direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C77, C3⋊C154, C231⋊7C2, C33⋊3C14, C21⋊3C22, SmallGroup(462,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C77 |
Generators and relations for S3×C77
G = < a,b,c | a77=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C77
►On 231 points
Generators in S231
(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)
(1 143 228)(2 144 229)(3 145 230)(4 146 231)(5 147 155)(6 148 156)(7 149 157)(8 150 158)(9 151 159)(10 152 160)(11 153 161)(12 154 162)(13 78 163)(14 79 164)(15 80 165)(16 81 166)(17 82 167)(18 83 168)(19 84 169)(20 85 170)(21 86 171)(22 87 172)(23 88 173)(24 89 174)(25 90 175)(26 91 176)(27 92 177)(28 93 178)(29 94 179)(30 95 180)(31 96 181)(32 97 182)(33 98 183)(34 99 184)(35 100 185)(36 101 186)(37 102 187)(38 103 188)(39 104 189)(40 105 190)(41 106 191)(42 107 192)(43 108 193)(44 109 194)(45 110 195)(46 111 196)(47 112 197)(48 113 198)(49 114 199)(50 115 200)(51 116 201)(52 117 202)(53 118 203)(54 119 204)(55 120 205)(56 121 206)(57 122 207)(58 123 208)(59 124 209)(60 125 210)(61 126 211)(62 127 212)(63 128 213)(64 129 214)(65 130 215)(66 131 216)(67 132 217)(68 133 218)(69 134 219)(70 135 220)(71 136 221)(72 137 222)(73 138 223)(74 139 224)(75 140 225)(76 141 226)(77 142 227)
(78 163)(79 164)(80 165)(81 166)(82 167)(83 168)(84 169)(85 170)(86 171)(87 172)(88 173)(89 174)(90 175)(91 176)(92 177)(93 178)(94 179)(95 180)(96 181)(97 182)(98 183)(99 184)(100 185)(101 186)(102 187)(103 188)(104 189)(105 190)(106 191)(107 192)(108 193)(109 194)(110 195)(111 196)(112 197)(113 198)(114 199)(115 200)(116 201)(117 202)(118 203)(119 204)(120 205)(121 206)(122 207)(123 208)(124 209)(125 210)(126 211)(127 212)(128 213)(129 214)(130 215)(131 216)(132 217)(133 218)(134 219)(135 220)(136 221)(137 222)(138 223)(139 224)(140 225)(141 226)(142 227)(143 228)(144 229)(145 230)(146 231)(147 155)(148 156)(149 157)(150 158)(151 159)(152 160)(153 161)(154 162)
G:=sub<Sym(231)| (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), (1,143,228)(2,144,229)(3,145,230)(4,146,231)(5,147,155)(6,148,156)(7,149,157)(8,150,158)(9,151,159)(10,152,160)(11,153,161)(12,154,162)(13,78,163)(14,79,164)(15,80,165)(16,81,166)(17,82,167)(18,83,168)(19,84,169)(20,85,170)(21,86,171)(22,87,172)(23,88,173)(24,89,174)(25,90,175)(26,91,176)(27,92,177)(28,93,178)(29,94,179)(30,95,180)(31,96,181)(32,97,182)(33,98,183)(34,99,184)(35,100,185)(36,101,186)(37,102,187)(38,103,188)(39,104,189)(40,105,190)(41,106,191)(42,107,192)(43,108,193)(44,109,194)(45,110,195)(46,111,196)(47,112,197)(48,113,198)(49,114,199)(50,115,200)(51,116,201)(52,117,202)(53,118,203)(54,119,204)(55,120,205)(56,121,206)(57,122,207)(58,123,208)(59,124,209)(60,125,210)(61,126,211)(62,127,212)(63,128,213)(64,129,214)(65,130,215)(66,131,216)(67,132,217)(68,133,218)(69,134,219)(70,135,220)(71,136,221)(72,137,222)(73,138,223)(74,139,224)(75,140,225)(76,141,226)(77,142,227), (78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,169)(85,170)(86,171)(87,172)(88,173)(89,174)(90,175)(91,176)(92,177)(93,178)(94,179)(95,180)(96,181)(97,182)(98,183)(99,184)(100,185)(101,186)(102,187)(103,188)(104,189)(105,190)(106,191)(107,192)(108,193)(109,194)(110,195)(111,196)(112,197)(113,198)(114,199)(115,200)(116,201)(117,202)(118,203)(119,204)(120,205)(121,206)(122,207)(123,208)(124,209)(125,210)(126,211)(127,212)(128,213)(129,214)(130,215)(131,216)(132,217)(133,218)(134,219)(135,220)(136,221)(137,222)(138,223)(139,224)(140,225)(141,226)(142,227)(143,228)(144,229)(145,230)(146,231)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(153,161)(154,162)>;
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), (1,143,228)(2,144,229)(3,145,230)(4,146,231)(5,147,155)(6,148,156)(7,149,157)(8,150,158)(9,151,159)(10,152,160)(11,153,161)(12,154,162)(13,78,163)(14,79,164)(15,80,165)(16,81,166)(17,82,167)(18,83,168)(19,84,169)(20,85,170)(21,86,171)(22,87,172)(23,88,173)(24,89,174)(25,90,175)(26,91,176)(27,92,177)(28,93,178)(29,94,179)(30,95,180)(31,96,181)(32,97,182)(33,98,183)(34,99,184)(35,100,185)(36,101,186)(37,102,187)(38,103,188)(39,104,189)(40,105,190)(41,106,191)(42,107,192)(43,108,193)(44,109,194)(45,110,195)(46,111,196)(47,112,197)(48,113,198)(49,114,199)(50,115,200)(51,116,201)(52,117,202)(53,118,203)(54,119,204)(55,120,205)(56,121,206)(57,122,207)(58,123,208)(59,124,209)(60,125,210)(61,126,211)(62,127,212)(63,128,213)(64,129,214)(65,130,215)(66,131,216)(67,132,217)(68,133,218)(69,134,219)(70,135,220)(71,136,221)(72,137,222)(73,138,223)(74,139,224)(75,140,225)(76,141,226)(77,142,227), (78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,169)(85,170)(86,171)(87,172)(88,173)(89,174)(90,175)(91,176)(92,177)(93,178)(94,179)(95,180)(96,181)(97,182)(98,183)(99,184)(100,185)(101,186)(102,187)(103,188)(104,189)(105,190)(106,191)(107,192)(108,193)(109,194)(110,195)(111,196)(112,197)(113,198)(114,199)(115,200)(116,201)(117,202)(118,203)(119,204)(120,205)(121,206)(122,207)(123,208)(124,209)(125,210)(126,211)(127,212)(128,213)(129,214)(130,215)(131,216)(132,217)(133,218)(134,219)(135,220)(136,221)(137,222)(138,223)(139,224)(140,225)(141,226)(142,227)(143,228)(144,229)(145,230)(146,231)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(153,161)(154,162) );
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)], [(1,143,228),(2,144,229),(3,145,230),(4,146,231),(5,147,155),(6,148,156),(7,149,157),(8,150,158),(9,151,159),(10,152,160),(11,153,161),(12,154,162),(13,78,163),(14,79,164),(15,80,165),(16,81,166),(17,82,167),(18,83,168),(19,84,169),(20,85,170),(21,86,171),(22,87,172),(23,88,173),(24,89,174),(25,90,175),(26,91,176),(27,92,177),(28,93,178),(29,94,179),(30,95,180),(31,96,181),(32,97,182),(33,98,183),(34,99,184),(35,100,185),(36,101,186),(37,102,187),(38,103,188),(39,104,189),(40,105,190),(41,106,191),(42,107,192),(43,108,193),(44,109,194),(45,110,195),(46,111,196),(47,112,197),(48,113,198),(49,114,199),(50,115,200),(51,116,201),(52,117,202),(53,118,203),(54,119,204),(55,120,205),(56,121,206),(57,122,207),(58,123,208),(59,124,209),(60,125,210),(61,126,211),(62,127,212),(63,128,213),(64,129,214),(65,130,215),(66,131,216),(67,132,217),(68,133,218),(69,134,219),(70,135,220),(71,136,221),(72,137,222),(73,138,223),(74,139,224),(75,140,225),(76,141,226),(77,142,227)], [(78,163),(79,164),(80,165),(81,166),(82,167),(83,168),(84,169),(85,170),(86,171),(87,172),(88,173),(89,174),(90,175),(91,176),(92,177),(93,178),(94,179),(95,180),(96,181),(97,182),(98,183),(99,184),(100,185),(101,186),(102,187),(103,188),(104,189),(105,190),(106,191),(107,192),(108,193),(109,194),(110,195),(111,196),(112,197),(113,198),(114,199),(115,200),(116,201),(117,202),(118,203),(119,204),(120,205),(121,206),(122,207),(123,208),(124,209),(125,210),(126,211),(127,212),(128,213),(129,214),(130,215),(131,216),(132,217),(133,218),(134,219),(135,220),(136,221),(137,222),(138,223),(139,224),(140,225),(141,226),(142,227),(143,228),(144,229),(145,230),(146,231),(147,155),(148,156),(149,157),(150,158),(151,159),(152,160),(153,161),(154,162)]])
231 conjugacy classes
| class | 1 | 2 | 3 | 7A | ··· | 7F | 11A | ··· | 11J | 14A | ··· | 14F | 21A | ··· | 21F | 22A | ··· | 22J | 33A | ··· | 33J | 77A | ··· | 77BH | 154A | ··· | 154BH | 231A | ··· | 231BH |
| order | 1 | 2 | 3 | 7 | ··· | 7 | 11 | ··· | 11 | 14 | ··· | 14 | 21 | ··· | 21 | 22 | ··· | 22 | 33 | ··· | 33 | 77 | ··· | 77 | 154 | ··· | 154 | 231 | ··· | 231 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
231 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C7 | C11 | C14 | C22 | C77 | C154 | S3 | S3×C7 | S3×C11 | S3×C77 |
| kernel | S3×C77 | C231 | S3×C11 | S3×C7 | C33 | C21 | S3 | C3 | C77 | C11 | C7 | C1 |
| # reps | 1 | 1 | 6 | 10 | 6 | 10 | 60 | 60 | 1 | 6 | 10 | 60 |
Matrix representation of S3×C77 ►in GL2(𝔽463) generated by
| 159 | 0 |
| 0 | 159 |
| 462 | 462 |
| 1 | 0 |
| 1 | 0 |
| 462 | 462 |
G:=sub<GL(2,GF(463))| [159,0,0,159],[462,1,462,0],[1,462,0,462] >;
S3×C77 in GAP, Magma, Sage, TeX
S_3\times C_{77} % in TeX
G:=Group("S3xC77"); // GroupNames label
G:=SmallGroup(462,8);
// by ID
G=gap.SmallGroup(462,8);
# by ID
G:=PCGroup([4,-2,-7,-11,-3,4931]);
// Polycyclic
G:=Group<a,b,c|a^77=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export