direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C6×D13, C39⋊3C23, C78⋊3C22, C26⋊3(C2×C6), (C2×C78)⋊5C2, (C2×C26)⋊11C6, C13⋊3(C22×C6), SmallGroup(312,58)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×C6×D13 |
Generators and relations for C2×C6×D13
G = < a,b,c,d | a2=b6=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 328 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C6, C6, C23, C2×C6, C2×C6, C13, C22×C6, D13, C26, C39, D26, C2×C26, C3×D13, C78, C22×D13, C6×D13, C2×C78, C2×C6×D13
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C22×C6, D13, D26, C3×D13, C22×D13, C6×D13, C2×C6×D13
►On 156 points
Generators in S156
(1 127)(2 128)(3 129)(4 130)(5 118)(6 119)(7 120)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 137)(15 138)(16 139)(17 140)(18 141)(19 142)(20 143)(21 131)(22 132)(23 133)(24 134)(25 135)(26 136)(27 146)(28 147)(29 148)(30 149)(31 150)(32 151)(33 152)(34 153)(35 154)(36 155)(37 156)(38 144)(39 145)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 79)(53 97)(54 98)(55 99)(56 100)(57 101)(58 102)(59 103)(60 104)(61 92)(62 93)(63 94)(64 95)(65 96)(66 117)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 111)(74 112)(75 113)(76 114)(77 115)(78 116)
(1 59 31 51 15 67)(2 60 32 52 16 68)(3 61 33 40 17 69)(4 62 34 41 18 70)(5 63 35 42 19 71)(6 64 36 43 20 72)(7 65 37 44 21 73)(8 53 38 45 22 74)(9 54 39 46 23 75)(10 55 27 47 24 76)(11 56 28 48 25 77)(12 57 29 49 26 78)(13 58 30 50 14 66)(79 139 106 128 104 151)(80 140 107 129 92 152)(81 141 108 130 93 153)(82 142 109 118 94 154)(83 143 110 119 95 155)(84 131 111 120 96 156)(85 132 112 121 97 144)(86 133 113 122 98 145)(87 134 114 123 99 146)(88 135 115 124 100 147)(89 136 116 125 101 148)(90 137 117 126 102 149)(91 138 105 127 103 150)
(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 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 52)(13 51)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 65)(22 64)(23 63)(24 62)(25 61)(26 60)(27 70)(28 69)(29 68)(30 67)(31 66)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(79 125)(80 124)(81 123)(82 122)(83 121)(84 120)(85 119)(86 118)(87 130)(88 129)(89 128)(90 127)(91 126)(92 135)(93 134)(94 133)(95 132)(96 131)(97 143)(98 142)(99 141)(100 140)(101 139)(102 138)(103 137)(104 136)(105 149)(106 148)(107 147)(108 146)(109 145)(110 144)(111 156)(112 155)(113 154)(114 153)(115 152)(116 151)(117 150)
G:=sub<Sym(156)| (1,127)(2,128)(3,129)(4,130)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,137)(15,138)(16,139)(17,140)(18,141)(19,142)(20,143)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(27,146)(28,147)(29,148)(30,149)(31,150)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,144)(39,145)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,79)(53,97)(54,98)(55,99)(56,100)(57,101)(58,102)(59,103)(60,104)(61,92)(62,93)(63,94)(64,95)(65,96)(66,117)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116), (1,59,31,51,15,67)(2,60,32,52,16,68)(3,61,33,40,17,69)(4,62,34,41,18,70)(5,63,35,42,19,71)(6,64,36,43,20,72)(7,65,37,44,21,73)(8,53,38,45,22,74)(9,54,39,46,23,75)(10,55,27,47,24,76)(11,56,28,48,25,77)(12,57,29,49,26,78)(13,58,30,50,14,66)(79,139,106,128,104,151)(80,140,107,129,92,152)(81,141,108,130,93,153)(82,142,109,118,94,154)(83,143,110,119,95,155)(84,131,111,120,96,156)(85,132,112,121,97,144)(86,133,113,122,98,145)(87,134,114,123,99,146)(88,135,115,124,100,147)(89,136,116,125,101,148)(90,137,117,126,102,149)(91,138,105,127,103,150), (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), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,52)(13,51)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,65)(22,64)(23,63)(24,62)(25,61)(26,60)(27,70)(28,69)(29,68)(30,67)(31,66)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(79,125)(80,124)(81,123)(82,122)(83,121)(84,120)(85,119)(86,118)(87,130)(88,129)(89,128)(90,127)(91,126)(92,135)(93,134)(94,133)(95,132)(96,131)(97,143)(98,142)(99,141)(100,140)(101,139)(102,138)(103,137)(104,136)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)>;
G:=Group( (1,127)(2,128)(3,129)(4,130)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,137)(15,138)(16,139)(17,140)(18,141)(19,142)(20,143)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(27,146)(28,147)(29,148)(30,149)(31,150)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,144)(39,145)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,79)(53,97)(54,98)(55,99)(56,100)(57,101)(58,102)(59,103)(60,104)(61,92)(62,93)(63,94)(64,95)(65,96)(66,117)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116), (1,59,31,51,15,67)(2,60,32,52,16,68)(3,61,33,40,17,69)(4,62,34,41,18,70)(5,63,35,42,19,71)(6,64,36,43,20,72)(7,65,37,44,21,73)(8,53,38,45,22,74)(9,54,39,46,23,75)(10,55,27,47,24,76)(11,56,28,48,25,77)(12,57,29,49,26,78)(13,58,30,50,14,66)(79,139,106,128,104,151)(80,140,107,129,92,152)(81,141,108,130,93,153)(82,142,109,118,94,154)(83,143,110,119,95,155)(84,131,111,120,96,156)(85,132,112,121,97,144)(86,133,113,122,98,145)(87,134,114,123,99,146)(88,135,115,124,100,147)(89,136,116,125,101,148)(90,137,117,126,102,149)(91,138,105,127,103,150), (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), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,52)(13,51)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,65)(22,64)(23,63)(24,62)(25,61)(26,60)(27,70)(28,69)(29,68)(30,67)(31,66)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(79,125)(80,124)(81,123)(82,122)(83,121)(84,120)(85,119)(86,118)(87,130)(88,129)(89,128)(90,127)(91,126)(92,135)(93,134)(94,133)(95,132)(96,131)(97,143)(98,142)(99,141)(100,140)(101,139)(102,138)(103,137)(104,136)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150) );
G=PermutationGroup([[(1,127),(2,128),(3,129),(4,130),(5,118),(6,119),(7,120),(8,121),(9,122),(10,123),(11,124),(12,125),(13,126),(14,137),(15,138),(16,139),(17,140),(18,141),(19,142),(20,143),(21,131),(22,132),(23,133),(24,134),(25,135),(26,136),(27,146),(28,147),(29,148),(30,149),(31,150),(32,151),(33,152),(34,153),(35,154),(36,155),(37,156),(38,144),(39,145),(40,80),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(50,90),(51,91),(52,79),(53,97),(54,98),(55,99),(56,100),(57,101),(58,102),(59,103),(60,104),(61,92),(62,93),(63,94),(64,95),(65,96),(66,117),(67,105),(68,106),(69,107),(70,108),(71,109),(72,110),(73,111),(74,112),(75,113),(76,114),(77,115),(78,116)], [(1,59,31,51,15,67),(2,60,32,52,16,68),(3,61,33,40,17,69),(4,62,34,41,18,70),(5,63,35,42,19,71),(6,64,36,43,20,72),(7,65,37,44,21,73),(8,53,38,45,22,74),(9,54,39,46,23,75),(10,55,27,47,24,76),(11,56,28,48,25,77),(12,57,29,49,26,78),(13,58,30,50,14,66),(79,139,106,128,104,151),(80,140,107,129,92,152),(81,141,108,130,93,153),(82,142,109,118,94,154),(83,143,110,119,95,155),(84,131,111,120,96,156),(85,132,112,121,97,144),(86,133,113,122,98,145),(87,134,114,123,99,146),(88,135,115,124,100,147),(89,136,116,125,101,148),(90,137,117,126,102,149),(91,138,105,127,103,150)], [(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)], [(1,50),(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,40),(12,52),(13,51),(14,59),(15,58),(16,57),(17,56),(18,55),(19,54),(20,53),(21,65),(22,64),(23,63),(24,62),(25,61),(26,60),(27,70),(28,69),(29,68),(30,67),(31,66),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(79,125),(80,124),(81,123),(82,122),(83,121),(84,120),(85,119),(86,118),(87,130),(88,129),(89,128),(90,127),(91,126),(92,135),(93,134),(94,133),(95,132),(96,131),(97,143),(98,142),(99,141),(100,140),(101,139),(102,138),(103,137),(104,136),(105,149),(106,148),(107,147),(108,146),(109,145),(110,144),(111,156),(112,155),(113,154),(114,153),(115,152),(116,151),(117,150)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 6A | ··· | 6F | 6G | ··· | 6N | 13A | ··· | 13F | 26A | ··· | 26R | 39A | ··· | 39L | 78A | ··· | 78AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 1 | 1 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D13 | D26 | C3×D13 | C6×D13 |
| kernel | C2×C6×D13 | C6×D13 | C2×C78 | C22×D13 | D26 | C2×C26 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 6 | 1 | 2 | 12 | 2 | 6 | 18 | 12 | 36 |
Matrix representation of C2×C6×D13 ►in GL4(𝔽79) generated by
| 1 | 0 | 0 | 0 |
| 0 | 78 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 78 | 0 | 0 | 0 |
| 0 | 55 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 45 | 1 |
| 0 | 0 | 61 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 78 | 0 | 0 |
| 0 | 0 | 40 | 78 |
| 0 | 0 | 19 | 39 |
G:=sub<GL(4,GF(79))| [1,0,0,0,0,78,0,0,0,0,1,0,0,0,0,1],[78,0,0,0,0,55,0,0,0,0,23,0,0,0,0,23],[1,0,0,0,0,1,0,0,0,0,45,61,0,0,1,40],[1,0,0,0,0,78,0,0,0,0,40,19,0,0,78,39] >;
C2×C6×D13 in GAP, Magma, Sage, TeX
C_2\times C_6\times D_{13} % in TeX
G:=Group("C2xC6xD13"); // GroupNames label
G:=SmallGroup(312,58);
// by ID
G=gap.SmallGroup(312,58);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,7204]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^13=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations