Copied to
clipboard

G = Q8×C41order 328 = 23·41

Direct product of C41 and Q8

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

C1C2 — Q8×C41
C1C2C82C164 — Q8×C41
C1C2 — Q8×C41
C1C82 — Q8×C41

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)
(1 303 183 270)(2 304 184 271)(3 305 185 272)(4 306 186 273)(5 307 187 274)(6 308 188 275)(7 309 189 276)(8 310 190 277)(9 311 191 278)(10 312 192 279)(11 313 193 280)(12 314 194 281)(13 315 195 282)(14 316 196 283)(15 317 197 284)(16 318 198 285)(17 319 199 286)(18 320 200 287)(19 321 201 247)(20 322 202 248)(21 323 203 249)(22 324 204 250)(23 325 205 251)(24 326 165 252)(25 327 166 253)(26 328 167 254)(27 288 168 255)(28 289 169 256)(29 290 170 257)(30 291 171 258)(31 292 172 259)(32 293 173 260)(33 294 174 261)(34 295 175 262)(35 296 176 263)(36 297 177 264)(37 298 178 265)(38 299 179 266)(39 300 180 267)(40 301 181 268)(41 302 182 269)(42 223 88 137)(43 224 89 138)(44 225 90 139)(45 226 91 140)(46 227 92 141)(47 228 93 142)(48 229 94 143)(49 230 95 144)(50 231 96 145)(51 232 97 146)(52 233 98 147)(53 234 99 148)(54 235 100 149)(55 236 101 150)(56 237 102 151)(57 238 103 152)(58 239 104 153)(59 240 105 154)(60 241 106 155)(61 242 107 156)(62 243 108 157)(63 244 109 158)(64 245 110 159)(65 246 111 160)(66 206 112 161)(67 207 113 162)(68 208 114 163)(69 209 115 164)(70 210 116 124)(71 211 117 125)(72 212 118 126)(73 213 119 127)(74 214 120 128)(75 215 121 129)(76 216 122 130)(77 217 123 131)(78 218 83 132)(79 219 84 133)(80 220 85 134)(81 221 86 135)(82 222 87 136)
(1 86 183 81)(2 87 184 82)(3 88 185 42)(4 89 186 43)(5 90 187 44)(6 91 188 45)(7 92 189 46)(8 93 190 47)(9 94 191 48)(10 95 192 49)(11 96 193 50)(12 97 194 51)(13 98 195 52)(14 99 196 53)(15 100 197 54)(16 101 198 55)(17 102 199 56)(18 103 200 57)(19 104 201 58)(20 105 202 59)(21 106 203 60)(22 107 204 61)(23 108 205 62)(24 109 165 63)(25 110 166 64)(26 111 167 65)(27 112 168 66)(28 113 169 67)(29 114 170 68)(30 115 171 69)(31 116 172 70)(32 117 173 71)(33 118 174 72)(34 119 175 73)(35 120 176 74)(36 121 177 75)(37 122 178 76)(38 123 179 77)(39 83 180 78)(40 84 181 79)(41 85 182 80)(124 292 210 259)(125 293 211 260)(126 294 212 261)(127 295 213 262)(128 296 214 263)(129 297 215 264)(130 298 216 265)(131 299 217 266)(132 300 218 267)(133 301 219 268)(134 302 220 269)(135 303 221 270)(136 304 222 271)(137 305 223 272)(138 306 224 273)(139 307 225 274)(140 308 226 275)(141 309 227 276)(142 310 228 277)(143 311 229 278)(144 312 230 279)(145 313 231 280)(146 314 232 281)(147 315 233 282)(148 316 234 283)(149 317 235 284)(150 318 236 285)(151 319 237 286)(152 320 238 287)(153 321 239 247)(154 322 240 248)(155 323 241 249)(156 324 242 250)(157 325 243 251)(158 326 244 252)(159 327 245 253)(160 328 246 254)(161 288 206 255)(162 289 207 256)(163 290 208 257)(164 291 209 258)

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), (1,303,183,270)(2,304,184,271)(3,305,185,272)(4,306,186,273)(5,307,187,274)(6,308,188,275)(7,309,189,276)(8,310,190,277)(9,311,191,278)(10,312,192,279)(11,313,193,280)(12,314,194,281)(13,315,195,282)(14,316,196,283)(15,317,197,284)(16,318,198,285)(17,319,199,286)(18,320,200,287)(19,321,201,247)(20,322,202,248)(21,323,203,249)(22,324,204,250)(23,325,205,251)(24,326,165,252)(25,327,166,253)(26,328,167,254)(27,288,168,255)(28,289,169,256)(29,290,170,257)(30,291,171,258)(31,292,172,259)(32,293,173,260)(33,294,174,261)(34,295,175,262)(35,296,176,263)(36,297,177,264)(37,298,178,265)(38,299,179,266)(39,300,180,267)(40,301,181,268)(41,302,182,269)(42,223,88,137)(43,224,89,138)(44,225,90,139)(45,226,91,140)(46,227,92,141)(47,228,93,142)(48,229,94,143)(49,230,95,144)(50,231,96,145)(51,232,97,146)(52,233,98,147)(53,234,99,148)(54,235,100,149)(55,236,101,150)(56,237,102,151)(57,238,103,152)(58,239,104,153)(59,240,105,154)(60,241,106,155)(61,242,107,156)(62,243,108,157)(63,244,109,158)(64,245,110,159)(65,246,111,160)(66,206,112,161)(67,207,113,162)(68,208,114,163)(69,209,115,164)(70,210,116,124)(71,211,117,125)(72,212,118,126)(73,213,119,127)(74,214,120,128)(75,215,121,129)(76,216,122,130)(77,217,123,131)(78,218,83,132)(79,219,84,133)(80,220,85,134)(81,221,86,135)(82,222,87,136), (1,86,183,81)(2,87,184,82)(3,88,185,42)(4,89,186,43)(5,90,187,44)(6,91,188,45)(7,92,189,46)(8,93,190,47)(9,94,191,48)(10,95,192,49)(11,96,193,50)(12,97,194,51)(13,98,195,52)(14,99,196,53)(15,100,197,54)(16,101,198,55)(17,102,199,56)(18,103,200,57)(19,104,201,58)(20,105,202,59)(21,106,203,60)(22,107,204,61)(23,108,205,62)(24,109,165,63)(25,110,166,64)(26,111,167,65)(27,112,168,66)(28,113,169,67)(29,114,170,68)(30,115,171,69)(31,116,172,70)(32,117,173,71)(33,118,174,72)(34,119,175,73)(35,120,176,74)(36,121,177,75)(37,122,178,76)(38,123,179,77)(39,83,180,78)(40,84,181,79)(41,85,182,80)(124,292,210,259)(125,293,211,260)(126,294,212,261)(127,295,213,262)(128,296,214,263)(129,297,215,264)(130,298,216,265)(131,299,217,266)(132,300,218,267)(133,301,219,268)(134,302,220,269)(135,303,221,270)(136,304,222,271)(137,305,223,272)(138,306,224,273)(139,307,225,274)(140,308,226,275)(141,309,227,276)(142,310,228,277)(143,311,229,278)(144,312,230,279)(145,313,231,280)(146,314,232,281)(147,315,233,282)(148,316,234,283)(149,317,235,284)(150,318,236,285)(151,319,237,286)(152,320,238,287)(153,321,239,247)(154,322,240,248)(155,323,241,249)(156,324,242,250)(157,325,243,251)(158,326,244,252)(159,327,245,253)(160,328,246,254)(161,288,206,255)(162,289,207,256)(163,290,208,257)(164,291,209,258)>;

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), (1,303,183,270)(2,304,184,271)(3,305,185,272)(4,306,186,273)(5,307,187,274)(6,308,188,275)(7,309,189,276)(8,310,190,277)(9,311,191,278)(10,312,192,279)(11,313,193,280)(12,314,194,281)(13,315,195,282)(14,316,196,283)(15,317,197,284)(16,318,198,285)(17,319,199,286)(18,320,200,287)(19,321,201,247)(20,322,202,248)(21,323,203,249)(22,324,204,250)(23,325,205,251)(24,326,165,252)(25,327,166,253)(26,328,167,254)(27,288,168,255)(28,289,169,256)(29,290,170,257)(30,291,171,258)(31,292,172,259)(32,293,173,260)(33,294,174,261)(34,295,175,262)(35,296,176,263)(36,297,177,264)(37,298,178,265)(38,299,179,266)(39,300,180,267)(40,301,181,268)(41,302,182,269)(42,223,88,137)(43,224,89,138)(44,225,90,139)(45,226,91,140)(46,227,92,141)(47,228,93,142)(48,229,94,143)(49,230,95,144)(50,231,96,145)(51,232,97,146)(52,233,98,147)(53,234,99,148)(54,235,100,149)(55,236,101,150)(56,237,102,151)(57,238,103,152)(58,239,104,153)(59,240,105,154)(60,241,106,155)(61,242,107,156)(62,243,108,157)(63,244,109,158)(64,245,110,159)(65,246,111,160)(66,206,112,161)(67,207,113,162)(68,208,114,163)(69,209,115,164)(70,210,116,124)(71,211,117,125)(72,212,118,126)(73,213,119,127)(74,214,120,128)(75,215,121,129)(76,216,122,130)(77,217,123,131)(78,218,83,132)(79,219,84,133)(80,220,85,134)(81,221,86,135)(82,222,87,136), (1,86,183,81)(2,87,184,82)(3,88,185,42)(4,89,186,43)(5,90,187,44)(6,91,188,45)(7,92,189,46)(8,93,190,47)(9,94,191,48)(10,95,192,49)(11,96,193,50)(12,97,194,51)(13,98,195,52)(14,99,196,53)(15,100,197,54)(16,101,198,55)(17,102,199,56)(18,103,200,57)(19,104,201,58)(20,105,202,59)(21,106,203,60)(22,107,204,61)(23,108,205,62)(24,109,165,63)(25,110,166,64)(26,111,167,65)(27,112,168,66)(28,113,169,67)(29,114,170,68)(30,115,171,69)(31,116,172,70)(32,117,173,71)(33,118,174,72)(34,119,175,73)(35,120,176,74)(36,121,177,75)(37,122,178,76)(38,123,179,77)(39,83,180,78)(40,84,181,79)(41,85,182,80)(124,292,210,259)(125,293,211,260)(126,294,212,261)(127,295,213,262)(128,296,214,263)(129,297,215,264)(130,298,216,265)(131,299,217,266)(132,300,218,267)(133,301,219,268)(134,302,220,269)(135,303,221,270)(136,304,222,271)(137,305,223,272)(138,306,224,273)(139,307,225,274)(140,308,226,275)(141,309,227,276)(142,310,228,277)(143,311,229,278)(144,312,230,279)(145,313,231,280)(146,314,232,281)(147,315,233,282)(148,316,234,283)(149,317,235,284)(150,318,236,285)(151,319,237,286)(152,320,238,287)(153,321,239,247)(154,322,240,248)(155,323,241,249)(156,324,242,250)(157,325,243,251)(158,326,244,252)(159,327,245,253)(160,328,246,254)(161,288,206,255)(162,289,207,256)(163,290,208,257)(164,291,209,258) );

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)], [(1,303,183,270),(2,304,184,271),(3,305,185,272),(4,306,186,273),(5,307,187,274),(6,308,188,275),(7,309,189,276),(8,310,190,277),(9,311,191,278),(10,312,192,279),(11,313,193,280),(12,314,194,281),(13,315,195,282),(14,316,196,283),(15,317,197,284),(16,318,198,285),(17,319,199,286),(18,320,200,287),(19,321,201,247),(20,322,202,248),(21,323,203,249),(22,324,204,250),(23,325,205,251),(24,326,165,252),(25,327,166,253),(26,328,167,254),(27,288,168,255),(28,289,169,256),(29,290,170,257),(30,291,171,258),(31,292,172,259),(32,293,173,260),(33,294,174,261),(34,295,175,262),(35,296,176,263),(36,297,177,264),(37,298,178,265),(38,299,179,266),(39,300,180,267),(40,301,181,268),(41,302,182,269),(42,223,88,137),(43,224,89,138),(44,225,90,139),(45,226,91,140),(46,227,92,141),(47,228,93,142),(48,229,94,143),(49,230,95,144),(50,231,96,145),(51,232,97,146),(52,233,98,147),(53,234,99,148),(54,235,100,149),(55,236,101,150),(56,237,102,151),(57,238,103,152),(58,239,104,153),(59,240,105,154),(60,241,106,155),(61,242,107,156),(62,243,108,157),(63,244,109,158),(64,245,110,159),(65,246,111,160),(66,206,112,161),(67,207,113,162),(68,208,114,163),(69,209,115,164),(70,210,116,124),(71,211,117,125),(72,212,118,126),(73,213,119,127),(74,214,120,128),(75,215,121,129),(76,216,122,130),(77,217,123,131),(78,218,83,132),(79,219,84,133),(80,220,85,134),(81,221,86,135),(82,222,87,136)], [(1,86,183,81),(2,87,184,82),(3,88,185,42),(4,89,186,43),(5,90,187,44),(6,91,188,45),(7,92,189,46),(8,93,190,47),(9,94,191,48),(10,95,192,49),(11,96,193,50),(12,97,194,51),(13,98,195,52),(14,99,196,53),(15,100,197,54),(16,101,198,55),(17,102,199,56),(18,103,200,57),(19,104,201,58),(20,105,202,59),(21,106,203,60),(22,107,204,61),(23,108,205,62),(24,109,165,63),(25,110,166,64),(26,111,167,65),(27,112,168,66),(28,113,169,67),(29,114,170,68),(30,115,171,69),(31,116,172,70),(32,117,173,71),(33,118,174,72),(34,119,175,73),(35,120,176,74),(36,121,177,75),(37,122,178,76),(38,123,179,77),(39,83,180,78),(40,84,181,79),(41,85,182,80),(124,292,210,259),(125,293,211,260),(126,294,212,261),(127,295,213,262),(128,296,214,263),(129,297,215,264),(130,298,216,265),(131,299,217,266),(132,300,218,267),(133,301,219,268),(134,302,220,269),(135,303,221,270),(136,304,222,271),(137,305,223,272),(138,306,224,273),(139,307,225,274),(140,308,226,275),(141,309,227,276),(142,310,228,277),(143,311,229,278),(144,312,230,279),(145,313,231,280),(146,314,232,281),(147,315,233,282),(148,316,234,283),(149,317,235,284),(150,318,236,285),(151,319,237,286),(152,320,238,287),(153,321,239,247),(154,322,240,248),(155,323,241,249),(156,324,242,250),(157,325,243,251),(158,326,244,252),(159,327,245,253),(160,328,246,254),(161,288,206,255),(162,289,207,256),(163,290,208,257),(164,291,209,258)])

205 conjugacy classes

class 1  2 4A4B4C41A···41AN82A···82AN164A···164DP
order1244441···4182···82164···164
size112221···11···12···2

205 irreducible representations

dim111122
type++-
imageC1C2C41C82Q8Q8×C41
kernelQ8×C41C164Q8C4C41C1
# reps1340120140

Matrix representation of Q8×C41 in GL2(𝔽821) generated by

1650
0165
,
447819
564374
,
249486
413572
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

Subgroup lattice of Q8×C41 in TeX

׿
×
𝔽