direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C79, C3⋊C158, C237⋊3C2, SmallGroup(474,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C79 |
Generators and relations for S3×C79
G = < a,b,c | a79=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C79
►On 237 points
Generators in S237
(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)(159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237)
(1 188 130)(2 189 131)(3 190 132)(4 191 133)(5 192 134)(6 193 135)(7 194 136)(8 195 137)(9 196 138)(10 197 139)(11 198 140)(12 199 141)(13 200 142)(14 201 143)(15 202 144)(16 203 145)(17 204 146)(18 205 147)(19 206 148)(20 207 149)(21 208 150)(22 209 151)(23 210 152)(24 211 153)(25 212 154)(26 213 155)(27 214 156)(28 215 157)(29 216 158)(30 217 80)(31 218 81)(32 219 82)(33 220 83)(34 221 84)(35 222 85)(36 223 86)(37 224 87)(38 225 88)(39 226 89)(40 227 90)(41 228 91)(42 229 92)(43 230 93)(44 231 94)(45 232 95)(46 233 96)(47 234 97)(48 235 98)(49 236 99)(50 237 100)(51 159 101)(52 160 102)(53 161 103)(54 162 104)(55 163 105)(56 164 106)(57 165 107)(58 166 108)(59 167 109)(60 168 110)(61 169 111)(62 170 112)(63 171 113)(64 172 114)(65 173 115)(66 174 116)(67 175 117)(68 176 118)(69 177 119)(70 178 120)(71 179 121)(72 180 122)(73 181 123)(74 182 124)(75 183 125)(76 184 126)(77 185 127)(78 186 128)(79 187 129)
(80 217)(81 218)(82 219)(83 220)(84 221)(85 222)(86 223)(87 224)(88 225)(89 226)(90 227)(91 228)(92 229)(93 230)(94 231)(95 232)(96 233)(97 234)(98 235)(99 236)(100 237)(101 159)(102 160)(103 161)(104 162)(105 163)(106 164)(107 165)(108 166)(109 167)(110 168)(111 169)(112 170)(113 171)(114 172)(115 173)(116 174)(117 175)(118 176)(119 177)(120 178)(121 179)(122 180)(123 181)(124 182)(125 183)(126 184)(127 185)(128 186)(129 187)(130 188)(131 189)(132 190)(133 191)(134 192)(135 193)(136 194)(137 195)(138 196)(139 197)(140 198)(141 199)(142 200)(143 201)(144 202)(145 203)(146 204)(147 205)(148 206)(149 207)(150 208)(151 209)(152 210)(153 211)(154 212)(155 213)(156 214)(157 215)(158 216)
G:=sub<Sym(237)| (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)(159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237), (1,188,130)(2,189,131)(3,190,132)(4,191,133)(5,192,134)(6,193,135)(7,194,136)(8,195,137)(9,196,138)(10,197,139)(11,198,140)(12,199,141)(13,200,142)(14,201,143)(15,202,144)(16,203,145)(17,204,146)(18,205,147)(19,206,148)(20,207,149)(21,208,150)(22,209,151)(23,210,152)(24,211,153)(25,212,154)(26,213,155)(27,214,156)(28,215,157)(29,216,158)(30,217,80)(31,218,81)(32,219,82)(33,220,83)(34,221,84)(35,222,85)(36,223,86)(37,224,87)(38,225,88)(39,226,89)(40,227,90)(41,228,91)(42,229,92)(43,230,93)(44,231,94)(45,232,95)(46,233,96)(47,234,97)(48,235,98)(49,236,99)(50,237,100)(51,159,101)(52,160,102)(53,161,103)(54,162,104)(55,163,105)(56,164,106)(57,165,107)(58,166,108)(59,167,109)(60,168,110)(61,169,111)(62,170,112)(63,171,113)(64,172,114)(65,173,115)(66,174,116)(67,175,117)(68,176,118)(69,177,119)(70,178,120)(71,179,121)(72,180,122)(73,181,123)(74,182,124)(75,183,125)(76,184,126)(77,185,127)(78,186,128)(79,187,129), (80,217)(81,218)(82,219)(83,220)(84,221)(85,222)(86,223)(87,224)(88,225)(89,226)(90,227)(91,228)(92,229)(93,230)(94,231)(95,232)(96,233)(97,234)(98,235)(99,236)(100,237)(101,159)(102,160)(103,161)(104,162)(105,163)(106,164)(107,165)(108,166)(109,167)(110,168)(111,169)(112,170)(113,171)(114,172)(115,173)(116,174)(117,175)(118,176)(119,177)(120,178)(121,179)(122,180)(123,181)(124,182)(125,183)(126,184)(127,185)(128,186)(129,187)(130,188)(131,189)(132,190)(133,191)(134,192)(135,193)(136,194)(137,195)(138,196)(139,197)(140,198)(141,199)(142,200)(143,201)(144,202)(145,203)(146,204)(147,205)(148,206)(149,207)(150,208)(151,209)(152,210)(153,211)(154,212)(155,213)(156,214)(157,215)(158,216)>;
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)(159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237), (1,188,130)(2,189,131)(3,190,132)(4,191,133)(5,192,134)(6,193,135)(7,194,136)(8,195,137)(9,196,138)(10,197,139)(11,198,140)(12,199,141)(13,200,142)(14,201,143)(15,202,144)(16,203,145)(17,204,146)(18,205,147)(19,206,148)(20,207,149)(21,208,150)(22,209,151)(23,210,152)(24,211,153)(25,212,154)(26,213,155)(27,214,156)(28,215,157)(29,216,158)(30,217,80)(31,218,81)(32,219,82)(33,220,83)(34,221,84)(35,222,85)(36,223,86)(37,224,87)(38,225,88)(39,226,89)(40,227,90)(41,228,91)(42,229,92)(43,230,93)(44,231,94)(45,232,95)(46,233,96)(47,234,97)(48,235,98)(49,236,99)(50,237,100)(51,159,101)(52,160,102)(53,161,103)(54,162,104)(55,163,105)(56,164,106)(57,165,107)(58,166,108)(59,167,109)(60,168,110)(61,169,111)(62,170,112)(63,171,113)(64,172,114)(65,173,115)(66,174,116)(67,175,117)(68,176,118)(69,177,119)(70,178,120)(71,179,121)(72,180,122)(73,181,123)(74,182,124)(75,183,125)(76,184,126)(77,185,127)(78,186,128)(79,187,129), (80,217)(81,218)(82,219)(83,220)(84,221)(85,222)(86,223)(87,224)(88,225)(89,226)(90,227)(91,228)(92,229)(93,230)(94,231)(95,232)(96,233)(97,234)(98,235)(99,236)(100,237)(101,159)(102,160)(103,161)(104,162)(105,163)(106,164)(107,165)(108,166)(109,167)(110,168)(111,169)(112,170)(113,171)(114,172)(115,173)(116,174)(117,175)(118,176)(119,177)(120,178)(121,179)(122,180)(123,181)(124,182)(125,183)(126,184)(127,185)(128,186)(129,187)(130,188)(131,189)(132,190)(133,191)(134,192)(135,193)(136,194)(137,195)(138,196)(139,197)(140,198)(141,199)(142,200)(143,201)(144,202)(145,203)(146,204)(147,205)(148,206)(149,207)(150,208)(151,209)(152,210)(153,211)(154,212)(155,213)(156,214)(157,215)(158,216) );
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),(159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237)], [(1,188,130),(2,189,131),(3,190,132),(4,191,133),(5,192,134),(6,193,135),(7,194,136),(8,195,137),(9,196,138),(10,197,139),(11,198,140),(12,199,141),(13,200,142),(14,201,143),(15,202,144),(16,203,145),(17,204,146),(18,205,147),(19,206,148),(20,207,149),(21,208,150),(22,209,151),(23,210,152),(24,211,153),(25,212,154),(26,213,155),(27,214,156),(28,215,157),(29,216,158),(30,217,80),(31,218,81),(32,219,82),(33,220,83),(34,221,84),(35,222,85),(36,223,86),(37,224,87),(38,225,88),(39,226,89),(40,227,90),(41,228,91),(42,229,92),(43,230,93),(44,231,94),(45,232,95),(46,233,96),(47,234,97),(48,235,98),(49,236,99),(50,237,100),(51,159,101),(52,160,102),(53,161,103),(54,162,104),(55,163,105),(56,164,106),(57,165,107),(58,166,108),(59,167,109),(60,168,110),(61,169,111),(62,170,112),(63,171,113),(64,172,114),(65,173,115),(66,174,116),(67,175,117),(68,176,118),(69,177,119),(70,178,120),(71,179,121),(72,180,122),(73,181,123),(74,182,124),(75,183,125),(76,184,126),(77,185,127),(78,186,128),(79,187,129)], [(80,217),(81,218),(82,219),(83,220),(84,221),(85,222),(86,223),(87,224),(88,225),(89,226),(90,227),(91,228),(92,229),(93,230),(94,231),(95,232),(96,233),(97,234),(98,235),(99,236),(100,237),(101,159),(102,160),(103,161),(104,162),(105,163),(106,164),(107,165),(108,166),(109,167),(110,168),(111,169),(112,170),(113,171),(114,172),(115,173),(116,174),(117,175),(118,176),(119,177),(120,178),(121,179),(122,180),(123,181),(124,182),(125,183),(126,184),(127,185),(128,186),(129,187),(130,188),(131,189),(132,190),(133,191),(134,192),(135,193),(136,194),(137,195),(138,196),(139,197),(140,198),(141,199),(142,200),(143,201),(144,202),(145,203),(146,204),(147,205),(148,206),(149,207),(150,208),(151,209),(152,210),(153,211),(154,212),(155,213),(156,214),(157,215),(158,216)]])
237 conjugacy classes
| class | 1 | 2 | 3 | 79A | ··· | 79BZ | 158A | ··· | 158BZ | 237A | ··· | 237BZ |
| order | 1 | 2 | 3 | 79 | ··· | 79 | 158 | ··· | 158 | 237 | ··· | 237 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
237 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C79 | C158 | S3 | S3×C79 |
| kernel | S3×C79 | C237 | S3 | C3 | C79 | C1 |
| # reps | 1 | 1 | 78 | 78 | 1 | 78 |
Matrix representation of S3×C79 ►in GL2(𝔽1423) generated by
| 836 | 0 |
| 0 | 836 |
| 1422 | 1422 |
| 1 | 0 |
| 1 | 0 |
| 1422 | 1422 |
G:=sub<GL(2,GF(1423))| [836,0,0,836],[1422,1,1422,0],[1,1422,0,1422] >;
S3×C79 in GAP, Magma, Sage, TeX
S_3\times C_{79} % in TeX
G:=Group("S3xC79"); // GroupNames label
G:=SmallGroup(474,3);
// by ID
G=gap.SmallGroup(474,3);
# by ID
G:=PCGroup([3,-2,-79,-3,2846]);
// Polycyclic
G:=Group<a,b,c|a^79=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export