136,118,149)(106,135,119,148)(107,134,120,147)(108,133,121,146)(109,132,122,145)(110,131,123,144)(111,156,124,143)(112,155,125,142)(113,154,126,141)(114,153,127,140)(115,152,128,139)(116,151,129,138)(117,150,130,137)>;
G:=Group( (1,123,76)(2,124,77)(3,125,78)(4,126,53)(5,127,54)(6,128,55)(7,129,56)(8,130,57)(9,105,58)(10,106,59)(11,107,60)(12,108,61)(13,109,62)(14,110,63)(15,111,64)(16,112,65)(17,113,66)(18,114,67)(19,115,68)(20,116,69)(21,117,70)(22,118,71)(23,119,72)(24,120,73)(25,121,74)(26,122,75)(27,102,131)(28,103,132)(29,104,133)(30,79,134)(31,80,135)(32,81,136)(33,82,137)(34,83,138)(35,84,139)(36,85,140)(37,86,141)(38,87,142)(39,88,143)(40,89,144)(41,90,145)(42,91,146)(43,92,147)(44,93,148)(45,94,149)(46,95,150)(47,96,151)(48,97,152)(49,98,153)(50,99,154)(51,100,155)(52,101,156), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,144)(28,145)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,131)(41,132)(42,133)(43,134)(44,135)(45,136)(46,137)(47,138)(48,139)(49,140)(50,141)(51,142)(52,143)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104), (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), (1,89,14,102)(2,88,15,101)(3,87,16,100)(4,86,17,99)(5,85,18,98)(6,84,19,97)(7,83,20,96)(8,82,21,95)(9,81,22,94)(10,80,23,93)(11,79,24,92)(12,104,25,91)(13,103,26,90)(27,76,40,63)(28,75,41,62)(29,74,42,61)(30,73,43,60)(31,72,44,59)(32,71,45,58)(33,70,46,57)(34,69,47,56)(35,68,48,55)(36,67,49,54)(37,66,50,53)(38,65,51,78)(39,64,52,77)(105,136,118,149)(106,135,119,148)(107,134,120,147)(108,133,121,146)(109,132,122,145)(110,131,123,144)(111,156,124,143)(112,155,125,142)(113,154,126,141)(114,153,127,140)(115,152,128,139)(116,151,129,138)(117,150,130,137) );
G=PermutationGroup([[(1,123,76),(2,124,77),(3,125,78),(4,126,53),(5,127,54),(6,128,55),(7,129,56),(8,130,57),(9,105,58),(10,106,59),(11,107,60),(12,108,61),(13,109,62),(14,110,63),(15,111,64),(16,112,65),(17,113,66),(18,114,67),(19,115,68),(20,116,69),(21,117,70),(22,118,71),(23,119,72),(24,120,73),(25,121,74),(26,122,75),(27,102,131),(28,103,132),(29,104,133),(30,79,134),(31,80,135),(32,81,136),(33,82,137),(34,83,138),(35,84,139),(36,85,140),(37,86,141),(38,87,142),(39,88,143),(40,89,144),(41,90,145),(42,91,146),(43,92,147),(44,93,148),(45,94,149),(46,95,150),(47,96,151),(48,97,152),(49,98,153),(50,99,154),(51,100,155),(52,101,156)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(27,144),(28,145),(29,146),(30,147),(31,148),(32,149),(33,150),(34,151),(35,152),(36,153),(37,154),(38,155),(39,156),(40,131),(41,132),(42,133),(43,134),(44,135),(45,136),(46,137),(47,138),(48,139),(49,140),(50,141),(51,142),(52,143),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(61,121),(62,122),(63,123),(64,124),(65,125),(66,126),(67,127),(68,128),(69,129),(70,130),(71,105),(72,106),(73,107),(74,108),(75,109),(76,110),(77,111),(78,112),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(85,98),(86,99),(87,100),(88,101),(89,102),(90,103),(91,104)], [(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)], [(1,89,14,102),(2,88,15,101),(3,87,16,100),(4,86,17,99),(5,85,18,98),(6,84,19,97),(7,83,20,96),(8,82,21,95),(9,81,22,94),(10,80,23,93),(11,79,24,92),(12,104,25,91),(13,103,26,90),(27,76,40,63),(28,75,41,62),(29,74,42,61),(30,73,43,60),(31,72,44,59),(32,71,45,58),(33,70,46,57),(34,69,47,56),(35,68,48,55),(36,67,49,54),(37,66,50,53),(38,65,51,78),(39,64,52,77),(105,136,118,149),(106,135,119,148),(107,134,120,147),(108,133,121,146),(109,132,122,145),(110,131,123,144),(111,156,124,143),(112,155,125,142),(113,154,126,141),(114,153,127,140),(115,152,128,139),(116,151,129,138),(117,150,130,137)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26R | 39A | ··· | 39F | 78A | ··· | 78F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 3 | 3 | 2 | 13 | 13 | 39 | 39 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | C4×S3 | D13 | Dic13 | D26 | S3×D13 | S3×Dic13 |
| kernel | S3×Dic13 | C3×Dic13 | Dic39 | S3×C26 | S3×C13 | Dic13 | C26 | C13 | D6 | S3 | C6 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 6 | 12 | 6 | 6 | 6 |
Matrix representation of S3×Dic13 ►in GL4(𝔽157) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 156 |
| 0 | 0 | 1 | 156 |
| 156 | 0 | 0 | 0 |
| 0 | 156 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 156 | 0 | 0 |
| 1 | 42 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 140 | 54 | 0 | 0 |
| 140 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(157))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,156,156],[156,0,0,0,0,156,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,156,42,0,0,0,0,1,0,0,0,0,1],[140,140,0,0,54,17,0,0,0,0,1,0,0,0,0,1] >;
S3×Dic13 in GAP, Magma, Sage, TeX
S_3\times {\rm Dic}_{13} % in TeX
G:=Group("S3xDic13"); // GroupNames label
G:=SmallGroup(312,16);
// by ID
G=gap.SmallGroup(312,16);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,20,168,7204]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^26=1,d^2=c^13,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export