direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3:S3xC19, C57:3S3, C32:2C38, C3:(S3xC19), (C3xC57):5C2, SmallGroup(342,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3:S3xC19 |
Generators and relations for C3:S3xC19
G = < a,b,c,d | a19=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Quotients: C1, C2, S3, C3:S3, C19, C38, S3xC19, C3:S3xC19
Smallest permutation representation of C3:S3xC19
►On 171 points
Generators in S171
(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)
(1 123 166)(2 124 167)(3 125 168)(4 126 169)(5 127 170)(6 128 171)(7 129 153)(8 130 154)(9 131 155)(10 132 156)(11 133 157)(12 115 158)(13 116 159)(14 117 160)(15 118 161)(16 119 162)(17 120 163)(18 121 164)(19 122 165)(20 53 76)(21 54 58)(22 55 59)(23 56 60)(24 57 61)(25 39 62)(26 40 63)(27 41 64)(28 42 65)(29 43 66)(30 44 67)(31 45 68)(32 46 69)(33 47 70)(34 48 71)(35 49 72)(36 50 73)(37 51 74)(38 52 75)(77 100 147)(78 101 148)(79 102 149)(80 103 150)(81 104 151)(82 105 152)(83 106 134)(84 107 135)(85 108 136)(86 109 137)(87 110 138)(88 111 139)(89 112 140)(90 113 141)(91 114 142)(92 96 143)(93 97 144)(94 98 145)(95 99 146)
(1 26 106)(2 27 107)(3 28 108)(4 29 109)(5 30 110)(6 31 111)(7 32 112)(8 33 113)(9 34 114)(10 35 96)(11 36 97)(12 37 98)(13 38 99)(14 20 100)(15 21 101)(16 22 102)(17 23 103)(18 24 104)(19 25 105)(39 152 122)(40 134 123)(41 135 124)(42 136 125)(43 137 126)(44 138 127)(45 139 128)(46 140 129)(47 141 130)(48 142 131)(49 143 132)(50 144 133)(51 145 115)(52 146 116)(53 147 117)(54 148 118)(55 149 119)(56 150 120)(57 151 121)(58 78 161)(59 79 162)(60 80 163)(61 81 164)(62 82 165)(63 83 166)(64 84 167)(65 85 168)(66 86 169)(67 87 170)(68 88 171)(69 89 153)(70 90 154)(71 91 155)(72 92 156)(73 93 157)(74 94 158)(75 95 159)(76 77 160)
(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 96)(36 97)(37 98)(38 99)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)(53 77)(54 78)(55 79)(56 80)(57 81)(58 148)(59 149)(60 150)(61 151)(62 152)(63 134)(64 135)(65 136)(66 137)(67 138)(68 139)(69 140)(70 141)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(115 158)(116 159)(117 160)(118 161)(119 162)(120 163)(121 164)(122 165)(123 166)(124 167)(125 168)(126 169)(127 170)(128 171)(129 153)(130 154)(131 155)(132 156)(133 157)
G:=sub<Sym(171)| (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), (1,123,166)(2,124,167)(3,125,168)(4,126,169)(5,127,170)(6,128,171)(7,129,153)(8,130,154)(9,131,155)(10,132,156)(11,133,157)(12,115,158)(13,116,159)(14,117,160)(15,118,161)(16,119,162)(17,120,163)(18,121,164)(19,122,165)(20,53,76)(21,54,58)(22,55,59)(23,56,60)(24,57,61)(25,39,62)(26,40,63)(27,41,64)(28,42,65)(29,43,66)(30,44,67)(31,45,68)(32,46,69)(33,47,70)(34,48,71)(35,49,72)(36,50,73)(37,51,74)(38,52,75)(77,100,147)(78,101,148)(79,102,149)(80,103,150)(81,104,151)(82,105,152)(83,106,134)(84,107,135)(85,108,136)(86,109,137)(87,110,138)(88,111,139)(89,112,140)(90,113,141)(91,114,142)(92,96,143)(93,97,144)(94,98,145)(95,99,146), (1,26,106)(2,27,107)(3,28,108)(4,29,109)(5,30,110)(6,31,111)(7,32,112)(8,33,113)(9,34,114)(10,35,96)(11,36,97)(12,37,98)(13,38,99)(14,20,100)(15,21,101)(16,22,102)(17,23,103)(18,24,104)(19,25,105)(39,152,122)(40,134,123)(41,135,124)(42,136,125)(43,137,126)(44,138,127)(45,139,128)(46,140,129)(47,141,130)(48,142,131)(49,143,132)(50,144,133)(51,145,115)(52,146,116)(53,147,117)(54,148,118)(55,149,119)(56,150,120)(57,151,121)(58,78,161)(59,79,162)(60,80,163)(61,81,164)(62,82,165)(63,83,166)(64,84,167)(65,85,168)(66,86,169)(67,87,170)(68,88,171)(69,89,153)(70,90,154)(71,91,155)(72,92,156)(73,93,157)(74,94,158)(75,95,159)(76,77,160), (20,100)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,96)(36,97)(37,98)(38,99)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,77)(54,78)(55,79)(56,80)(57,81)(58,148)(59,149)(60,150)(61,151)(62,152)(63,134)(64,135)(65,136)(66,137)(67,138)(68,139)(69,140)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(115,158)(116,159)(117,160)(118,161)(119,162)(120,163)(121,164)(122,165)(123,166)(124,167)(125,168)(126,169)(127,170)(128,171)(129,153)(130,154)(131,155)(132,156)(133,157)>;
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)(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171), (1,123,166)(2,124,167)(3,125,168)(4,126,169)(5,127,170)(6,128,171)(7,129,153)(8,130,154)(9,131,155)(10,132,156)(11,133,157)(12,115,158)(13,116,159)(14,117,160)(15,118,161)(16,119,162)(17,120,163)(18,121,164)(19,122,165)(20,53,76)(21,54,58)(22,55,59)(23,56,60)(24,57,61)(25,39,62)(26,40,63)(27,41,64)(28,42,65)(29,43,66)(30,44,67)(31,45,68)(32,46,69)(33,47,70)(34,48,71)(35,49,72)(36,50,73)(37,51,74)(38,52,75)(77,100,147)(78,101,148)(79,102,149)(80,103,150)(81,104,151)(82,105,152)(83,106,134)(84,107,135)(85,108,136)(86,109,137)(87,110,138)(88,111,139)(89,112,140)(90,113,141)(91,114,142)(92,96,143)(93,97,144)(94,98,145)(95,99,146), (1,26,106)(2,27,107)(3,28,108)(4,29,109)(5,30,110)(6,31,111)(7,32,112)(8,33,113)(9,34,114)(10,35,96)(11,36,97)(12,37,98)(13,38,99)(14,20,100)(15,21,101)(16,22,102)(17,23,103)(18,24,104)(19,25,105)(39,152,122)(40,134,123)(41,135,124)(42,136,125)(43,137,126)(44,138,127)(45,139,128)(46,140,129)(47,141,130)(48,142,131)(49,143,132)(50,144,133)(51,145,115)(52,146,116)(53,147,117)(54,148,118)(55,149,119)(56,150,120)(57,151,121)(58,78,161)(59,79,162)(60,80,163)(61,81,164)(62,82,165)(63,83,166)(64,84,167)(65,85,168)(66,86,169)(67,87,170)(68,88,171)(69,89,153)(70,90,154)(71,91,155)(72,92,156)(73,93,157)(74,94,158)(75,95,159)(76,77,160), (20,100)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,96)(36,97)(37,98)(38,99)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,77)(54,78)(55,79)(56,80)(57,81)(58,148)(59,149)(60,150)(61,151)(62,152)(63,134)(64,135)(65,136)(66,137)(67,138)(68,139)(69,140)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(115,158)(116,159)(117,160)(118,161)(119,162)(120,163)(121,164)(122,165)(123,166)(124,167)(125,168)(126,169)(127,170)(128,171)(129,153)(130,154)(131,155)(132,156)(133,157) );
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),(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171)], [(1,123,166),(2,124,167),(3,125,168),(4,126,169),(5,127,170),(6,128,171),(7,129,153),(8,130,154),(9,131,155),(10,132,156),(11,133,157),(12,115,158),(13,116,159),(14,117,160),(15,118,161),(16,119,162),(17,120,163),(18,121,164),(19,122,165),(20,53,76),(21,54,58),(22,55,59),(23,56,60),(24,57,61),(25,39,62),(26,40,63),(27,41,64),(28,42,65),(29,43,66),(30,44,67),(31,45,68),(32,46,69),(33,47,70),(34,48,71),(35,49,72),(36,50,73),(37,51,74),(38,52,75),(77,100,147),(78,101,148),(79,102,149),(80,103,150),(81,104,151),(82,105,152),(83,106,134),(84,107,135),(85,108,136),(86,109,137),(87,110,138),(88,111,139),(89,112,140),(90,113,141),(91,114,142),(92,96,143),(93,97,144),(94,98,145),(95,99,146)], [(1,26,106),(2,27,107),(3,28,108),(4,29,109),(5,30,110),(6,31,111),(7,32,112),(8,33,113),(9,34,114),(10,35,96),(11,36,97),(12,37,98),(13,38,99),(14,20,100),(15,21,101),(16,22,102),(17,23,103),(18,24,104),(19,25,105),(39,152,122),(40,134,123),(41,135,124),(42,136,125),(43,137,126),(44,138,127),(45,139,128),(46,140,129),(47,141,130),(48,142,131),(49,143,132),(50,144,133),(51,145,115),(52,146,116),(53,147,117),(54,148,118),(55,149,119),(56,150,120),(57,151,121),(58,78,161),(59,79,162),(60,80,163),(61,81,164),(62,82,165),(63,83,166),(64,84,167),(65,85,168),(66,86,169),(67,87,170),(68,88,171),(69,89,153),(70,90,154),(71,91,155),(72,92,156),(73,93,157),(74,94,158),(75,95,159),(76,77,160)], [(20,100),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,96),(36,97),(37,98),(38,99),(39,82),(40,83),(41,84),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,92),(50,93),(51,94),(52,95),(53,77),(54,78),(55,79),(56,80),(57,81),(58,148),(59,149),(60,150),(61,151),(62,152),(63,134),(64,135),(65,136),(66,137),(67,138),(68,139),(69,140),(70,141),(71,142),(72,143),(73,144),(74,145),(75,146),(76,147),(115,158),(116,159),(117,160),(118,161),(119,162),(120,163),(121,164),(122,165),(123,166),(124,167),(125,168),(126,169),(127,170),(128,171),(129,153),(130,154),(131,155),(132,156),(133,157)]])
114 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 19A | ··· | 19R | 38A | ··· | 38R | 57A | ··· | 57BT |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 |
| size | 1 | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C19 | C38 | S3 | S3xC19 |
| kernel | C3:S3xC19 | C3xC57 | C3:S3 | C32 | C57 | C3 |
| # reps | 1 | 1 | 18 | 18 | 4 | 72 |
Matrix representation of C3:S3xC19 ►in GL4(F229) generated by
| 57 | 0 | 0 | 0 |
| 0 | 57 | 0 | 0 |
| 0 | 0 | 104 | 0 |
| 0 | 0 | 0 | 104 |
| 0 | 1 | 0 | 0 |
| 228 | 228 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 228 | 228 | 0 | 0 |
| 0 | 0 | 1 | 226 |
| 0 | 0 | 1 | 227 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 228 | 3 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(229))| [57,0,0,0,0,57,0,0,0,0,104,0,0,0,0,104],[0,228,0,0,1,228,0,0,0,0,1,0,0,0,0,1],[0,228,0,0,1,228,0,0,0,0,1,1,0,0,226,227],[0,1,0,0,1,0,0,0,0,0,228,0,0,0,3,1] >;
C3:S3xC19 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\times C_{19} % in TeX
G:=Group("C3:S3xC19"); // GroupNames label
G:=SmallGroup(342,16);
// by ID
G=gap.SmallGroup(342,16);
# by ID
G:=PCGroup([4,-2,-19,-3,-3,914,3651]);
// Polycyclic
G:=Group<a,b,c,d|a^19=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export