metabelian, supersoluble, monomial, A-group
Aliases: C48.9S3, C32⋊4C32, C24.10Dic3, C3⋊(C3⋊C32), C12.7(C3⋊C8), C6.3(C3⋊C16), (C3×C24).9C4, (C3×C48).5C2, (C3×C6).4C16, C16.2(C3⋊S3), (C3×C12).10C8, C2.(C24.S3), C8.3(C3⋊Dic3), C4.2(C32⋊4C8), SmallGroup(288,65)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C48.S3 |
Generators and relations for C48.S3
G = < a,b,c | a48=b3=1, c2=a33, ab=ba, cac-1=a17, cbc-1=b-1 >
9C32
Smallest permutation representation of C48.S3
►Regular action on 288 points
Generators in S288
(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)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288)
(1 119 275)(2 120 276)(3 121 277)(4 122 278)(5 123 279)(6 124 280)(7 125 281)(8 126 282)(9 127 283)(10 128 284)(11 129 285)(12 130 286)(13 131 287)(14 132 288)(15 133 241)(16 134 242)(17 135 243)(18 136 244)(19 137 245)(20 138 246)(21 139 247)(22 140 248)(23 141 249)(24 142 250)(25 143 251)(26 144 252)(27 97 253)(28 98 254)(29 99 255)(30 100 256)(31 101 257)(32 102 258)(33 103 259)(34 104 260)(35 105 261)(36 106 262)(37 107 263)(38 108 264)(39 109 265)(40 110 266)(41 111 267)(42 112 268)(43 113 269)(44 114 270)(45 115 271)(46 116 272)(47 117 273)(48 118 274)(49 222 175)(50 223 176)(51 224 177)(52 225 178)(53 226 179)(54 227 180)(55 228 181)(56 229 182)(57 230 183)(58 231 184)(59 232 185)(60 233 186)(61 234 187)(62 235 188)(63 236 189)(64 237 190)(65 238 191)(66 239 192)(67 240 145)(68 193 146)(69 194 147)(70 195 148)(71 196 149)(72 197 150)(73 198 151)(74 199 152)(75 200 153)(76 201 154)(77 202 155)(78 203 156)(79 204 157)(80 205 158)(81 206 159)(82 207 160)(83 208 161)(84 209 162)(85 210 163)(86 211 164)(87 212 165)(88 213 166)(89 214 167)(90 215 168)(91 216 169)(92 217 170)(93 218 171)(94 219 172)(95 220 173)(96 221 174)
(1 188 34 173 19 158 4 191 37 176 22 161 7 146 40 179 25 164 10 149 43 182 28 167 13 152 46 185 31 170 16 155)(2 157 35 190 20 175 5 160 38 145 23 178 8 163 41 148 26 181 11 166 44 151 29 184 14 169 47 154 32 187 17 172)(3 174 36 159 21 192 6 177 39 162 24 147 9 180 42 165 27 150 12 183 45 168 30 153 15 186 48 171 33 156 18 189)(49 279 82 264 67 249 52 282 85 267 70 252 55 285 88 270 73 255 58 288 91 273 76 258 61 243 94 276 79 261 64 246)(50 248 83 281 68 266 53 251 86 284 71 269 56 254 89 287 74 272 59 257 92 242 77 275 62 260 95 245 80 278 65 263)(51 265 84 250 69 283 54 268 87 253 72 286 57 271 90 256 75 241 60 274 93 259 78 244 63 277 96 262 81 247 66 280)(97 197 130 230 115 215 100 200 133 233 118 218 103 203 136 236 121 221 106 206 139 239 124 224 109 209 142 194 127 227 112 212)(98 214 131 199 116 232 101 217 134 202 119 235 104 220 137 205 122 238 107 223 140 208 125 193 110 226 143 211 128 196 113 229)(99 231 132 216 117 201 102 234 135 219 120 204 105 237 138 222 123 207 108 240 141 225 126 210 111 195 144 228 129 213 114 198)
G:=sub<Sym(288)| (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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288), (1,119,275)(2,120,276)(3,121,277)(4,122,278)(5,123,279)(6,124,280)(7,125,281)(8,126,282)(9,127,283)(10,128,284)(11,129,285)(12,130,286)(13,131,287)(14,132,288)(15,133,241)(16,134,242)(17,135,243)(18,136,244)(19,137,245)(20,138,246)(21,139,247)(22,140,248)(23,141,249)(24,142,250)(25,143,251)(26,144,252)(27,97,253)(28,98,254)(29,99,255)(30,100,256)(31,101,257)(32,102,258)(33,103,259)(34,104,260)(35,105,261)(36,106,262)(37,107,263)(38,108,264)(39,109,265)(40,110,266)(41,111,267)(42,112,268)(43,113,269)(44,114,270)(45,115,271)(46,116,272)(47,117,273)(48,118,274)(49,222,175)(50,223,176)(51,224,177)(52,225,178)(53,226,179)(54,227,180)(55,228,181)(56,229,182)(57,230,183)(58,231,184)(59,232,185)(60,233,186)(61,234,187)(62,235,188)(63,236,189)(64,237,190)(65,238,191)(66,239,192)(67,240,145)(68,193,146)(69,194,147)(70,195,148)(71,196,149)(72,197,150)(73,198,151)(74,199,152)(75,200,153)(76,201,154)(77,202,155)(78,203,156)(79,204,157)(80,205,158)(81,206,159)(82,207,160)(83,208,161)(84,209,162)(85,210,163)(86,211,164)(87,212,165)(88,213,166)(89,214,167)(90,215,168)(91,216,169)(92,217,170)(93,218,171)(94,219,172)(95,220,173)(96,221,174), (1,188,34,173,19,158,4,191,37,176,22,161,7,146,40,179,25,164,10,149,43,182,28,167,13,152,46,185,31,170,16,155)(2,157,35,190,20,175,5,160,38,145,23,178,8,163,41,148,26,181,11,166,44,151,29,184,14,169,47,154,32,187,17,172)(3,174,36,159,21,192,6,177,39,162,24,147,9,180,42,165,27,150,12,183,45,168,30,153,15,186,48,171,33,156,18,189)(49,279,82,264,67,249,52,282,85,267,70,252,55,285,88,270,73,255,58,288,91,273,76,258,61,243,94,276,79,261,64,246)(50,248,83,281,68,266,53,251,86,284,71,269,56,254,89,287,74,272,59,257,92,242,77,275,62,260,95,245,80,278,65,263)(51,265,84,250,69,283,54,268,87,253,72,286,57,271,90,256,75,241,60,274,93,259,78,244,63,277,96,262,81,247,66,280)(97,197,130,230,115,215,100,200,133,233,118,218,103,203,136,236,121,221,106,206,139,239,124,224,109,209,142,194,127,227,112,212)(98,214,131,199,116,232,101,217,134,202,119,235,104,220,137,205,122,238,107,223,140,208,125,193,110,226,143,211,128,196,113,229)(99,231,132,216,117,201,102,234,135,219,120,204,105,237,138,222,123,207,108,240,141,225,126,210,111,195,144,228,129,213,114,198)>;
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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288), (1,119,275)(2,120,276)(3,121,277)(4,122,278)(5,123,279)(6,124,280)(7,125,281)(8,126,282)(9,127,283)(10,128,284)(11,129,285)(12,130,286)(13,131,287)(14,132,288)(15,133,241)(16,134,242)(17,135,243)(18,136,244)(19,137,245)(20,138,246)(21,139,247)(22,140,248)(23,141,249)(24,142,250)(25,143,251)(26,144,252)(27,97,253)(28,98,254)(29,99,255)(30,100,256)(31,101,257)(32,102,258)(33,103,259)(34,104,260)(35,105,261)(36,106,262)(37,107,263)(38,108,264)(39,109,265)(40,110,266)(41,111,267)(42,112,268)(43,113,269)(44,114,270)(45,115,271)(46,116,272)(47,117,273)(48,118,274)(49,222,175)(50,223,176)(51,224,177)(52,225,178)(53,226,179)(54,227,180)(55,228,181)(56,229,182)(57,230,183)(58,231,184)(59,232,185)(60,233,186)(61,234,187)(62,235,188)(63,236,189)(64,237,190)(65,238,191)(66,239,192)(67,240,145)(68,193,146)(69,194,147)(70,195,148)(71,196,149)(72,197,150)(73,198,151)(74,199,152)(75,200,153)(76,201,154)(77,202,155)(78,203,156)(79,204,157)(80,205,158)(81,206,159)(82,207,160)(83,208,161)(84,209,162)(85,210,163)(86,211,164)(87,212,165)(88,213,166)(89,214,167)(90,215,168)(91,216,169)(92,217,170)(93,218,171)(94,219,172)(95,220,173)(96,221,174), (1,188,34,173,19,158,4,191,37,176,22,161,7,146,40,179,25,164,10,149,43,182,28,167,13,152,46,185,31,170,16,155)(2,157,35,190,20,175,5,160,38,145,23,178,8,163,41,148,26,181,11,166,44,151,29,184,14,169,47,154,32,187,17,172)(3,174,36,159,21,192,6,177,39,162,24,147,9,180,42,165,27,150,12,183,45,168,30,153,15,186,48,171,33,156,18,189)(49,279,82,264,67,249,52,282,85,267,70,252,55,285,88,270,73,255,58,288,91,273,76,258,61,243,94,276,79,261,64,246)(50,248,83,281,68,266,53,251,86,284,71,269,56,254,89,287,74,272,59,257,92,242,77,275,62,260,95,245,80,278,65,263)(51,265,84,250,69,283,54,268,87,253,72,286,57,271,90,256,75,241,60,274,93,259,78,244,63,277,96,262,81,247,66,280)(97,197,130,230,115,215,100,200,133,233,118,218,103,203,136,236,121,221,106,206,139,239,124,224,109,209,142,194,127,227,112,212)(98,214,131,199,116,232,101,217,134,202,119,235,104,220,137,205,122,238,107,223,140,208,125,193,110,226,143,211,128,196,113,229)(99,231,132,216,117,201,102,234,135,219,120,204,105,237,138,222,123,207,108,240,141,225,126,210,111,195,144,228,129,213,114,198) );
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),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288)], [(1,119,275),(2,120,276),(3,121,277),(4,122,278),(5,123,279),(6,124,280),(7,125,281),(8,126,282),(9,127,283),(10,128,284),(11,129,285),(12,130,286),(13,131,287),(14,132,288),(15,133,241),(16,134,242),(17,135,243),(18,136,244),(19,137,245),(20,138,246),(21,139,247),(22,140,248),(23,141,249),(24,142,250),(25,143,251),(26,144,252),(27,97,253),(28,98,254),(29,99,255),(30,100,256),(31,101,257),(32,102,258),(33,103,259),(34,104,260),(35,105,261),(36,106,262),(37,107,263),(38,108,264),(39,109,265),(40,110,266),(41,111,267),(42,112,268),(43,113,269),(44,114,270),(45,115,271),(46,116,272),(47,117,273),(48,118,274),(49,222,175),(50,223,176),(51,224,177),(52,225,178),(53,226,179),(54,227,180),(55,228,181),(56,229,182),(57,230,183),(58,231,184),(59,232,185),(60,233,186),(61,234,187),(62,235,188),(63,236,189),(64,237,190),(65,238,191),(66,239,192),(67,240,145),(68,193,146),(69,194,147),(70,195,148),(71,196,149),(72,197,150),(73,198,151),(74,199,152),(75,200,153),(76,201,154),(77,202,155),(78,203,156),(79,204,157),(80,205,158),(81,206,159),(82,207,160),(83,208,161),(84,209,162),(85,210,163),(86,211,164),(87,212,165),(88,213,166),(89,214,167),(90,215,168),(91,216,169),(92,217,170),(93,218,171),(94,219,172),(95,220,173),(96,221,174)], [(1,188,34,173,19,158,4,191,37,176,22,161,7,146,40,179,25,164,10,149,43,182,28,167,13,152,46,185,31,170,16,155),(2,157,35,190,20,175,5,160,38,145,23,178,8,163,41,148,26,181,11,166,44,151,29,184,14,169,47,154,32,187,17,172),(3,174,36,159,21,192,6,177,39,162,24,147,9,180,42,165,27,150,12,183,45,168,30,153,15,186,48,171,33,156,18,189),(49,279,82,264,67,249,52,282,85,267,70,252,55,285,88,270,73,255,58,288,91,273,76,258,61,243,94,276,79,261,64,246),(50,248,83,281,68,266,53,251,86,284,71,269,56,254,89,287,74,272,59,257,92,242,77,275,62,260,95,245,80,278,65,263),(51,265,84,250,69,283,54,268,87,253,72,286,57,271,90,256,75,241,60,274,93,259,78,244,63,277,96,262,81,247,66,280),(97,197,130,230,115,215,100,200,133,233,118,218,103,203,136,236,121,221,106,206,139,239,124,224,109,209,142,194,127,227,112,212),(98,214,131,199,116,232,101,217,134,202,119,235,104,220,137,205,122,238,107,223,140,208,125,193,110,226,143,211,128,196,113,229),(99,231,132,216,117,201,102,234,135,219,120,204,105,237,138,222,123,207,108,240,141,225,126,210,111,195,144,228,129,213,114,198)]])
96 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 16A | ··· | 16H | 24A | ··· | 24P | 32A | ··· | 32P | 48A | ··· | 48AF |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 32 | ··· | 32 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | S3 | Dic3 | C3⋊C8 | C3⋊C16 | C3⋊C32 |
| kernel | C48.S3 | C3×C48 | C3×C24 | C3×C12 | C3×C6 | C32 | C48 | C24 | C12 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 4 | 4 | 8 | 16 | 32 |
Matrix representation of C48.S3 ►in GL4(𝔽97) generated by
| 32 | 0 | 0 | 0 |
| 0 | 53 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 5 | 86 |
| 35 | 0 | 0 | 0 |
| 0 | 61 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 39 | 46 |
| 0 | 0 | 62 | 58 |
G:=sub<GL(4,GF(97))| [32,0,0,0,0,53,0,0,0,0,3,5,0,0,0,86],[35,0,0,0,0,61,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,39,62,0,0,46,58] >;
C48.S3 in GAP, Magma, Sage, TeX
C_{48}.S_3 % in TeX
G:=Group("C48.S3"); // GroupNames label
G:=SmallGroup(288,65);
// by ID
G=gap.SmallGroup(288,65);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,14,36,58,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^48=b^3=1,c^2=a^33,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=b^-1>;
// generators/relations
Export