direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D52, C39⋊5D4, C52⋊5C6, C156⋊3C2, D26⋊4C6, C12⋊3D13, C6.15D26, C78.15C22, C4⋊(C3×D13), C13⋊4(C3×D4), (C6×D13)⋊4C2, C2.4(C6×D13), C26.11(C2×C6), SmallGroup(312,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D52
G = < a,b,c | a3=b52=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D52
►On 156 points
Generators in S156
(1 80 130)(2 81 131)(3 82 132)(4 83 133)(5 84 134)(6 85 135)(7 86 136)(8 87 137)(9 88 138)(10 89 139)(11 90 140)(12 91 141)(13 92 142)(14 93 143)(15 94 144)(16 95 145)(17 96 146)(18 97 147)(19 98 148)(20 99 149)(21 100 150)(22 101 151)(23 102 152)(24 103 153)(25 104 154)(26 53 155)(27 54 156)(28 55 105)(29 56 106)(30 57 107)(31 58 108)(32 59 109)(33 60 110)(34 61 111)(35 62 112)(36 63 113)(37 64 114)(38 65 115)(39 66 116)(40 67 117)(41 68 118)(42 69 119)(43 70 120)(44 71 121)(45 72 122)(46 73 123)(47 74 124)(48 75 125)(49 76 126)(50 77 127)(51 78 128)(52 79 129)
(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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(53 54)(55 104)(56 103)(57 102)(58 101)(59 100)(60 99)(61 98)(62 97)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(76 83)(77 82)(78 81)(79 80)(105 154)(106 153)(107 152)(108 151)(109 150)(110 149)(111 148)(112 147)(113 146)(114 145)(115 144)(116 143)(117 142)(118 141)(119 140)(120 139)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 130)(155 156)
G:=sub<Sym(156)| (1,80,130)(2,81,131)(3,82,132)(4,83,133)(5,84,134)(6,85,135)(7,86,136)(8,87,137)(9,88,138)(10,89,139)(11,90,140)(12,91,141)(13,92,142)(14,93,143)(15,94,144)(16,95,145)(17,96,146)(18,97,147)(19,98,148)(20,99,149)(21,100,150)(22,101,151)(23,102,152)(24,103,153)(25,104,154)(26,53,155)(27,54,156)(28,55,105)(29,56,106)(30,57,107)(31,58,108)(32,59,109)(33,60,110)(34,61,111)(35,62,112)(36,63,113)(37,64,114)(38,65,115)(39,66,116)(40,67,117)(41,68,118)(42,69,119)(43,70,120)(44,71,121)(45,72,122)(46,73,123)(47,74,124)(48,75,125)(49,76,126)(50,77,127)(51,78,128)(52,79,129), (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(53,54)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)(105,154)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,147)(113,146)(114,145)(115,144)(116,143)(117,142)(118,141)(119,140)(120,139)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(155,156)>;
G:=Group( (1,80,130)(2,81,131)(3,82,132)(4,83,133)(5,84,134)(6,85,135)(7,86,136)(8,87,137)(9,88,138)(10,89,139)(11,90,140)(12,91,141)(13,92,142)(14,93,143)(15,94,144)(16,95,145)(17,96,146)(18,97,147)(19,98,148)(20,99,149)(21,100,150)(22,101,151)(23,102,152)(24,103,153)(25,104,154)(26,53,155)(27,54,156)(28,55,105)(29,56,106)(30,57,107)(31,58,108)(32,59,109)(33,60,110)(34,61,111)(35,62,112)(36,63,113)(37,64,114)(38,65,115)(39,66,116)(40,67,117)(41,68,118)(42,69,119)(43,70,120)(44,71,121)(45,72,122)(46,73,123)(47,74,124)(48,75,125)(49,76,126)(50,77,127)(51,78,128)(52,79,129), (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(53,54)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)(105,154)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,147)(113,146)(114,145)(115,144)(116,143)(117,142)(118,141)(119,140)(120,139)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(155,156) );
G=PermutationGroup([[(1,80,130),(2,81,131),(3,82,132),(4,83,133),(5,84,134),(6,85,135),(7,86,136),(8,87,137),(9,88,138),(10,89,139),(11,90,140),(12,91,141),(13,92,142),(14,93,143),(15,94,144),(16,95,145),(17,96,146),(18,97,147),(19,98,148),(20,99,149),(21,100,150),(22,101,151),(23,102,152),(24,103,153),(25,104,154),(26,53,155),(27,54,156),(28,55,105),(29,56,106),(30,57,107),(31,58,108),(32,59,109),(33,60,110),(34,61,111),(35,62,112),(36,63,113),(37,64,114),(38,65,115),(39,66,116),(40,67,117),(41,68,118),(42,69,119),(43,70,120),(44,71,121),(45,72,122),(46,73,123),(47,74,124),(48,75,125),(49,76,126),(50,77,127),(51,78,128),(52,79,129)], [(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)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(53,54),(55,104),(56,103),(57,102),(58,101),(59,100),(60,99),(61,98),(62,97),(63,96),(64,95),(65,94),(66,93),(67,92),(68,91),(69,90),(70,89),(71,88),(72,87),(73,86),(74,85),(75,84),(76,83),(77,82),(78,81),(79,80),(105,154),(106,153),(107,152),(108,151),(109,150),(110,149),(111,148),(112,147),(113,146),(114,145),(115,144),(116,143),(117,142),(118,141),(119,140),(120,139),(121,138),(122,137),(123,136),(124,135),(125,134),(126,133),(127,132),(128,131),(129,130),(155,156)]])
87 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39L | 52A | ··· | 52L | 78A | ··· | 78L | 156A | ··· | 156X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 26 | 26 | 1 | 1 | 2 | 1 | 1 | 26 | 26 | 26 | 26 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
87 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | C3×D4 | D13 | D26 | C3×D13 | D52 | C6×D13 | C3×D52 |
| kernel | C3×D52 | C156 | C6×D13 | D52 | C52 | D26 | C39 | C13 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 6 | 6 | 12 | 12 | 12 | 24 |
Matrix representation of C3×D52 ►in GL4(𝔽157) generated by
| 144 | 0 | 0 | 0 |
| 0 | 144 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 150 | 96 | 0 | 0 |
| 36 | 22 | 0 | 0 |
| 0 | 0 | 74 | 116 |
| 0 | 0 | 52 | 20 |
| 150 | 51 | 0 | 0 |
| 36 | 7 | 0 | 0 |
| 0 | 0 | 76 | 52 |
| 0 | 0 | 55 | 81 |
G:=sub<GL(4,GF(157))| [144,0,0,0,0,144,0,0,0,0,1,0,0,0,0,1],[150,36,0,0,96,22,0,0,0,0,74,52,0,0,116,20],[150,36,0,0,51,7,0,0,0,0,76,55,0,0,52,81] >;
C3×D52 in GAP, Magma, Sage, TeX
C_3\times D_{52} % in TeX
G:=Group("C3xD52"); // GroupNames label
G:=SmallGroup(312,29);
// by ID
G=gap.SmallGroup(312,29);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-13,141,66,7204]);
// Polycyclic
G:=Group<a,b,c|a^3=b^52=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export