metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C39⋊2D4, C3⋊2D52, D78⋊3C2, D26⋊2S3, Dic3⋊D13, C6.5D26, C26.5D6, C78.5C22, (C6×D13)⋊2C2, C13⋊1(C3⋊D4), C2.5(S3×D13), (Dic3×C13)⋊3C2, SmallGroup(312,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D52
G = < a,b,c | a3=b52=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C3⋊D52
►On 156 points
Generators in S156
(1 146 103)(2 104 147)(3 148 53)(4 54 149)(5 150 55)(6 56 151)(7 152 57)(8 58 153)(9 154 59)(10 60 155)(11 156 61)(12 62 105)(13 106 63)(14 64 107)(15 108 65)(16 66 109)(17 110 67)(18 68 111)(19 112 69)(20 70 113)(21 114 71)(22 72 115)(23 116 73)(24 74 117)(25 118 75)(26 76 119)(27 120 77)(28 78 121)(29 122 79)(30 80 123)(31 124 81)(32 82 125)(33 126 83)(34 84 127)(35 128 85)(36 86 129)(37 130 87)(38 88 131)(39 132 89)(40 90 133)(41 134 91)(42 92 135)(43 136 93)(44 94 137)(45 138 95)(46 96 139)(47 140 97)(48 98 141)(49 142 99)(50 100 143)(51 144 101)(52 102 145)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 130)(54 129)(55 128)(56 127)(57 126)(58 125)(59 124)(60 123)(61 122)(62 121)(63 120)(64 119)(65 118)(66 117)(67 116)(68 115)(69 114)(70 113)(71 112)(72 111)(73 110)(74 109)(75 108)(76 107)(77 106)(78 105)(79 156)(80 155)(81 154)(82 153)(83 152)(84 151)(85 150)(86 149)(87 148)(88 147)(89 146)(90 145)(91 144)(92 143)(93 142)(94 141)(95 140)(96 139)(97 138)(98 137)(99 136)(100 135)(101 134)(102 133)(103 132)(104 131)
G:=sub<Sym(156)| (1,146,103)(2,104,147)(3,148,53)(4,54,149)(5,150,55)(6,56,151)(7,152,57)(8,58,153)(9,154,59)(10,60,155)(11,156,61)(12,62,105)(13,106,63)(14,64,107)(15,108,65)(16,66,109)(17,110,67)(18,68,111)(19,112,69)(20,70,113)(21,114,71)(22,72,115)(23,116,73)(24,74,117)(25,118,75)(26,76,119)(27,120,77)(28,78,121)(29,122,79)(30,80,123)(31,124,81)(32,82,125)(33,126,83)(34,84,127)(35,128,85)(36,86,129)(37,130,87)(38,88,131)(39,132,89)(40,90,133)(41,134,91)(42,92,135)(43,136,93)(44,94,137)(45,138,95)(46,96,139)(47,140,97)(48,98,141)(49,142,99)(50,100,143)(51,144,101)(52,102,145), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,130)(54,129)(55,128)(56,127)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,156)(80,155)(81,154)(82,153)(83,152)(84,151)(85,150)(86,149)(87,148)(88,147)(89,146)(90,145)(91,144)(92,143)(93,142)(94,141)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,132)(104,131)>;
G:=Group( (1,146,103)(2,104,147)(3,148,53)(4,54,149)(5,150,55)(6,56,151)(7,152,57)(8,58,153)(9,154,59)(10,60,155)(11,156,61)(12,62,105)(13,106,63)(14,64,107)(15,108,65)(16,66,109)(17,110,67)(18,68,111)(19,112,69)(20,70,113)(21,114,71)(22,72,115)(23,116,73)(24,74,117)(25,118,75)(26,76,119)(27,120,77)(28,78,121)(29,122,79)(30,80,123)(31,124,81)(32,82,125)(33,126,83)(34,84,127)(35,128,85)(36,86,129)(37,130,87)(38,88,131)(39,132,89)(40,90,133)(41,134,91)(42,92,135)(43,136,93)(44,94,137)(45,138,95)(46,96,139)(47,140,97)(48,98,141)(49,142,99)(50,100,143)(51,144,101)(52,102,145), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,130)(54,129)(55,128)(56,127)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,156)(80,155)(81,154)(82,153)(83,152)(84,151)(85,150)(86,149)(87,148)(88,147)(89,146)(90,145)(91,144)(92,143)(93,142)(94,141)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,132)(104,131) );
G=PermutationGroup([[(1,146,103),(2,104,147),(3,148,53),(4,54,149),(5,150,55),(6,56,151),(7,152,57),(8,58,153),(9,154,59),(10,60,155),(11,156,61),(12,62,105),(13,106,63),(14,64,107),(15,108,65),(16,66,109),(17,110,67),(18,68,111),(19,112,69),(20,70,113),(21,114,71),(22,72,115),(23,116,73),(24,74,117),(25,118,75),(26,76,119),(27,120,77),(28,78,121),(29,122,79),(30,80,123),(31,124,81),(32,82,125),(33,126,83),(34,84,127),(35,128,85),(36,86,129),(37,130,87),(38,88,131),(39,132,89),(40,90,133),(41,134,91),(42,92,135),(43,136,93),(44,94,137),(45,138,95),(46,96,139),(47,140,97),(48,98,141),(49,142,99),(50,100,143),(51,144,101),(52,102,145)], [(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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(53,130),(54,129),(55,128),(56,127),(57,126),(58,125),(59,124),(60,123),(61,122),(62,121),(63,120),(64,119),(65,118),(66,117),(67,116),(68,115),(69,114),(70,113),(71,112),(72,111),(73,110),(74,109),(75,108),(76,107),(77,106),(78,105),(79,156),(80,155),(81,154),(82,153),(83,152),(84,151),(85,150),(86,149),(87,148),(88,147),(89,146),(90,145),(91,144),(92,143),(93,142),(94,141),(95,140),(96,139),(97,138),(98,137),(99,136),(100,135),(101,134),(102,133),(103,132),(104,131)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39F | 52A | ··· | 52L | 78A | ··· | 78F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 |
| size | 1 | 1 | 26 | 78 | 2 | 6 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 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 | C3⋊D4 | D13 | D26 | D52 | S3×D13 | C3⋊D52 |
| kernel | C3⋊D52 | Dic3×C13 | C6×D13 | D78 | D26 | C39 | C26 | C13 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 6 | 6 |
Matrix representation of C3⋊D52 ►in GL4(𝔽157) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 155 | 19 |
| 0 | 0 | 99 | 1 |
| 74 | 37 | 0 | 0 |
| 120 | 147 | 0 | 0 |
| 0 | 0 | 25 | 51 |
| 0 | 0 | 77 | 132 |
| 142 | 90 | 0 | 0 |
| 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 138 |
| 0 | 0 | 0 | 156 |
G:=sub<GL(4,GF(157))| [1,0,0,0,0,1,0,0,0,0,155,99,0,0,19,1],[74,120,0,0,37,147,0,0,0,0,25,77,0,0,51,132],[142,1,0,0,90,15,0,0,0,0,1,0,0,0,138,156] >;
C3⋊D52 in GAP, Magma, Sage, TeX
C_3\rtimes D_{52} % in TeX
G:=Group("C3:D52"); // GroupNames label
G:=SmallGroup(312,19);
// by ID
G=gap.SmallGroup(312,19);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,61,26,168,7204]);
// Polycyclic
G:=Group<a,b,c|a^3=b^52=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export