metabelian, supersoluble, monomial, A-group
Aliases: C3⋊D51, C51⋊1S3, C32⋊2D17, C17⋊(C3⋊S3), (C3×C51)⋊1C2, SmallGroup(306,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C51 — C3⋊D51 |
Generators and relations for C3⋊D51
G = < a,b,c | a3=b51=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C3⋊D51
►On 153 points
Generators in S153
(1 122 73)(2 123 74)(3 124 75)(4 125 76)(5 126 77)(6 127 78)(7 128 79)(8 129 80)(9 130 81)(10 131 82)(11 132 83)(12 133 84)(13 134 85)(14 135 86)(15 136 87)(16 137 88)(17 138 89)(18 139 90)(19 140 91)(20 141 92)(21 142 93)(22 143 94)(23 144 95)(24 145 96)(25 146 97)(26 147 98)(27 148 99)(28 149 100)(29 150 101)(30 151 102)(31 152 52)(32 153 53)(33 103 54)(34 104 55)(35 105 56)(36 106 57)(37 107 58)(38 108 59)(39 109 60)(40 110 61)(41 111 62)(42 112 63)(43 113 64)(44 114 65)(45 115 66)(46 116 67)(47 117 68)(48 118 69)(49 119 70)(50 120 71)(51 121 72)
(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)
(1 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(52 142)(53 141)(54 140)(55 139)(56 138)(57 137)(58 136)(59 135)(60 134)(61 133)(62 132)(63 131)(64 130)(65 129)(66 128)(67 127)(68 126)(69 125)(70 124)(71 123)(72 122)(73 121)(74 120)(75 119)(76 118)(77 117)(78 116)(79 115)(80 114)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 153)(93 152)(94 151)(95 150)(96 149)(97 148)(98 147)(99 146)(100 145)(101 144)(102 143)
G:=sub<Sym(153)| (1,122,73)(2,123,74)(3,124,75)(4,125,76)(5,126,77)(6,127,78)(7,128,79)(8,129,80)(9,130,81)(10,131,82)(11,132,83)(12,133,84)(13,134,85)(14,135,86)(15,136,87)(16,137,88)(17,138,89)(18,139,90)(19,140,91)(20,141,92)(21,142,93)(22,143,94)(23,144,95)(24,145,96)(25,146,97)(26,147,98)(27,148,99)(28,149,100)(29,150,101)(30,151,102)(31,152,52)(32,153,53)(33,103,54)(34,104,55)(35,105,56)(36,106,57)(37,107,58)(38,108,59)(39,109,60)(40,110,61)(41,111,62)(42,112,63)(43,113,64)(44,114,65)(45,115,66)(46,116,67)(47,117,68)(48,118,69)(49,119,70)(50,120,71)(51,121,72), (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), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,39)(14,38)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(52,142)(53,141)(54,140)(55,139)(56,138)(57,137)(58,136)(59,135)(60,134)(61,133)(62,132)(63,131)(64,130)(65,129)(66,128)(67,127)(68,126)(69,125)(70,124)(71,123)(72,122)(73,121)(74,120)(75,119)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145)(101,144)(102,143)>;
G:=Group( (1,122,73)(2,123,74)(3,124,75)(4,125,76)(5,126,77)(6,127,78)(7,128,79)(8,129,80)(9,130,81)(10,131,82)(11,132,83)(12,133,84)(13,134,85)(14,135,86)(15,136,87)(16,137,88)(17,138,89)(18,139,90)(19,140,91)(20,141,92)(21,142,93)(22,143,94)(23,144,95)(24,145,96)(25,146,97)(26,147,98)(27,148,99)(28,149,100)(29,150,101)(30,151,102)(31,152,52)(32,153,53)(33,103,54)(34,104,55)(35,105,56)(36,106,57)(37,107,58)(38,108,59)(39,109,60)(40,110,61)(41,111,62)(42,112,63)(43,113,64)(44,114,65)(45,115,66)(46,116,67)(47,117,68)(48,118,69)(49,119,70)(50,120,71)(51,121,72), (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), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,39)(14,38)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(52,142)(53,141)(54,140)(55,139)(56,138)(57,137)(58,136)(59,135)(60,134)(61,133)(62,132)(63,131)(64,130)(65,129)(66,128)(67,127)(68,126)(69,125)(70,124)(71,123)(72,122)(73,121)(74,120)(75,119)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145)(101,144)(102,143) );
G=PermutationGroup([[(1,122,73),(2,123,74),(3,124,75),(4,125,76),(5,126,77),(6,127,78),(7,128,79),(8,129,80),(9,130,81),(10,131,82),(11,132,83),(12,133,84),(13,134,85),(14,135,86),(15,136,87),(16,137,88),(17,138,89),(18,139,90),(19,140,91),(20,141,92),(21,142,93),(22,143,94),(23,144,95),(24,145,96),(25,146,97),(26,147,98),(27,148,99),(28,149,100),(29,150,101),(30,151,102),(31,152,52),(32,153,53),(33,103,54),(34,104,55),(35,105,56),(36,106,57),(37,107,58),(38,108,59),(39,109,60),(40,110,61),(41,111,62),(42,112,63),(43,113,64),(44,114,65),(45,115,66),(46,116,67),(47,117,68),(48,118,69),(49,119,70),(50,120,71),(51,121,72)], [(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)], [(1,51),(2,50),(3,49),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,40),(13,39),(14,38),(15,37),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28),(25,27),(52,142),(53,141),(54,140),(55,139),(56,138),(57,137),(58,136),(59,135),(60,134),(61,133),(62,132),(63,131),(64,130),(65,129),(66,128),(67,127),(68,126),(69,125),(70,124),(71,123),(72,122),(73,121),(74,120),(75,119),(76,118),(77,117),(78,116),(79,115),(80,114),(81,113),(82,112),(83,111),(84,110),(85,109),(86,108),(87,107),(88,106),(89,105),(90,104),(91,103),(92,153),(93,152),(94,151),(95,150),(96,149),(97,148),(98,147),(99,146),(100,145),(101,144),(102,143)]])
78 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 17A | ··· | 17H | 51A | ··· | 51BL |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 17 | ··· | 17 | 51 | ··· | 51 |
| size | 1 | 153 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | D17 | D51 |
| kernel | C3⋊D51 | C3×C51 | C51 | C32 | C3 |
| # reps | 1 | 1 | 4 | 8 | 64 |
Matrix representation of C3⋊D51 ►in GL4(𝔽103) generated by
| 75 | 25 | 0 | 0 |
| 48 | 27 | 0 | 0 |
| 0 | 0 | 70 | 68 |
| 0 | 0 | 92 | 32 |
| 43 | 97 | 0 | 0 |
| 75 | 71 | 0 | 0 |
| 0 | 0 | 66 | 63 |
| 0 | 0 | 61 | 52 |
| 44 | 37 | 0 | 0 |
| 34 | 59 | 0 | 0 |
| 0 | 0 | 41 | 66 |
| 0 | 0 | 12 | 62 |
G:=sub<GL(4,GF(103))| [75,48,0,0,25,27,0,0,0,0,70,92,0,0,68,32],[43,75,0,0,97,71,0,0,0,0,66,61,0,0,63,52],[44,34,0,0,37,59,0,0,0,0,41,12,0,0,66,62] >;
C3⋊D51 in GAP, Magma, Sage, TeX
C_3\rtimes D_{51} % in TeX
G:=Group("C3:D51"); // GroupNames label
G:=SmallGroup(306,9);
// by ID
G=gap.SmallGroup(306,9);
# by ID
G:=PCGroup([4,-2,-3,-3,-17,33,146,4611]);
// Polycyclic
G:=Group<a,b,c|a^3=b^51=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export