direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3⋊S3×C19, C57⋊3S3, C32⋊2C38, C3⋊(S3×C19), (C3×C57)⋊5C2, SmallGroup(342,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3⋊S3×C19 |
Generators and relations for C3⋊S3×C19
G = < a,b,c,d | a19=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Smallest permutation representation of C3⋊S3×C19
►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 | S3×C19 |
| kernel | C3⋊S3×C19 | C3×C57 | C3⋊S3 | C32 | C57 | C3 |
| # reps | 1 | 1 | 18 | 18 | 4 | 72 |
Matrix representation of C3⋊S3×C19 ►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⋊S3×C19 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
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