direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary
Aliases: C5×Q64, C32.C10, Q32.C10, C160.2C2, C40.70D4, C20.41D8, C10.17D16, C80.21C22, C8.7(C5×D4), C4.3(C5×D8), C2.5(C5×D16), C16.4(C2×C10), (C5×Q32).2C2, SmallGroup(320,178)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q64
G = < a,b,c | a5=b32=1, c2=b16, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Q64
►Regular action on 320 points
Generators in S320
(1 70 289 179 261)(2 71 290 180 262)(3 72 291 181 263)(4 73 292 182 264)(5 74 293 183 265)(6 75 294 184 266)(7 76 295 185 267)(8 77 296 186 268)(9 78 297 187 269)(10 79 298 188 270)(11 80 299 189 271)(12 81 300 190 272)(13 82 301 191 273)(14 83 302 192 274)(15 84 303 161 275)(16 85 304 162 276)(17 86 305 163 277)(18 87 306 164 278)(19 88 307 165 279)(20 89 308 166 280)(21 90 309 167 281)(22 91 310 168 282)(23 92 311 169 283)(24 93 312 170 284)(25 94 313 171 285)(26 95 314 172 286)(27 96 315 173 287)(28 65 316 174 288)(29 66 317 175 257)(30 67 318 176 258)(31 68 319 177 259)(32 69 320 178 260)(33 119 255 150 210)(34 120 256 151 211)(35 121 225 152 212)(36 122 226 153 213)(37 123 227 154 214)(38 124 228 155 215)(39 125 229 156 216)(40 126 230 157 217)(41 127 231 158 218)(42 128 232 159 219)(43 97 233 160 220)(44 98 234 129 221)(45 99 235 130 222)(46 100 236 131 223)(47 101 237 132 224)(48 102 238 133 193)(49 103 239 134 194)(50 104 240 135 195)(51 105 241 136 196)(52 106 242 137 197)(53 107 243 138 198)(54 108 244 139 199)(55 109 245 140 200)(56 110 246 141 201)(57 111 247 142 202)(58 112 248 143 203)(59 113 249 144 204)(60 114 250 145 205)(61 115 251 146 206)(62 116 252 147 207)(63 117 253 148 208)(64 118 254 149 209)
(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)
(1 118 17 102)(2 117 18 101)(3 116 19 100)(4 115 20 99)(5 114 21 98)(6 113 22 97)(7 112 23 128)(8 111 24 127)(9 110 25 126)(10 109 26 125)(11 108 27 124)(12 107 28 123)(13 106 29 122)(14 105 30 121)(15 104 31 120)(16 103 32 119)(33 276 49 260)(34 275 50 259)(35 274 51 258)(36 273 52 257)(37 272 53 288)(38 271 54 287)(39 270 55 286)(40 269 56 285)(41 268 57 284)(42 267 58 283)(43 266 59 282)(44 265 60 281)(45 264 61 280)(46 263 62 279)(47 262 63 278)(48 261 64 277)(65 227 81 243)(66 226 82 242)(67 225 83 241)(68 256 84 240)(69 255 85 239)(70 254 86 238)(71 253 87 237)(72 252 88 236)(73 251 89 235)(74 250 90 234)(75 249 91 233)(76 248 92 232)(77 247 93 231)(78 246 94 230)(79 245 95 229)(80 244 96 228)(129 293 145 309)(130 292 146 308)(131 291 147 307)(132 290 148 306)(133 289 149 305)(134 320 150 304)(135 319 151 303)(136 318 152 302)(137 317 153 301)(138 316 154 300)(139 315 155 299)(140 314 156 298)(141 313 157 297)(142 312 158 296)(143 311 159 295)(144 310 160 294)(161 195 177 211)(162 194 178 210)(163 193 179 209)(164 224 180 208)(165 223 181 207)(166 222 182 206)(167 221 183 205)(168 220 184 204)(169 219 185 203)(170 218 186 202)(171 217 187 201)(172 216 188 200)(173 215 189 199)(174 214 190 198)(175 213 191 197)(176 212 192 196)
G:=sub<Sym(320)| (1,70,289,179,261)(2,71,290,180,262)(3,72,291,181,263)(4,73,292,182,264)(5,74,293,183,265)(6,75,294,184,266)(7,76,295,185,267)(8,77,296,186,268)(9,78,297,187,269)(10,79,298,188,270)(11,80,299,189,271)(12,81,300,190,272)(13,82,301,191,273)(14,83,302,192,274)(15,84,303,161,275)(16,85,304,162,276)(17,86,305,163,277)(18,87,306,164,278)(19,88,307,165,279)(20,89,308,166,280)(21,90,309,167,281)(22,91,310,168,282)(23,92,311,169,283)(24,93,312,170,284)(25,94,313,171,285)(26,95,314,172,286)(27,96,315,173,287)(28,65,316,174,288)(29,66,317,175,257)(30,67,318,176,258)(31,68,319,177,259)(32,69,320,178,260)(33,119,255,150,210)(34,120,256,151,211)(35,121,225,152,212)(36,122,226,153,213)(37,123,227,154,214)(38,124,228,155,215)(39,125,229,156,216)(40,126,230,157,217)(41,127,231,158,218)(42,128,232,159,219)(43,97,233,160,220)(44,98,234,129,221)(45,99,235,130,222)(46,100,236,131,223)(47,101,237,132,224)(48,102,238,133,193)(49,103,239,134,194)(50,104,240,135,195)(51,105,241,136,196)(52,106,242,137,197)(53,107,243,138,198)(54,108,244,139,199)(55,109,245,140,200)(56,110,246,141,201)(57,111,247,142,202)(58,112,248,143,203)(59,113,249,144,204)(60,114,250,145,205)(61,115,251,146,206)(62,116,252,147,207)(63,117,253,148,208)(64,118,254,149,209), (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), (1,118,17,102)(2,117,18,101)(3,116,19,100)(4,115,20,99)(5,114,21,98)(6,113,22,97)(7,112,23,128)(8,111,24,127)(9,110,25,126)(10,109,26,125)(11,108,27,124)(12,107,28,123)(13,106,29,122)(14,105,30,121)(15,104,31,120)(16,103,32,119)(33,276,49,260)(34,275,50,259)(35,274,51,258)(36,273,52,257)(37,272,53,288)(38,271,54,287)(39,270,55,286)(40,269,56,285)(41,268,57,284)(42,267,58,283)(43,266,59,282)(44,265,60,281)(45,264,61,280)(46,263,62,279)(47,262,63,278)(48,261,64,277)(65,227,81,243)(66,226,82,242)(67,225,83,241)(68,256,84,240)(69,255,85,239)(70,254,86,238)(71,253,87,237)(72,252,88,236)(73,251,89,235)(74,250,90,234)(75,249,91,233)(76,248,92,232)(77,247,93,231)(78,246,94,230)(79,245,95,229)(80,244,96,228)(129,293,145,309)(130,292,146,308)(131,291,147,307)(132,290,148,306)(133,289,149,305)(134,320,150,304)(135,319,151,303)(136,318,152,302)(137,317,153,301)(138,316,154,300)(139,315,155,299)(140,314,156,298)(141,313,157,297)(142,312,158,296)(143,311,159,295)(144,310,160,294)(161,195,177,211)(162,194,178,210)(163,193,179,209)(164,224,180,208)(165,223,181,207)(166,222,182,206)(167,221,183,205)(168,220,184,204)(169,219,185,203)(170,218,186,202)(171,217,187,201)(172,216,188,200)(173,215,189,199)(174,214,190,198)(175,213,191,197)(176,212,192,196)>;
G:=Group( (1,70,289,179,261)(2,71,290,180,262)(3,72,291,181,263)(4,73,292,182,264)(5,74,293,183,265)(6,75,294,184,266)(7,76,295,185,267)(8,77,296,186,268)(9,78,297,187,269)(10,79,298,188,270)(11,80,299,189,271)(12,81,300,190,272)(13,82,301,191,273)(14,83,302,192,274)(15,84,303,161,275)(16,85,304,162,276)(17,86,305,163,277)(18,87,306,164,278)(19,88,307,165,279)(20,89,308,166,280)(21,90,309,167,281)(22,91,310,168,282)(23,92,311,169,283)(24,93,312,170,284)(25,94,313,171,285)(26,95,314,172,286)(27,96,315,173,287)(28,65,316,174,288)(29,66,317,175,257)(30,67,318,176,258)(31,68,319,177,259)(32,69,320,178,260)(33,119,255,150,210)(34,120,256,151,211)(35,121,225,152,212)(36,122,226,153,213)(37,123,227,154,214)(38,124,228,155,215)(39,125,229,156,216)(40,126,230,157,217)(41,127,231,158,218)(42,128,232,159,219)(43,97,233,160,220)(44,98,234,129,221)(45,99,235,130,222)(46,100,236,131,223)(47,101,237,132,224)(48,102,238,133,193)(49,103,239,134,194)(50,104,240,135,195)(51,105,241,136,196)(52,106,242,137,197)(53,107,243,138,198)(54,108,244,139,199)(55,109,245,140,200)(56,110,246,141,201)(57,111,247,142,202)(58,112,248,143,203)(59,113,249,144,204)(60,114,250,145,205)(61,115,251,146,206)(62,116,252,147,207)(63,117,253,148,208)(64,118,254,149,209), (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), (1,118,17,102)(2,117,18,101)(3,116,19,100)(4,115,20,99)(5,114,21,98)(6,113,22,97)(7,112,23,128)(8,111,24,127)(9,110,25,126)(10,109,26,125)(11,108,27,124)(12,107,28,123)(13,106,29,122)(14,105,30,121)(15,104,31,120)(16,103,32,119)(33,276,49,260)(34,275,50,259)(35,274,51,258)(36,273,52,257)(37,272,53,288)(38,271,54,287)(39,270,55,286)(40,269,56,285)(41,268,57,284)(42,267,58,283)(43,266,59,282)(44,265,60,281)(45,264,61,280)(46,263,62,279)(47,262,63,278)(48,261,64,277)(65,227,81,243)(66,226,82,242)(67,225,83,241)(68,256,84,240)(69,255,85,239)(70,254,86,238)(71,253,87,237)(72,252,88,236)(73,251,89,235)(74,250,90,234)(75,249,91,233)(76,248,92,232)(77,247,93,231)(78,246,94,230)(79,245,95,229)(80,244,96,228)(129,293,145,309)(130,292,146,308)(131,291,147,307)(132,290,148,306)(133,289,149,305)(134,320,150,304)(135,319,151,303)(136,318,152,302)(137,317,153,301)(138,316,154,300)(139,315,155,299)(140,314,156,298)(141,313,157,297)(142,312,158,296)(143,311,159,295)(144,310,160,294)(161,195,177,211)(162,194,178,210)(163,193,179,209)(164,224,180,208)(165,223,181,207)(166,222,182,206)(167,221,183,205)(168,220,184,204)(169,219,185,203)(170,218,186,202)(171,217,187,201)(172,216,188,200)(173,215,189,199)(174,214,190,198)(175,213,191,197)(176,212,192,196) );
G=PermutationGroup([[(1,70,289,179,261),(2,71,290,180,262),(3,72,291,181,263),(4,73,292,182,264),(5,74,293,183,265),(6,75,294,184,266),(7,76,295,185,267),(8,77,296,186,268),(9,78,297,187,269),(10,79,298,188,270),(11,80,299,189,271),(12,81,300,190,272),(13,82,301,191,273),(14,83,302,192,274),(15,84,303,161,275),(16,85,304,162,276),(17,86,305,163,277),(18,87,306,164,278),(19,88,307,165,279),(20,89,308,166,280),(21,90,309,167,281),(22,91,310,168,282),(23,92,311,169,283),(24,93,312,170,284),(25,94,313,171,285),(26,95,314,172,286),(27,96,315,173,287),(28,65,316,174,288),(29,66,317,175,257),(30,67,318,176,258),(31,68,319,177,259),(32,69,320,178,260),(33,119,255,150,210),(34,120,256,151,211),(35,121,225,152,212),(36,122,226,153,213),(37,123,227,154,214),(38,124,228,155,215),(39,125,229,156,216),(40,126,230,157,217),(41,127,231,158,218),(42,128,232,159,219),(43,97,233,160,220),(44,98,234,129,221),(45,99,235,130,222),(46,100,236,131,223),(47,101,237,132,224),(48,102,238,133,193),(49,103,239,134,194),(50,104,240,135,195),(51,105,241,136,196),(52,106,242,137,197),(53,107,243,138,198),(54,108,244,139,199),(55,109,245,140,200),(56,110,246,141,201),(57,111,247,142,202),(58,112,248,143,203),(59,113,249,144,204),(60,114,250,145,205),(61,115,251,146,206),(62,116,252,147,207),(63,117,253,148,208),(64,118,254,149,209)], [(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)], [(1,118,17,102),(2,117,18,101),(3,116,19,100),(4,115,20,99),(5,114,21,98),(6,113,22,97),(7,112,23,128),(8,111,24,127),(9,110,25,126),(10,109,26,125),(11,108,27,124),(12,107,28,123),(13,106,29,122),(14,105,30,121),(15,104,31,120),(16,103,32,119),(33,276,49,260),(34,275,50,259),(35,274,51,258),(36,273,52,257),(37,272,53,288),(38,271,54,287),(39,270,55,286),(40,269,56,285),(41,268,57,284),(42,267,58,283),(43,266,59,282),(44,265,60,281),(45,264,61,280),(46,263,62,279),(47,262,63,278),(48,261,64,277),(65,227,81,243),(66,226,82,242),(67,225,83,241),(68,256,84,240),(69,255,85,239),(70,254,86,238),(71,253,87,237),(72,252,88,236),(73,251,89,235),(74,250,90,234),(75,249,91,233),(76,248,92,232),(77,247,93,231),(78,246,94,230),(79,245,95,229),(80,244,96,228),(129,293,145,309),(130,292,146,308),(131,291,147,307),(132,290,148,306),(133,289,149,305),(134,320,150,304),(135,319,151,303),(136,318,152,302),(137,317,153,301),(138,316,154,300),(139,315,155,299),(140,314,156,298),(141,313,157,297),(142,312,158,296),(143,311,159,295),(144,310,160,294),(161,195,177,211),(162,194,178,210),(163,193,179,209),(164,224,180,208),(165,223,181,207),(166,222,182,206),(167,221,183,205),(168,220,184,204),(169,219,185,203),(170,218,186,202),(171,217,187,201),(172,216,188,200),(173,215,189,199),(174,214,190,198),(175,213,191,197),(176,212,192,196)]])
95 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 32A | ··· | 32H | 40A | ··· | 40H | 80A | ··· | 80P | 160A | ··· | 160AF |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 32 | ··· | 32 | 40 | ··· | 40 | 80 | ··· | 80 | 160 | ··· | 160 |
| size | 1 | 1 | 2 | 16 | 16 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | ··· | 16 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D4 | D8 | D16 | C5×D4 | Q64 | C5×D8 | C5×D16 | C5×Q64 |
| kernel | C5×Q64 | C160 | C5×Q32 | Q64 | C32 | Q32 | C40 | C20 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 2 | 4 | 4 | 8 | 8 | 16 | 32 |
Matrix representation of C5×Q64 ►in GL2(𝔽31) generated by
| 2 | 0 |
| 0 | 2 |
| 18 | 12 |
| 24 | 4 |
| 0 | 27 |
| 8 | 0 |
G:=sub<GL(2,GF(31))| [2,0,0,2],[18,24,12,4],[0,8,27,0] >;
C5×Q64 in GAP, Magma, Sage, TeX
C_5\times Q_{64} % in TeX
G:=Group("C5xQ64"); // GroupNames label
G:=SmallGroup(320,178);
// by ID
G=gap.SmallGroup(320,178);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,1120,309,1128,1683,850,192,4204,2111,242,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c|a^5=b^32=1,c^2=b^16,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export