direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3xC78, C6:C78, C78:7C6, C3:(C2xC78), (C3xC6):1C26, (C3xC78):4C2, C39:9(C2xC6), (C3xC39):9C22, C32:2(C2xC26), SmallGroup(468,51)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3xC78 |
Generators and relations for S3xC78
G = < a,b,c | a78=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C3, C22, S3, C6, D6, C2xC6, C13, C3xS3, C26, S3xC6, C39, C2xC26, S3xC13, C78, S3xC26, C2xC78, S3xC39, S3xC78
Smallest permutation representation of S3xC78
►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 27 53)(2 28 54)(3 29 55)(4 30 56)(5 31 57)(6 32 58)(7 33 59)(8 34 60)(9 35 61)(10 36 62)(11 37 63)(12 38 64)(13 39 65)(14 40 66)(15 41 67)(16 42 68)(17 43 69)(18 44 70)(19 45 71)(20 46 72)(21 47 73)(22 48 74)(23 49 75)(24 50 76)(25 51 77)(26 52 78)(79 131 105)(80 132 106)(81 133 107)(82 134 108)(83 135 109)(84 136 110)(85 137 111)(86 138 112)(87 139 113)(88 140 114)(89 141 115)(90 142 116)(91 143 117)(92 144 118)(93 145 119)(94 146 120)(95 147 121)(96 148 122)(97 149 123)(98 150 124)(99 151 125)(100 152 126)(101 153 127)(102 154 128)(103 155 129)(104 156 130)
(1 119)(2 120)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 133)(16 134)(17 135)(18 136)(19 137)(20 138)(21 139)(22 140)(23 141)(24 142)(25 143)(26 144)(27 145)(28 146)(29 147)(30 148)(31 149)(32 150)(33 151)(34 152)(35 153)(36 154)(37 155)(38 156)(39 79)(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 92)(53 93)(54 94)(55 95)(56 96)(57 97)(58 98)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)
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), (1,27,53)(2,28,54)(3,29,55)(4,30,56)(5,31,57)(6,32,58)(7,33,59)(8,34,60)(9,35,61)(10,36,62)(11,37,63)(12,38,64)(13,39,65)(14,40,66)(15,41,67)(16,42,68)(17,43,69)(18,44,70)(19,45,71)(20,46,72)(21,47,73)(22,48,74)(23,49,75)(24,50,76)(25,51,77)(26,52,78)(79,131,105)(80,132,106)(81,133,107)(82,134,108)(83,135,109)(84,136,110)(85,137,111)(86,138,112)(87,139,113)(88,140,114)(89,141,115)(90,142,116)(91,143,117)(92,144,118)(93,145,119)(94,146,120)(95,147,121)(96,148,122)(97,149,123)(98,150,124)(99,151,125)(100,152,126)(101,153,127)(102,154,128)(103,155,129)(104,156,130), (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)>;
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), (1,27,53)(2,28,54)(3,29,55)(4,30,56)(5,31,57)(6,32,58)(7,33,59)(8,34,60)(9,35,61)(10,36,62)(11,37,63)(12,38,64)(13,39,65)(14,40,66)(15,41,67)(16,42,68)(17,43,69)(18,44,70)(19,45,71)(20,46,72)(21,47,73)(22,48,74)(23,49,75)(24,50,76)(25,51,77)(26,52,78)(79,131,105)(80,132,106)(81,133,107)(82,134,108)(83,135,109)(84,136,110)(85,137,111)(86,138,112)(87,139,113)(88,140,114)(89,141,115)(90,142,116)(91,143,117)(92,144,118)(93,145,119)(94,146,120)(95,147,121)(96,148,122)(97,149,123)(98,150,124)(99,151,125)(100,152,126)(101,153,127)(102,154,128)(103,155,129)(104,156,130), (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118) );
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)], [(1,27,53),(2,28,54),(3,29,55),(4,30,56),(5,31,57),(6,32,58),(7,33,59),(8,34,60),(9,35,61),(10,36,62),(11,37,63),(12,38,64),(13,39,65),(14,40,66),(15,41,67),(16,42,68),(17,43,69),(18,44,70),(19,45,71),(20,46,72),(21,47,73),(22,48,74),(23,49,75),(24,50,76),(25,51,77),(26,52,78),(79,131,105),(80,132,106),(81,133,107),(82,134,108),(83,135,109),(84,136,110),(85,137,111),(86,138,112),(87,139,113),(88,140,114),(89,141,115),(90,142,116),(91,143,117),(92,144,118),(93,145,119),(94,146,120),(95,147,121),(96,148,122),(97,149,123),(98,150,124),(99,151,125),(100,152,126),(101,153,127),(102,154,128),(103,155,129),(104,156,130)], [(1,119),(2,120),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,133),(16,134),(17,135),(18,136),(19,137),(20,138),(21,139),(22,140),(23,141),(24,142),(25,143),(26,144),(27,145),(28,146),(29,147),(30,148),(31,149),(32,150),(33,151),(34,152),(35,153),(36,154),(37,155),(38,156),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(56,96),(57,97),(58,98),(59,99),(60,100),(61,101),(62,102),(63,103),(64,104),(65,105),(66,106),(67,107),(68,108),(69,109),(70,110),(71,111),(72,112),(73,113),(74,114),(75,115),(76,116),(77,117),(78,118)]])
234 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26AJ | 39A | ··· | 39X | 39Y | ··· | 39BH | 78A | ··· | 78X | 78Y | ··· | 78BH | 78BI | ··· | 78DD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 39 | ··· | 39 | 78 | ··· | 78 | 78 | ··· | 78 | 78 | ··· | 78 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
234 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C13 | C26 | C26 | C39 | C78 | C78 | S3 | D6 | C3xS3 | S3xC6 | S3xC13 | S3xC26 | S3xC39 | S3xC78 |
| kernel | S3xC78 | S3xC39 | C3xC78 | S3xC26 | S3xC13 | C78 | S3xC6 | C3xS3 | C3xC6 | D6 | S3 | C6 | C78 | C39 | C26 | C13 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 12 | 24 | 12 | 24 | 48 | 24 | 1 | 1 | 2 | 2 | 12 | 12 | 24 | 24 |
Matrix representation of S3xC78 ►in GL3(F79) generated by
| 56 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 1 | 0 | 0 |
| 0 | 55 | 0 |
| 0 | 29 | 23 |
| 78 | 0 | 0 |
| 0 | 64 | 22 |
| 0 | 76 | 15 |
G:=sub<GL(3,GF(79))| [56,0,0,0,11,0,0,0,11],[1,0,0,0,55,29,0,0,23],[78,0,0,0,64,76,0,22,15] >;
S3xC78 in GAP, Magma, Sage, TeX
S_3\times C_{78} % in TeX
G:=Group("S3xC78"); // GroupNames label
G:=SmallGroup(468,51);
// by ID
G=gap.SmallGroup(468,51);
# by ID
G:=PCGroup([5,-2,-2,-3,-13,-3,7804]);
// Polycyclic
G:=Group<a,b,c|a^78=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export