direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C43, C4.C86, C172.3C2, C86.7C22, C2.2(C2×C86), SmallGroup(344,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C43
G = < a,b,c | a43=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C43
►Regular action on 344 points
Generators in S344
(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)(259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301)(302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344)
(1 250 174 314)(2 251 175 315)(3 252 176 316)(4 253 177 317)(5 254 178 318)(6 255 179 319)(7 256 180 320)(8 257 181 321)(9 258 182 322)(10 216 183 323)(11 217 184 324)(12 218 185 325)(13 219 186 326)(14 220 187 327)(15 221 188 328)(16 222 189 329)(17 223 190 330)(18 224 191 331)(19 225 192 332)(20 226 193 333)(21 227 194 334)(22 228 195 335)(23 229 196 336)(24 230 197 337)(25 231 198 338)(26 232 199 339)(27 233 200 340)(28 234 201 341)(29 235 202 342)(30 236 203 343)(31 237 204 344)(32 238 205 302)(33 239 206 303)(34 240 207 304)(35 241 208 305)(36 242 209 306)(37 243 210 307)(38 244 211 308)(39 245 212 309)(40 246 213 310)(41 247 214 311)(42 248 215 312)(43 249 173 313)(44 126 167 282)(45 127 168 283)(46 128 169 284)(47 129 170 285)(48 87 171 286)(49 88 172 287)(50 89 130 288)(51 90 131 289)(52 91 132 290)(53 92 133 291)(54 93 134 292)(55 94 135 293)(56 95 136 294)(57 96 137 295)(58 97 138 296)(59 98 139 297)(60 99 140 298)(61 100 141 299)(62 101 142 300)(63 102 143 301)(64 103 144 259)(65 104 145 260)(66 105 146 261)(67 106 147 262)(68 107 148 263)(69 108 149 264)(70 109 150 265)(71 110 151 266)(72 111 152 267)(73 112 153 268)(74 113 154 269)(75 114 155 270)(76 115 156 271)(77 116 157 272)(78 117 158 273)(79 118 159 274)(80 119 160 275)(81 120 161 276)(82 121 162 277)(83 122 163 278)(84 123 164 279)(85 124 165 280)(86 125 166 281)
(1 70 174 150)(2 71 175 151)(3 72 176 152)(4 73 177 153)(5 74 178 154)(6 75 179 155)(7 76 180 156)(8 77 181 157)(9 78 182 158)(10 79 183 159)(11 80 184 160)(12 81 185 161)(13 82 186 162)(14 83 187 163)(15 84 188 164)(16 85 189 165)(17 86 190 166)(18 44 191 167)(19 45 192 168)(20 46 193 169)(21 47 194 170)(22 48 195 171)(23 49 196 172)(24 50 197 130)(25 51 198 131)(26 52 199 132)(27 53 200 133)(28 54 201 134)(29 55 202 135)(30 56 203 136)(31 57 204 137)(32 58 205 138)(33 59 206 139)(34 60 207 140)(35 61 208 141)(36 62 209 142)(37 63 210 143)(38 64 211 144)(39 65 212 145)(40 66 213 146)(41 67 214 147)(42 68 215 148)(43 69 173 149)(87 228 286 335)(88 229 287 336)(89 230 288 337)(90 231 289 338)(91 232 290 339)(92 233 291 340)(93 234 292 341)(94 235 293 342)(95 236 294 343)(96 237 295 344)(97 238 296 302)(98 239 297 303)(99 240 298 304)(100 241 299 305)(101 242 300 306)(102 243 301 307)(103 244 259 308)(104 245 260 309)(105 246 261 310)(106 247 262 311)(107 248 263 312)(108 249 264 313)(109 250 265 314)(110 251 266 315)(111 252 267 316)(112 253 268 317)(113 254 269 318)(114 255 270 319)(115 256 271 320)(116 257 272 321)(117 258 273 322)(118 216 274 323)(119 217 275 324)(120 218 276 325)(121 219 277 326)(122 220 278 327)(123 221 279 328)(124 222 280 329)(125 223 281 330)(126 224 282 331)(127 225 283 332)(128 226 284 333)(129 227 285 334)
G:=sub<Sym(344)| (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)(259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301)(302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344), (1,250,174,314)(2,251,175,315)(3,252,176,316)(4,253,177,317)(5,254,178,318)(6,255,179,319)(7,256,180,320)(8,257,181,321)(9,258,182,322)(10,216,183,323)(11,217,184,324)(12,218,185,325)(13,219,186,326)(14,220,187,327)(15,221,188,328)(16,222,189,329)(17,223,190,330)(18,224,191,331)(19,225,192,332)(20,226,193,333)(21,227,194,334)(22,228,195,335)(23,229,196,336)(24,230,197,337)(25,231,198,338)(26,232,199,339)(27,233,200,340)(28,234,201,341)(29,235,202,342)(30,236,203,343)(31,237,204,344)(32,238,205,302)(33,239,206,303)(34,240,207,304)(35,241,208,305)(36,242,209,306)(37,243,210,307)(38,244,211,308)(39,245,212,309)(40,246,213,310)(41,247,214,311)(42,248,215,312)(43,249,173,313)(44,126,167,282)(45,127,168,283)(46,128,169,284)(47,129,170,285)(48,87,171,286)(49,88,172,287)(50,89,130,288)(51,90,131,289)(52,91,132,290)(53,92,133,291)(54,93,134,292)(55,94,135,293)(56,95,136,294)(57,96,137,295)(58,97,138,296)(59,98,139,297)(60,99,140,298)(61,100,141,299)(62,101,142,300)(63,102,143,301)(64,103,144,259)(65,104,145,260)(66,105,146,261)(67,106,147,262)(68,107,148,263)(69,108,149,264)(70,109,150,265)(71,110,151,266)(72,111,152,267)(73,112,153,268)(74,113,154,269)(75,114,155,270)(76,115,156,271)(77,116,157,272)(78,117,158,273)(79,118,159,274)(80,119,160,275)(81,120,161,276)(82,121,162,277)(83,122,163,278)(84,123,164,279)(85,124,165,280)(86,125,166,281), (1,70,174,150)(2,71,175,151)(3,72,176,152)(4,73,177,153)(5,74,178,154)(6,75,179,155)(7,76,180,156)(8,77,181,157)(9,78,182,158)(10,79,183,159)(11,80,184,160)(12,81,185,161)(13,82,186,162)(14,83,187,163)(15,84,188,164)(16,85,189,165)(17,86,190,166)(18,44,191,167)(19,45,192,168)(20,46,193,169)(21,47,194,170)(22,48,195,171)(23,49,196,172)(24,50,197,130)(25,51,198,131)(26,52,199,132)(27,53,200,133)(28,54,201,134)(29,55,202,135)(30,56,203,136)(31,57,204,137)(32,58,205,138)(33,59,206,139)(34,60,207,140)(35,61,208,141)(36,62,209,142)(37,63,210,143)(38,64,211,144)(39,65,212,145)(40,66,213,146)(41,67,214,147)(42,68,215,148)(43,69,173,149)(87,228,286,335)(88,229,287,336)(89,230,288,337)(90,231,289,338)(91,232,290,339)(92,233,291,340)(93,234,292,341)(94,235,293,342)(95,236,294,343)(96,237,295,344)(97,238,296,302)(98,239,297,303)(99,240,298,304)(100,241,299,305)(101,242,300,306)(102,243,301,307)(103,244,259,308)(104,245,260,309)(105,246,261,310)(106,247,262,311)(107,248,263,312)(108,249,264,313)(109,250,265,314)(110,251,266,315)(111,252,267,316)(112,253,268,317)(113,254,269,318)(114,255,270,319)(115,256,271,320)(116,257,272,321)(117,258,273,322)(118,216,274,323)(119,217,275,324)(120,218,276,325)(121,219,277,326)(122,220,278,327)(123,221,279,328)(124,222,280,329)(125,223,281,330)(126,224,282,331)(127,225,283,332)(128,226,284,333)(129,227,285,334)>;
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)(259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301)(302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344), (1,250,174,314)(2,251,175,315)(3,252,176,316)(4,253,177,317)(5,254,178,318)(6,255,179,319)(7,256,180,320)(8,257,181,321)(9,258,182,322)(10,216,183,323)(11,217,184,324)(12,218,185,325)(13,219,186,326)(14,220,187,327)(15,221,188,328)(16,222,189,329)(17,223,190,330)(18,224,191,331)(19,225,192,332)(20,226,193,333)(21,227,194,334)(22,228,195,335)(23,229,196,336)(24,230,197,337)(25,231,198,338)(26,232,199,339)(27,233,200,340)(28,234,201,341)(29,235,202,342)(30,236,203,343)(31,237,204,344)(32,238,205,302)(33,239,206,303)(34,240,207,304)(35,241,208,305)(36,242,209,306)(37,243,210,307)(38,244,211,308)(39,245,212,309)(40,246,213,310)(41,247,214,311)(42,248,215,312)(43,249,173,313)(44,126,167,282)(45,127,168,283)(46,128,169,284)(47,129,170,285)(48,87,171,286)(49,88,172,287)(50,89,130,288)(51,90,131,289)(52,91,132,290)(53,92,133,291)(54,93,134,292)(55,94,135,293)(56,95,136,294)(57,96,137,295)(58,97,138,296)(59,98,139,297)(60,99,140,298)(61,100,141,299)(62,101,142,300)(63,102,143,301)(64,103,144,259)(65,104,145,260)(66,105,146,261)(67,106,147,262)(68,107,148,263)(69,108,149,264)(70,109,150,265)(71,110,151,266)(72,111,152,267)(73,112,153,268)(74,113,154,269)(75,114,155,270)(76,115,156,271)(77,116,157,272)(78,117,158,273)(79,118,159,274)(80,119,160,275)(81,120,161,276)(82,121,162,277)(83,122,163,278)(84,123,164,279)(85,124,165,280)(86,125,166,281), (1,70,174,150)(2,71,175,151)(3,72,176,152)(4,73,177,153)(5,74,178,154)(6,75,179,155)(7,76,180,156)(8,77,181,157)(9,78,182,158)(10,79,183,159)(11,80,184,160)(12,81,185,161)(13,82,186,162)(14,83,187,163)(15,84,188,164)(16,85,189,165)(17,86,190,166)(18,44,191,167)(19,45,192,168)(20,46,193,169)(21,47,194,170)(22,48,195,171)(23,49,196,172)(24,50,197,130)(25,51,198,131)(26,52,199,132)(27,53,200,133)(28,54,201,134)(29,55,202,135)(30,56,203,136)(31,57,204,137)(32,58,205,138)(33,59,206,139)(34,60,207,140)(35,61,208,141)(36,62,209,142)(37,63,210,143)(38,64,211,144)(39,65,212,145)(40,66,213,146)(41,67,214,147)(42,68,215,148)(43,69,173,149)(87,228,286,335)(88,229,287,336)(89,230,288,337)(90,231,289,338)(91,232,290,339)(92,233,291,340)(93,234,292,341)(94,235,293,342)(95,236,294,343)(96,237,295,344)(97,238,296,302)(98,239,297,303)(99,240,298,304)(100,241,299,305)(101,242,300,306)(102,243,301,307)(103,244,259,308)(104,245,260,309)(105,246,261,310)(106,247,262,311)(107,248,263,312)(108,249,264,313)(109,250,265,314)(110,251,266,315)(111,252,267,316)(112,253,268,317)(113,254,269,318)(114,255,270,319)(115,256,271,320)(116,257,272,321)(117,258,273,322)(118,216,274,323)(119,217,275,324)(120,218,276,325)(121,219,277,326)(122,220,278,327)(123,221,279,328)(124,222,280,329)(125,223,281,330)(126,224,282,331)(127,225,283,332)(128,226,284,333)(129,227,285,334) );
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),(259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301),(302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344)], [(1,250,174,314),(2,251,175,315),(3,252,176,316),(4,253,177,317),(5,254,178,318),(6,255,179,319),(7,256,180,320),(8,257,181,321),(9,258,182,322),(10,216,183,323),(11,217,184,324),(12,218,185,325),(13,219,186,326),(14,220,187,327),(15,221,188,328),(16,222,189,329),(17,223,190,330),(18,224,191,331),(19,225,192,332),(20,226,193,333),(21,227,194,334),(22,228,195,335),(23,229,196,336),(24,230,197,337),(25,231,198,338),(26,232,199,339),(27,233,200,340),(28,234,201,341),(29,235,202,342),(30,236,203,343),(31,237,204,344),(32,238,205,302),(33,239,206,303),(34,240,207,304),(35,241,208,305),(36,242,209,306),(37,243,210,307),(38,244,211,308),(39,245,212,309),(40,246,213,310),(41,247,214,311),(42,248,215,312),(43,249,173,313),(44,126,167,282),(45,127,168,283),(46,128,169,284),(47,129,170,285),(48,87,171,286),(49,88,172,287),(50,89,130,288),(51,90,131,289),(52,91,132,290),(53,92,133,291),(54,93,134,292),(55,94,135,293),(56,95,136,294),(57,96,137,295),(58,97,138,296),(59,98,139,297),(60,99,140,298),(61,100,141,299),(62,101,142,300),(63,102,143,301),(64,103,144,259),(65,104,145,260),(66,105,146,261),(67,106,147,262),(68,107,148,263),(69,108,149,264),(70,109,150,265),(71,110,151,266),(72,111,152,267),(73,112,153,268),(74,113,154,269),(75,114,155,270),(76,115,156,271),(77,116,157,272),(78,117,158,273),(79,118,159,274),(80,119,160,275),(81,120,161,276),(82,121,162,277),(83,122,163,278),(84,123,164,279),(85,124,165,280),(86,125,166,281)], [(1,70,174,150),(2,71,175,151),(3,72,176,152),(4,73,177,153),(5,74,178,154),(6,75,179,155),(7,76,180,156),(8,77,181,157),(9,78,182,158),(10,79,183,159),(11,80,184,160),(12,81,185,161),(13,82,186,162),(14,83,187,163),(15,84,188,164),(16,85,189,165),(17,86,190,166),(18,44,191,167),(19,45,192,168),(20,46,193,169),(21,47,194,170),(22,48,195,171),(23,49,196,172),(24,50,197,130),(25,51,198,131),(26,52,199,132),(27,53,200,133),(28,54,201,134),(29,55,202,135),(30,56,203,136),(31,57,204,137),(32,58,205,138),(33,59,206,139),(34,60,207,140),(35,61,208,141),(36,62,209,142),(37,63,210,143),(38,64,211,144),(39,65,212,145),(40,66,213,146),(41,67,214,147),(42,68,215,148),(43,69,173,149),(87,228,286,335),(88,229,287,336),(89,230,288,337),(90,231,289,338),(91,232,290,339),(92,233,291,340),(93,234,292,341),(94,235,293,342),(95,236,294,343),(96,237,295,344),(97,238,296,302),(98,239,297,303),(99,240,298,304),(100,241,299,305),(101,242,300,306),(102,243,301,307),(103,244,259,308),(104,245,260,309),(105,246,261,310),(106,247,262,311),(107,248,263,312),(108,249,264,313),(109,250,265,314),(110,251,266,315),(111,252,267,316),(112,253,268,317),(113,254,269,318),(114,255,270,319),(115,256,271,320),(116,257,272,321),(117,258,273,322),(118,216,274,323),(119,217,275,324),(120,218,276,325),(121,219,277,326),(122,220,278,327),(123,221,279,328),(124,222,280,329),(125,223,281,330),(126,224,282,331),(127,225,283,332),(128,226,284,333),(129,227,285,334)]])
215 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 43A | ··· | 43AP | 86A | ··· | 86AP | 172A | ··· | 172DV |
| order | 1 | 2 | 4 | 4 | 4 | 43 | ··· | 43 | 86 | ··· | 86 | 172 | ··· | 172 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
215 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C43 | C86 | Q8 | Q8×C43 |
| kernel | Q8×C43 | C172 | Q8 | C4 | C43 | C1 |
| # reps | 1 | 3 | 42 | 126 | 1 | 42 |
Matrix representation of Q8×C43 ►in GL2(𝔽173) generated by
| 96 | 0 |
| 0 | 96 |
| 1 | 171 |
| 1 | 172 |
| 116 | 51 |
| 55 | 57 |
G:=sub<GL(2,GF(173))| [96,0,0,96],[1,1,171,172],[116,55,51,57] >;
Q8×C43 in GAP, Magma, Sage, TeX
Q_8\times C_{43} % in TeX
G:=Group("Q8xC43"); // GroupNames label
G:=SmallGroup(344,10);
// by ID
G=gap.SmallGroup(344,10);
# by ID
G:=PCGroup([4,-2,-2,-43,-2,688,1393,693]);
// Polycyclic
G:=Group<a,b,c|a^43=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export