direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C41, C4.C82, C164.3C2, C82.7C22, C2.2(C2×C82), SmallGroup(328,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C41
G = < a,b,c | a41=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C41
►Regular action on 328 points
Generators in S328
(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)
(1 207 102 271)(2 208 103 272)(3 209 104 273)(4 210 105 274)(5 211 106 275)(6 212 107 276)(7 213 108 277)(8 214 109 278)(9 215 110 279)(10 216 111 280)(11 217 112 281)(12 218 113 282)(13 219 114 283)(14 220 115 284)(15 221 116 285)(16 222 117 286)(17 223 118 287)(18 224 119 247)(19 225 120 248)(20 226 121 249)(21 227 122 250)(22 228 123 251)(23 229 83 252)(24 230 84 253)(25 231 85 254)(26 232 86 255)(27 233 87 256)(28 234 88 257)(29 235 89 258)(30 236 90 259)(31 237 91 260)(32 238 92 261)(33 239 93 262)(34 240 94 263)(35 241 95 264)(36 242 96 265)(37 243 97 266)(38 244 98 267)(39 245 99 268)(40 246 100 269)(41 206 101 270)(42 294 188 150)(43 295 189 151)(44 296 190 152)(45 297 191 153)(46 298 192 154)(47 299 193 155)(48 300 194 156)(49 301 195 157)(50 302 196 158)(51 303 197 159)(52 304 198 160)(53 305 199 161)(54 306 200 162)(55 307 201 163)(56 308 202 164)(57 309 203 124)(58 310 204 125)(59 311 205 126)(60 312 165 127)(61 313 166 128)(62 314 167 129)(63 315 168 130)(64 316 169 131)(65 317 170 132)(66 318 171 133)(67 319 172 134)(68 320 173 135)(69 321 174 136)(70 322 175 137)(71 323 176 138)(72 324 177 139)(73 325 178 140)(74 326 179 141)(75 327 180 142)(76 328 181 143)(77 288 182 144)(78 289 183 145)(79 290 184 146)(80 291 185 147)(81 292 186 148)(82 293 187 149)
(1 183 102 78)(2 184 103 79)(3 185 104 80)(4 186 105 81)(5 187 106 82)(6 188 107 42)(7 189 108 43)(8 190 109 44)(9 191 110 45)(10 192 111 46)(11 193 112 47)(12 194 113 48)(13 195 114 49)(14 196 115 50)(15 197 116 51)(16 198 117 52)(17 199 118 53)(18 200 119 54)(19 201 120 55)(20 202 121 56)(21 203 122 57)(22 204 123 58)(23 205 83 59)(24 165 84 60)(25 166 85 61)(26 167 86 62)(27 168 87 63)(28 169 88 64)(29 170 89 65)(30 171 90 66)(31 172 91 67)(32 173 92 68)(33 174 93 69)(34 175 94 70)(35 176 95 71)(36 177 96 72)(37 178 97 73)(38 179 98 74)(39 180 99 75)(40 181 100 76)(41 182 101 77)(124 227 309 250)(125 228 310 251)(126 229 311 252)(127 230 312 253)(128 231 313 254)(129 232 314 255)(130 233 315 256)(131 234 316 257)(132 235 317 258)(133 236 318 259)(134 237 319 260)(135 238 320 261)(136 239 321 262)(137 240 322 263)(138 241 323 264)(139 242 324 265)(140 243 325 266)(141 244 326 267)(142 245 327 268)(143 246 328 269)(144 206 288 270)(145 207 289 271)(146 208 290 272)(147 209 291 273)(148 210 292 274)(149 211 293 275)(150 212 294 276)(151 213 295 277)(152 214 296 278)(153 215 297 279)(154 216 298 280)(155 217 299 281)(156 218 300 282)(157 219 301 283)(158 220 302 284)(159 221 303 285)(160 222 304 286)(161 223 305 287)(162 224 306 247)(163 225 307 248)(164 226 308 249)
G:=sub<Sym(328)| (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), (1,207,102,271)(2,208,103,272)(3,209,104,273)(4,210,105,274)(5,211,106,275)(6,212,107,276)(7,213,108,277)(8,214,109,278)(9,215,110,279)(10,216,111,280)(11,217,112,281)(12,218,113,282)(13,219,114,283)(14,220,115,284)(15,221,116,285)(16,222,117,286)(17,223,118,287)(18,224,119,247)(19,225,120,248)(20,226,121,249)(21,227,122,250)(22,228,123,251)(23,229,83,252)(24,230,84,253)(25,231,85,254)(26,232,86,255)(27,233,87,256)(28,234,88,257)(29,235,89,258)(30,236,90,259)(31,237,91,260)(32,238,92,261)(33,239,93,262)(34,240,94,263)(35,241,95,264)(36,242,96,265)(37,243,97,266)(38,244,98,267)(39,245,99,268)(40,246,100,269)(41,206,101,270)(42,294,188,150)(43,295,189,151)(44,296,190,152)(45,297,191,153)(46,298,192,154)(47,299,193,155)(48,300,194,156)(49,301,195,157)(50,302,196,158)(51,303,197,159)(52,304,198,160)(53,305,199,161)(54,306,200,162)(55,307,201,163)(56,308,202,164)(57,309,203,124)(58,310,204,125)(59,311,205,126)(60,312,165,127)(61,313,166,128)(62,314,167,129)(63,315,168,130)(64,316,169,131)(65,317,170,132)(66,318,171,133)(67,319,172,134)(68,320,173,135)(69,321,174,136)(70,322,175,137)(71,323,176,138)(72,324,177,139)(73,325,178,140)(74,326,179,141)(75,327,180,142)(76,328,181,143)(77,288,182,144)(78,289,183,145)(79,290,184,146)(80,291,185,147)(81,292,186,148)(82,293,187,149), (1,183,102,78)(2,184,103,79)(3,185,104,80)(4,186,105,81)(5,187,106,82)(6,188,107,42)(7,189,108,43)(8,190,109,44)(9,191,110,45)(10,192,111,46)(11,193,112,47)(12,194,113,48)(13,195,114,49)(14,196,115,50)(15,197,116,51)(16,198,117,52)(17,199,118,53)(18,200,119,54)(19,201,120,55)(20,202,121,56)(21,203,122,57)(22,204,123,58)(23,205,83,59)(24,165,84,60)(25,166,85,61)(26,167,86,62)(27,168,87,63)(28,169,88,64)(29,170,89,65)(30,171,90,66)(31,172,91,67)(32,173,92,68)(33,174,93,69)(34,175,94,70)(35,176,95,71)(36,177,96,72)(37,178,97,73)(38,179,98,74)(39,180,99,75)(40,181,100,76)(41,182,101,77)(124,227,309,250)(125,228,310,251)(126,229,311,252)(127,230,312,253)(128,231,313,254)(129,232,314,255)(130,233,315,256)(131,234,316,257)(132,235,317,258)(133,236,318,259)(134,237,319,260)(135,238,320,261)(136,239,321,262)(137,240,322,263)(138,241,323,264)(139,242,324,265)(140,243,325,266)(141,244,326,267)(142,245,327,268)(143,246,328,269)(144,206,288,270)(145,207,289,271)(146,208,290,272)(147,209,291,273)(148,210,292,274)(149,211,293,275)(150,212,294,276)(151,213,295,277)(152,214,296,278)(153,215,297,279)(154,216,298,280)(155,217,299,281)(156,218,300,282)(157,219,301,283)(158,220,302,284)(159,221,303,285)(160,222,304,286)(161,223,305,287)(162,224,306,247)(163,225,307,248)(164,226,308,249)>;
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), (1,207,102,271)(2,208,103,272)(3,209,104,273)(4,210,105,274)(5,211,106,275)(6,212,107,276)(7,213,108,277)(8,214,109,278)(9,215,110,279)(10,216,111,280)(11,217,112,281)(12,218,113,282)(13,219,114,283)(14,220,115,284)(15,221,116,285)(16,222,117,286)(17,223,118,287)(18,224,119,247)(19,225,120,248)(20,226,121,249)(21,227,122,250)(22,228,123,251)(23,229,83,252)(24,230,84,253)(25,231,85,254)(26,232,86,255)(27,233,87,256)(28,234,88,257)(29,235,89,258)(30,236,90,259)(31,237,91,260)(32,238,92,261)(33,239,93,262)(34,240,94,263)(35,241,95,264)(36,242,96,265)(37,243,97,266)(38,244,98,267)(39,245,99,268)(40,246,100,269)(41,206,101,270)(42,294,188,150)(43,295,189,151)(44,296,190,152)(45,297,191,153)(46,298,192,154)(47,299,193,155)(48,300,194,156)(49,301,195,157)(50,302,196,158)(51,303,197,159)(52,304,198,160)(53,305,199,161)(54,306,200,162)(55,307,201,163)(56,308,202,164)(57,309,203,124)(58,310,204,125)(59,311,205,126)(60,312,165,127)(61,313,166,128)(62,314,167,129)(63,315,168,130)(64,316,169,131)(65,317,170,132)(66,318,171,133)(67,319,172,134)(68,320,173,135)(69,321,174,136)(70,322,175,137)(71,323,176,138)(72,324,177,139)(73,325,178,140)(74,326,179,141)(75,327,180,142)(76,328,181,143)(77,288,182,144)(78,289,183,145)(79,290,184,146)(80,291,185,147)(81,292,186,148)(82,293,187,149), (1,183,102,78)(2,184,103,79)(3,185,104,80)(4,186,105,81)(5,187,106,82)(6,188,107,42)(7,189,108,43)(8,190,109,44)(9,191,110,45)(10,192,111,46)(11,193,112,47)(12,194,113,48)(13,195,114,49)(14,196,115,50)(15,197,116,51)(16,198,117,52)(17,199,118,53)(18,200,119,54)(19,201,120,55)(20,202,121,56)(21,203,122,57)(22,204,123,58)(23,205,83,59)(24,165,84,60)(25,166,85,61)(26,167,86,62)(27,168,87,63)(28,169,88,64)(29,170,89,65)(30,171,90,66)(31,172,91,67)(32,173,92,68)(33,174,93,69)(34,175,94,70)(35,176,95,71)(36,177,96,72)(37,178,97,73)(38,179,98,74)(39,180,99,75)(40,181,100,76)(41,182,101,77)(124,227,309,250)(125,228,310,251)(126,229,311,252)(127,230,312,253)(128,231,313,254)(129,232,314,255)(130,233,315,256)(131,234,316,257)(132,235,317,258)(133,236,318,259)(134,237,319,260)(135,238,320,261)(136,239,321,262)(137,240,322,263)(138,241,323,264)(139,242,324,265)(140,243,325,266)(141,244,326,267)(142,245,327,268)(143,246,328,269)(144,206,288,270)(145,207,289,271)(146,208,290,272)(147,209,291,273)(148,210,292,274)(149,211,293,275)(150,212,294,276)(151,213,295,277)(152,214,296,278)(153,215,297,279)(154,216,298,280)(155,217,299,281)(156,218,300,282)(157,219,301,283)(158,220,302,284)(159,221,303,285)(160,222,304,286)(161,223,305,287)(162,224,306,247)(163,225,307,248)(164,226,308,249) );
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)], [(1,207,102,271),(2,208,103,272),(3,209,104,273),(4,210,105,274),(5,211,106,275),(6,212,107,276),(7,213,108,277),(8,214,109,278),(9,215,110,279),(10,216,111,280),(11,217,112,281),(12,218,113,282),(13,219,114,283),(14,220,115,284),(15,221,116,285),(16,222,117,286),(17,223,118,287),(18,224,119,247),(19,225,120,248),(20,226,121,249),(21,227,122,250),(22,228,123,251),(23,229,83,252),(24,230,84,253),(25,231,85,254),(26,232,86,255),(27,233,87,256),(28,234,88,257),(29,235,89,258),(30,236,90,259),(31,237,91,260),(32,238,92,261),(33,239,93,262),(34,240,94,263),(35,241,95,264),(36,242,96,265),(37,243,97,266),(38,244,98,267),(39,245,99,268),(40,246,100,269),(41,206,101,270),(42,294,188,150),(43,295,189,151),(44,296,190,152),(45,297,191,153),(46,298,192,154),(47,299,193,155),(48,300,194,156),(49,301,195,157),(50,302,196,158),(51,303,197,159),(52,304,198,160),(53,305,199,161),(54,306,200,162),(55,307,201,163),(56,308,202,164),(57,309,203,124),(58,310,204,125),(59,311,205,126),(60,312,165,127),(61,313,166,128),(62,314,167,129),(63,315,168,130),(64,316,169,131),(65,317,170,132),(66,318,171,133),(67,319,172,134),(68,320,173,135),(69,321,174,136),(70,322,175,137),(71,323,176,138),(72,324,177,139),(73,325,178,140),(74,326,179,141),(75,327,180,142),(76,328,181,143),(77,288,182,144),(78,289,183,145),(79,290,184,146),(80,291,185,147),(81,292,186,148),(82,293,187,149)], [(1,183,102,78),(2,184,103,79),(3,185,104,80),(4,186,105,81),(5,187,106,82),(6,188,107,42),(7,189,108,43),(8,190,109,44),(9,191,110,45),(10,192,111,46),(11,193,112,47),(12,194,113,48),(13,195,114,49),(14,196,115,50),(15,197,116,51),(16,198,117,52),(17,199,118,53),(18,200,119,54),(19,201,120,55),(20,202,121,56),(21,203,122,57),(22,204,123,58),(23,205,83,59),(24,165,84,60),(25,166,85,61),(26,167,86,62),(27,168,87,63),(28,169,88,64),(29,170,89,65),(30,171,90,66),(31,172,91,67),(32,173,92,68),(33,174,93,69),(34,175,94,70),(35,176,95,71),(36,177,96,72),(37,178,97,73),(38,179,98,74),(39,180,99,75),(40,181,100,76),(41,182,101,77),(124,227,309,250),(125,228,310,251),(126,229,311,252),(127,230,312,253),(128,231,313,254),(129,232,314,255),(130,233,315,256),(131,234,316,257),(132,235,317,258),(133,236,318,259),(134,237,319,260),(135,238,320,261),(136,239,321,262),(137,240,322,263),(138,241,323,264),(139,242,324,265),(140,243,325,266),(141,244,326,267),(142,245,327,268),(143,246,328,269),(144,206,288,270),(145,207,289,271),(146,208,290,272),(147,209,291,273),(148,210,292,274),(149,211,293,275),(150,212,294,276),(151,213,295,277),(152,214,296,278),(153,215,297,279),(154,216,298,280),(155,217,299,281),(156,218,300,282),(157,219,301,283),(158,220,302,284),(159,221,303,285),(160,222,304,286),(161,223,305,287),(162,224,306,247),(163,225,307,248),(164,226,308,249)]])
205 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 41A | ··· | 41AN | 82A | ··· | 82AN | 164A | ··· | 164DP |
| order | 1 | 2 | 4 | 4 | 4 | 41 | ··· | 41 | 82 | ··· | 82 | 164 | ··· | 164 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
205 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C41 | C82 | Q8 | Q8×C41 |
| kernel | Q8×C41 | C164 | Q8 | C4 | C41 | C1 |
| # reps | 1 | 3 | 40 | 120 | 1 | 40 |
Matrix representation of Q8×C41 ►in GL2(𝔽821) generated by
| 165 | 0 |
| 0 | 165 |
| 447 | 819 |
| 564 | 374 |
| 249 | 486 |
| 413 | 572 |
G:=sub<GL(2,GF(821))| [165,0,0,165],[447,564,819,374],[249,413,486,572] >;
Q8×C41 in GAP, Magma, Sage, TeX
Q_8\times C_{41} % in TeX
G:=Group("Q8xC41"); // GroupNames label
G:=SmallGroup(328,11);
// by ID
G=gap.SmallGroup(328,11);
# by ID
G:=PCGroup([4,-2,-2,-41,-2,656,1329,661]);
// Polycyclic
G:=Group<a,b,c|a^41=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export