metabelian, supersoluble, monomial, A-group
Aliases: C3⋊D57, C57⋊1S3, C32⋊2D19, C19⋊(C3⋊S3), (C3×C57)⋊1C2, SmallGroup(342,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C57 — C3⋊D57 |
Generators and relations for C3⋊D57
G = < a,b,c | a3=b57=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C3⋊D57
►On 171 points
Generators in S171
(1 149 80)(2 150 81)(3 151 82)(4 152 83)(5 153 84)(6 154 85)(7 155 86)(8 156 87)(9 157 88)(10 158 89)(11 159 90)(12 160 91)(13 161 92)(14 162 93)(15 163 94)(16 164 95)(17 165 96)(18 166 97)(19 167 98)(20 168 99)(21 169 100)(22 170 101)(23 171 102)(24 115 103)(25 116 104)(26 117 105)(27 118 106)(28 119 107)(29 120 108)(30 121 109)(31 122 110)(32 123 111)(33 124 112)(34 125 113)(35 126 114)(36 127 58)(37 128 59)(38 129 60)(39 130 61)(40 131 62)(41 132 63)(42 133 64)(43 134 65)(44 135 66)(45 136 67)(46 137 68)(47 138 69)(48 139 70)(49 140 71)(50 141 72)(51 142 73)(52 143 74)(53 144 75)(54 145 76)(55 146 77)(56 147 78)(57 148 79)
(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)(115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171)
(1 57)(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 50)(9 49)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(58 170)(59 169)(60 168)(61 167)(62 166)(63 165)(64 164)(65 163)(66 162)(67 161)(68 160)(69 159)(70 158)(71 157)(72 156)(73 155)(74 154)(75 153)(76 152)(77 151)(78 150)(79 149)(80 148)(81 147)(82 146)(83 145)(84 144)(85 143)(86 142)(87 141)(88 140)(89 139)(90 138)(91 137)(92 136)(93 135)(94 134)(95 133)(96 132)(97 131)(98 130)(99 129)(100 128)(101 127)(102 126)(103 125)(104 124)(105 123)(106 122)(107 121)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(114 171)
G:=sub<Sym(171)| (1,149,80)(2,150,81)(3,151,82)(4,152,83)(5,153,84)(6,154,85)(7,155,86)(8,156,87)(9,157,88)(10,158,89)(11,159,90)(12,160,91)(13,161,92)(14,162,93)(15,163,94)(16,164,95)(17,165,96)(18,166,97)(19,167,98)(20,168,99)(21,169,100)(22,170,101)(23,171,102)(24,115,103)(25,116,104)(26,117,105)(27,118,106)(28,119,107)(29,120,108)(30,121,109)(31,122,110)(32,123,111)(33,124,112)(34,125,113)(35,126,114)(36,127,58)(37,128,59)(38,129,60)(39,130,61)(40,131,62)(41,132,63)(42,133,64)(43,134,65)(44,135,66)(45,136,67)(46,137,68)(47,138,69)(48,139,70)(49,140,71)(50,141,72)(51,142,73)(52,143,74)(53,144,75)(54,145,76)(55,146,77)(56,147,78)(57,148,79), (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)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171), (1,57)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(58,170)(59,169)(60,168)(61,167)(62,166)(63,165)(64,164)(65,163)(66,162)(67,161)(68,160)(69,159)(70,158)(71,157)(72,156)(73,155)(74,154)(75,153)(76,152)(77,151)(78,150)(79,149)(80,148)(81,147)(82,146)(83,145)(84,144)(85,143)(86,142)(87,141)(88,140)(89,139)(90,138)(91,137)(92,136)(93,135)(94,134)(95,133)(96,132)(97,131)(98,130)(99,129)(100,128)(101,127)(102,126)(103,125)(104,124)(105,123)(106,122)(107,121)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(114,171)>;
G:=Group( (1,149,80)(2,150,81)(3,151,82)(4,152,83)(5,153,84)(6,154,85)(7,155,86)(8,156,87)(9,157,88)(10,158,89)(11,159,90)(12,160,91)(13,161,92)(14,162,93)(15,163,94)(16,164,95)(17,165,96)(18,166,97)(19,167,98)(20,168,99)(21,169,100)(22,170,101)(23,171,102)(24,115,103)(25,116,104)(26,117,105)(27,118,106)(28,119,107)(29,120,108)(30,121,109)(31,122,110)(32,123,111)(33,124,112)(34,125,113)(35,126,114)(36,127,58)(37,128,59)(38,129,60)(39,130,61)(40,131,62)(41,132,63)(42,133,64)(43,134,65)(44,135,66)(45,136,67)(46,137,68)(47,138,69)(48,139,70)(49,140,71)(50,141,72)(51,142,73)(52,143,74)(53,144,75)(54,145,76)(55,146,77)(56,147,78)(57,148,79), (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)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171), (1,57)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(58,170)(59,169)(60,168)(61,167)(62,166)(63,165)(64,164)(65,163)(66,162)(67,161)(68,160)(69,159)(70,158)(71,157)(72,156)(73,155)(74,154)(75,153)(76,152)(77,151)(78,150)(79,149)(80,148)(81,147)(82,146)(83,145)(84,144)(85,143)(86,142)(87,141)(88,140)(89,139)(90,138)(91,137)(92,136)(93,135)(94,134)(95,133)(96,132)(97,131)(98,130)(99,129)(100,128)(101,127)(102,126)(103,125)(104,124)(105,123)(106,122)(107,121)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(114,171) );
G=PermutationGroup([[(1,149,80),(2,150,81),(3,151,82),(4,152,83),(5,153,84),(6,154,85),(7,155,86),(8,156,87),(9,157,88),(10,158,89),(11,159,90),(12,160,91),(13,161,92),(14,162,93),(15,163,94),(16,164,95),(17,165,96),(18,166,97),(19,167,98),(20,168,99),(21,169,100),(22,170,101),(23,171,102),(24,115,103),(25,116,104),(26,117,105),(27,118,106),(28,119,107),(29,120,108),(30,121,109),(31,122,110),(32,123,111),(33,124,112),(34,125,113),(35,126,114),(36,127,58),(37,128,59),(38,129,60),(39,130,61),(40,131,62),(41,132,63),(42,133,64),(43,134,65),(44,135,66),(45,136,67),(46,137,68),(47,138,69),(48,139,70),(49,140,71),(50,141,72),(51,142,73),(52,143,74),(53,144,75),(54,145,76),(55,146,77),(56,147,78),(57,148,79)], [(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),(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171)], [(1,57),(2,56),(3,55),(4,54),(5,53),(6,52),(7,51),(8,50),(9,49),(10,48),(11,47),(12,46),(13,45),(14,44),(15,43),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(58,170),(59,169),(60,168),(61,167),(62,166),(63,165),(64,164),(65,163),(66,162),(67,161),(68,160),(69,159),(70,158),(71,157),(72,156),(73,155),(74,154),(75,153),(76,152),(77,151),(78,150),(79,149),(80,148),(81,147),(82,146),(83,145),(84,144),(85,143),(86,142),(87,141),(88,140),(89,139),(90,138),(91,137),(92,136),(93,135),(94,134),(95,133),(96,132),(97,131),(98,130),(99,129),(100,128),(101,127),(102,126),(103,125),(104,124),(105,123),(106,122),(107,121),(108,120),(109,119),(110,118),(111,117),(112,116),(113,115),(114,171)]])
87 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 19A | ··· | 19I | 57A | ··· | 57BT |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 19 | ··· | 19 | 57 | ··· | 57 |
| size | 1 | 171 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
87 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | D19 | D57 |
| kernel | C3⋊D57 | C3×C57 | C57 | C32 | C3 |
| # reps | 1 | 1 | 4 | 9 | 72 |
Matrix representation of C3⋊D57 ►in GL4(𝔽229) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 196 | 4 |
| 0 | 0 | 22 | 32 |
| 130 | 29 | 0 | 0 |
| 200 | 64 | 0 | 0 |
| 0 | 0 | 85 | 39 |
| 0 | 0 | 100 | 89 |
| 130 | 29 | 0 | 0 |
| 49 | 99 | 0 | 0 |
| 0 | 0 | 191 | 222 |
| 0 | 0 | 108 | 38 |
G:=sub<GL(4,GF(229))| [1,0,0,0,0,1,0,0,0,0,196,22,0,0,4,32],[130,200,0,0,29,64,0,0,0,0,85,100,0,0,39,89],[130,49,0,0,29,99,0,0,0,0,191,108,0,0,222,38] >;
C3⋊D57 in GAP, Magma, Sage, TeX
C_3\rtimes D_{57} % in TeX
G:=Group("C3:D57"); // GroupNames label
G:=SmallGroup(342,17);
// by ID
G=gap.SmallGroup(342,17);
# by ID
G:=PCGroup([4,-2,-3,-3,-19,33,146,5187]);
// Polycyclic
G:=Group<a,b,c|a^3=b^57=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export