direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C73, C3⋊C146, C219⋊3C2, SmallGroup(438,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C73 |
Generators and relations for S3×C73
G = < a,b,c | a73=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C73
►On 219 points
Generators in S219
(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)
(1 80 173)(2 81 174)(3 82 175)(4 83 176)(5 84 177)(6 85 178)(7 86 179)(8 87 180)(9 88 181)(10 89 182)(11 90 183)(12 91 184)(13 92 185)(14 93 186)(15 94 187)(16 95 188)(17 96 189)(18 97 190)(19 98 191)(20 99 192)(21 100 193)(22 101 194)(23 102 195)(24 103 196)(25 104 197)(26 105 198)(27 106 199)(28 107 200)(29 108 201)(30 109 202)(31 110 203)(32 111 204)(33 112 205)(34 113 206)(35 114 207)(36 115 208)(37 116 209)(38 117 210)(39 118 211)(40 119 212)(41 120 213)(42 121 214)(43 122 215)(44 123 216)(45 124 217)(46 125 218)(47 126 219)(48 127 147)(49 128 148)(50 129 149)(51 130 150)(52 131 151)(53 132 152)(54 133 153)(55 134 154)(56 135 155)(57 136 156)(58 137 157)(59 138 158)(60 139 159)(61 140 160)(62 141 161)(63 142 162)(64 143 163)(65 144 164)(66 145 165)(67 146 166)(68 74 167)(69 75 168)(70 76 169)(71 77 170)(72 78 171)(73 79 172)
(74 167)(75 168)(76 169)(77 170)(78 171)(79 172)(80 173)(81 174)(82 175)(83 176)(84 177)(85 178)(86 179)(87 180)(88 181)(89 182)(90 183)(91 184)(92 185)(93 186)(94 187)(95 188)(96 189)(97 190)(98 191)(99 192)(100 193)(101 194)(102 195)(103 196)(104 197)(105 198)(106 199)(107 200)(108 201)(109 202)(110 203)(111 204)(112 205)(113 206)(114 207)(115 208)(116 209)(117 210)(118 211)(119 212)(120 213)(121 214)(122 215)(123 216)(124 217)(125 218)(126 219)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)(141 161)(142 162)(143 163)(144 164)(145 165)(146 166)
G:=sub<Sym(219)| (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), (1,80,173)(2,81,174)(3,82,175)(4,83,176)(5,84,177)(6,85,178)(7,86,179)(8,87,180)(9,88,181)(10,89,182)(11,90,183)(12,91,184)(13,92,185)(14,93,186)(15,94,187)(16,95,188)(17,96,189)(18,97,190)(19,98,191)(20,99,192)(21,100,193)(22,101,194)(23,102,195)(24,103,196)(25,104,197)(26,105,198)(27,106,199)(28,107,200)(29,108,201)(30,109,202)(31,110,203)(32,111,204)(33,112,205)(34,113,206)(35,114,207)(36,115,208)(37,116,209)(38,117,210)(39,118,211)(40,119,212)(41,120,213)(42,121,214)(43,122,215)(44,123,216)(45,124,217)(46,125,218)(47,126,219)(48,127,147)(49,128,148)(50,129,149)(51,130,150)(52,131,151)(53,132,152)(54,133,153)(55,134,154)(56,135,155)(57,136,156)(58,137,157)(59,138,158)(60,139,159)(61,140,160)(62,141,161)(63,142,162)(64,143,163)(65,144,164)(66,145,165)(67,146,166)(68,74,167)(69,75,168)(70,76,169)(71,77,170)(72,78,171)(73,79,172), (74,167)(75,168)(76,169)(77,170)(78,171)(79,172)(80,173)(81,174)(82,175)(83,176)(84,177)(85,178)(86,179)(87,180)(88,181)(89,182)(90,183)(91,184)(92,185)(93,186)(94,187)(95,188)(96,189)(97,190)(98,191)(99,192)(100,193)(101,194)(102,195)(103,196)(104,197)(105,198)(106,199)(107,200)(108,201)(109,202)(110,203)(111,204)(112,205)(113,206)(114,207)(115,208)(116,209)(117,210)(118,211)(119,212)(120,213)(121,214)(122,215)(123,216)(124,217)(125,218)(126,219)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)(141,161)(142,162)(143,163)(144,164)(145,165)(146,166)>;
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), (1,80,173)(2,81,174)(3,82,175)(4,83,176)(5,84,177)(6,85,178)(7,86,179)(8,87,180)(9,88,181)(10,89,182)(11,90,183)(12,91,184)(13,92,185)(14,93,186)(15,94,187)(16,95,188)(17,96,189)(18,97,190)(19,98,191)(20,99,192)(21,100,193)(22,101,194)(23,102,195)(24,103,196)(25,104,197)(26,105,198)(27,106,199)(28,107,200)(29,108,201)(30,109,202)(31,110,203)(32,111,204)(33,112,205)(34,113,206)(35,114,207)(36,115,208)(37,116,209)(38,117,210)(39,118,211)(40,119,212)(41,120,213)(42,121,214)(43,122,215)(44,123,216)(45,124,217)(46,125,218)(47,126,219)(48,127,147)(49,128,148)(50,129,149)(51,130,150)(52,131,151)(53,132,152)(54,133,153)(55,134,154)(56,135,155)(57,136,156)(58,137,157)(59,138,158)(60,139,159)(61,140,160)(62,141,161)(63,142,162)(64,143,163)(65,144,164)(66,145,165)(67,146,166)(68,74,167)(69,75,168)(70,76,169)(71,77,170)(72,78,171)(73,79,172), (74,167)(75,168)(76,169)(77,170)(78,171)(79,172)(80,173)(81,174)(82,175)(83,176)(84,177)(85,178)(86,179)(87,180)(88,181)(89,182)(90,183)(91,184)(92,185)(93,186)(94,187)(95,188)(96,189)(97,190)(98,191)(99,192)(100,193)(101,194)(102,195)(103,196)(104,197)(105,198)(106,199)(107,200)(108,201)(109,202)(110,203)(111,204)(112,205)(113,206)(114,207)(115,208)(116,209)(117,210)(118,211)(119,212)(120,213)(121,214)(122,215)(123,216)(124,217)(125,218)(126,219)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)(141,161)(142,162)(143,163)(144,164)(145,165)(146,166) );
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)], [(1,80,173),(2,81,174),(3,82,175),(4,83,176),(5,84,177),(6,85,178),(7,86,179),(8,87,180),(9,88,181),(10,89,182),(11,90,183),(12,91,184),(13,92,185),(14,93,186),(15,94,187),(16,95,188),(17,96,189),(18,97,190),(19,98,191),(20,99,192),(21,100,193),(22,101,194),(23,102,195),(24,103,196),(25,104,197),(26,105,198),(27,106,199),(28,107,200),(29,108,201),(30,109,202),(31,110,203),(32,111,204),(33,112,205),(34,113,206),(35,114,207),(36,115,208),(37,116,209),(38,117,210),(39,118,211),(40,119,212),(41,120,213),(42,121,214),(43,122,215),(44,123,216),(45,124,217),(46,125,218),(47,126,219),(48,127,147),(49,128,148),(50,129,149),(51,130,150),(52,131,151),(53,132,152),(54,133,153),(55,134,154),(56,135,155),(57,136,156),(58,137,157),(59,138,158),(60,139,159),(61,140,160),(62,141,161),(63,142,162),(64,143,163),(65,144,164),(66,145,165),(67,146,166),(68,74,167),(69,75,168),(70,76,169),(71,77,170),(72,78,171),(73,79,172)], [(74,167),(75,168),(76,169),(77,170),(78,171),(79,172),(80,173),(81,174),(82,175),(83,176),(84,177),(85,178),(86,179),(87,180),(88,181),(89,182),(90,183),(91,184),(92,185),(93,186),(94,187),(95,188),(96,189),(97,190),(98,191),(99,192),(100,193),(101,194),(102,195),(103,196),(104,197),(105,198),(106,199),(107,200),(108,201),(109,202),(110,203),(111,204),(112,205),(113,206),(114,207),(115,208),(116,209),(117,210),(118,211),(119,212),(120,213),(121,214),(122,215),(123,216),(124,217),(125,218),(126,219),(127,147),(128,148),(129,149),(130,150),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160),(141,161),(142,162),(143,163),(144,164),(145,165),(146,166)]])
219 conjugacy classes
| class | 1 | 2 | 3 | 73A | ··· | 73BT | 146A | ··· | 146BT | 219A | ··· | 219BT |
| order | 1 | 2 | 3 | 73 | ··· | 73 | 146 | ··· | 146 | 219 | ··· | 219 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
219 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C73 | C146 | S3 | S3×C73 |
| kernel | S3×C73 | C219 | S3 | C3 | C73 | C1 |
| # reps | 1 | 1 | 72 | 72 | 1 | 72 |
Matrix representation of S3×C73 ►in GL2(𝔽439) generated by
| 433 | 0 |
| 0 | 433 |
| 0 | 438 |
| 1 | 438 |
| 1 | 438 |
| 0 | 438 |
G:=sub<GL(2,GF(439))| [433,0,0,433],[0,1,438,438],[1,0,438,438] >;
S3×C73 in GAP, Magma, Sage, TeX
S_3\times C_{73} % in TeX
G:=Group("S3xC73"); // GroupNames label
G:=SmallGroup(438,3);
// by ID
G=gap.SmallGroup(438,3);
# by ID
G:=PCGroup([3,-2,-73,-3,2630]);
// Polycyclic
G:=Group<a,b,c|a^73=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export