metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2Q64, Q32.D5, C20.8D8, C40.12D4, C16.7D10, C10.11D16, C80.5C22, Dic40.2C2, C5⋊2C32.1C2, C4.4(D4⋊D5), (C5×Q32).1C2, C2.7(C5⋊D16), C8.12(C5⋊D4), SmallGroup(320,80)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊Q64
G = < a,b,c | a5=b32=1, c2=b16, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5⋊Q64
►Regular action on 320 points
Generators in S320
(1 172 120 240 139)(2 140 241 121 173)(3 174 122 242 141)(4 142 243 123 175)(5 176 124 244 143)(6 144 245 125 177)(7 178 126 246 145)(8 146 247 127 179)(9 180 128 248 147)(10 148 249 97 181)(11 182 98 250 149)(12 150 251 99 183)(13 184 100 252 151)(14 152 253 101 185)(15 186 102 254 153)(16 154 255 103 187)(17 188 104 256 155)(18 156 225 105 189)(19 190 106 226 157)(20 158 227 107 191)(21 192 108 228 159)(22 160 229 109 161)(23 162 110 230 129)(24 130 231 111 163)(25 164 112 232 131)(26 132 233 113 165)(27 166 114 234 133)(28 134 235 115 167)(29 168 116 236 135)(30 136 237 117 169)(31 170 118 238 137)(32 138 239 119 171)(33 278 71 306 214)(34 215 307 72 279)(35 280 73 308 216)(36 217 309 74 281)(37 282 75 310 218)(38 219 311 76 283)(39 284 77 312 220)(40 221 313 78 285)(41 286 79 314 222)(42 223 315 80 287)(43 288 81 316 224)(44 193 317 82 257)(45 258 83 318 194)(46 195 319 84 259)(47 260 85 320 196)(48 197 289 86 261)(49 262 87 290 198)(50 199 291 88 263)(51 264 89 292 200)(52 201 293 90 265)(53 266 91 294 202)(54 203 295 92 267)(55 268 93 296 204)(56 205 297 94 269)(57 270 95 298 206)(58 207 299 96 271)(59 272 65 300 208)(60 209 301 66 273)(61 274 67 302 210)(62 211 303 68 275)(63 276 69 304 212)(64 213 305 70 277)
(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)
(1 290 17 306)(2 289 18 305)(3 320 19 304)(4 319 20 303)(5 318 21 302)(6 317 22 301)(7 316 23 300)(8 315 24 299)(9 314 25 298)(10 313 26 297)(11 312 27 296)(12 311 28 295)(13 310 29 294)(14 309 30 293)(15 308 31 292)(16 307 32 291)(33 120 49 104)(34 119 50 103)(35 118 51 102)(36 117 52 101)(37 116 53 100)(38 115 54 99)(39 114 55 98)(40 113 56 97)(41 112 57 128)(42 111 58 127)(43 110 59 126)(44 109 60 125)(45 108 61 124)(46 107 62 123)(47 106 63 122)(48 105 64 121)(65 145 81 129)(66 144 82 160)(67 143 83 159)(68 142 84 158)(69 141 85 157)(70 140 86 156)(71 139 87 155)(72 138 88 154)(73 137 89 153)(74 136 90 152)(75 135 91 151)(76 134 92 150)(77 133 93 149)(78 132 94 148)(79 131 95 147)(80 130 96 146)(161 209 177 193)(162 208 178 224)(163 207 179 223)(164 206 180 222)(165 205 181 221)(166 204 182 220)(167 203 183 219)(168 202 184 218)(169 201 185 217)(170 200 186 216)(171 199 187 215)(172 198 188 214)(173 197 189 213)(174 196 190 212)(175 195 191 211)(176 194 192 210)(225 277 241 261)(226 276 242 260)(227 275 243 259)(228 274 244 258)(229 273 245 257)(230 272 246 288)(231 271 247 287)(232 270 248 286)(233 269 249 285)(234 268 250 284)(235 267 251 283)(236 266 252 282)(237 265 253 281)(238 264 254 280)(239 263 255 279)(240 262 256 278)
G:=sub<Sym(320)| (1,172,120,240,139)(2,140,241,121,173)(3,174,122,242,141)(4,142,243,123,175)(5,176,124,244,143)(6,144,245,125,177)(7,178,126,246,145)(8,146,247,127,179)(9,180,128,248,147)(10,148,249,97,181)(11,182,98,250,149)(12,150,251,99,183)(13,184,100,252,151)(14,152,253,101,185)(15,186,102,254,153)(16,154,255,103,187)(17,188,104,256,155)(18,156,225,105,189)(19,190,106,226,157)(20,158,227,107,191)(21,192,108,228,159)(22,160,229,109,161)(23,162,110,230,129)(24,130,231,111,163)(25,164,112,232,131)(26,132,233,113,165)(27,166,114,234,133)(28,134,235,115,167)(29,168,116,236,135)(30,136,237,117,169)(31,170,118,238,137)(32,138,239,119,171)(33,278,71,306,214)(34,215,307,72,279)(35,280,73,308,216)(36,217,309,74,281)(37,282,75,310,218)(38,219,311,76,283)(39,284,77,312,220)(40,221,313,78,285)(41,286,79,314,222)(42,223,315,80,287)(43,288,81,316,224)(44,193,317,82,257)(45,258,83,318,194)(46,195,319,84,259)(47,260,85,320,196)(48,197,289,86,261)(49,262,87,290,198)(50,199,291,88,263)(51,264,89,292,200)(52,201,293,90,265)(53,266,91,294,202)(54,203,295,92,267)(55,268,93,296,204)(56,205,297,94,269)(57,270,95,298,206)(58,207,299,96,271)(59,272,65,300,208)(60,209,301,66,273)(61,274,67,302,210)(62,211,303,68,275)(63,276,69,304,212)(64,213,305,70,277), (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), (1,290,17,306)(2,289,18,305)(3,320,19,304)(4,319,20,303)(5,318,21,302)(6,317,22,301)(7,316,23,300)(8,315,24,299)(9,314,25,298)(10,313,26,297)(11,312,27,296)(12,311,28,295)(13,310,29,294)(14,309,30,293)(15,308,31,292)(16,307,32,291)(33,120,49,104)(34,119,50,103)(35,118,51,102)(36,117,52,101)(37,116,53,100)(38,115,54,99)(39,114,55,98)(40,113,56,97)(41,112,57,128)(42,111,58,127)(43,110,59,126)(44,109,60,125)(45,108,61,124)(46,107,62,123)(47,106,63,122)(48,105,64,121)(65,145,81,129)(66,144,82,160)(67,143,83,159)(68,142,84,158)(69,141,85,157)(70,140,86,156)(71,139,87,155)(72,138,88,154)(73,137,89,153)(74,136,90,152)(75,135,91,151)(76,134,92,150)(77,133,93,149)(78,132,94,148)(79,131,95,147)(80,130,96,146)(161,209,177,193)(162,208,178,224)(163,207,179,223)(164,206,180,222)(165,205,181,221)(166,204,182,220)(167,203,183,219)(168,202,184,218)(169,201,185,217)(170,200,186,216)(171,199,187,215)(172,198,188,214)(173,197,189,213)(174,196,190,212)(175,195,191,211)(176,194,192,210)(225,277,241,261)(226,276,242,260)(227,275,243,259)(228,274,244,258)(229,273,245,257)(230,272,246,288)(231,271,247,287)(232,270,248,286)(233,269,249,285)(234,268,250,284)(235,267,251,283)(236,266,252,282)(237,265,253,281)(238,264,254,280)(239,263,255,279)(240,262,256,278)>;
G:=Group( (1,172,120,240,139)(2,140,241,121,173)(3,174,122,242,141)(4,142,243,123,175)(5,176,124,244,143)(6,144,245,125,177)(7,178,126,246,145)(8,146,247,127,179)(9,180,128,248,147)(10,148,249,97,181)(11,182,98,250,149)(12,150,251,99,183)(13,184,100,252,151)(14,152,253,101,185)(15,186,102,254,153)(16,154,255,103,187)(17,188,104,256,155)(18,156,225,105,189)(19,190,106,226,157)(20,158,227,107,191)(21,192,108,228,159)(22,160,229,109,161)(23,162,110,230,129)(24,130,231,111,163)(25,164,112,232,131)(26,132,233,113,165)(27,166,114,234,133)(28,134,235,115,167)(29,168,116,236,135)(30,136,237,117,169)(31,170,118,238,137)(32,138,239,119,171)(33,278,71,306,214)(34,215,307,72,279)(35,280,73,308,216)(36,217,309,74,281)(37,282,75,310,218)(38,219,311,76,283)(39,284,77,312,220)(40,221,313,78,285)(41,286,79,314,222)(42,223,315,80,287)(43,288,81,316,224)(44,193,317,82,257)(45,258,83,318,194)(46,195,319,84,259)(47,260,85,320,196)(48,197,289,86,261)(49,262,87,290,198)(50,199,291,88,263)(51,264,89,292,200)(52,201,293,90,265)(53,266,91,294,202)(54,203,295,92,267)(55,268,93,296,204)(56,205,297,94,269)(57,270,95,298,206)(58,207,299,96,271)(59,272,65,300,208)(60,209,301,66,273)(61,274,67,302,210)(62,211,303,68,275)(63,276,69,304,212)(64,213,305,70,277), (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), (1,290,17,306)(2,289,18,305)(3,320,19,304)(4,319,20,303)(5,318,21,302)(6,317,22,301)(7,316,23,300)(8,315,24,299)(9,314,25,298)(10,313,26,297)(11,312,27,296)(12,311,28,295)(13,310,29,294)(14,309,30,293)(15,308,31,292)(16,307,32,291)(33,120,49,104)(34,119,50,103)(35,118,51,102)(36,117,52,101)(37,116,53,100)(38,115,54,99)(39,114,55,98)(40,113,56,97)(41,112,57,128)(42,111,58,127)(43,110,59,126)(44,109,60,125)(45,108,61,124)(46,107,62,123)(47,106,63,122)(48,105,64,121)(65,145,81,129)(66,144,82,160)(67,143,83,159)(68,142,84,158)(69,141,85,157)(70,140,86,156)(71,139,87,155)(72,138,88,154)(73,137,89,153)(74,136,90,152)(75,135,91,151)(76,134,92,150)(77,133,93,149)(78,132,94,148)(79,131,95,147)(80,130,96,146)(161,209,177,193)(162,208,178,224)(163,207,179,223)(164,206,180,222)(165,205,181,221)(166,204,182,220)(167,203,183,219)(168,202,184,218)(169,201,185,217)(170,200,186,216)(171,199,187,215)(172,198,188,214)(173,197,189,213)(174,196,190,212)(175,195,191,211)(176,194,192,210)(225,277,241,261)(226,276,242,260)(227,275,243,259)(228,274,244,258)(229,273,245,257)(230,272,246,288)(231,271,247,287)(232,270,248,286)(233,269,249,285)(234,268,250,284)(235,267,251,283)(236,266,252,282)(237,265,253,281)(238,264,254,280)(239,263,255,279)(240,262,256,278) );
G=PermutationGroup([[(1,172,120,240,139),(2,140,241,121,173),(3,174,122,242,141),(4,142,243,123,175),(5,176,124,244,143),(6,144,245,125,177),(7,178,126,246,145),(8,146,247,127,179),(9,180,128,248,147),(10,148,249,97,181),(11,182,98,250,149),(12,150,251,99,183),(13,184,100,252,151),(14,152,253,101,185),(15,186,102,254,153),(16,154,255,103,187),(17,188,104,256,155),(18,156,225,105,189),(19,190,106,226,157),(20,158,227,107,191),(21,192,108,228,159),(22,160,229,109,161),(23,162,110,230,129),(24,130,231,111,163),(25,164,112,232,131),(26,132,233,113,165),(27,166,114,234,133),(28,134,235,115,167),(29,168,116,236,135),(30,136,237,117,169),(31,170,118,238,137),(32,138,239,119,171),(33,278,71,306,214),(34,215,307,72,279),(35,280,73,308,216),(36,217,309,74,281),(37,282,75,310,218),(38,219,311,76,283),(39,284,77,312,220),(40,221,313,78,285),(41,286,79,314,222),(42,223,315,80,287),(43,288,81,316,224),(44,193,317,82,257),(45,258,83,318,194),(46,195,319,84,259),(47,260,85,320,196),(48,197,289,86,261),(49,262,87,290,198),(50,199,291,88,263),(51,264,89,292,200),(52,201,293,90,265),(53,266,91,294,202),(54,203,295,92,267),(55,268,93,296,204),(56,205,297,94,269),(57,270,95,298,206),(58,207,299,96,271),(59,272,65,300,208),(60,209,301,66,273),(61,274,67,302,210),(62,211,303,68,275),(63,276,69,304,212),(64,213,305,70,277)], [(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)], [(1,290,17,306),(2,289,18,305),(3,320,19,304),(4,319,20,303),(5,318,21,302),(6,317,22,301),(7,316,23,300),(8,315,24,299),(9,314,25,298),(10,313,26,297),(11,312,27,296),(12,311,28,295),(13,310,29,294),(14,309,30,293),(15,308,31,292),(16,307,32,291),(33,120,49,104),(34,119,50,103),(35,118,51,102),(36,117,52,101),(37,116,53,100),(38,115,54,99),(39,114,55,98),(40,113,56,97),(41,112,57,128),(42,111,58,127),(43,110,59,126),(44,109,60,125),(45,108,61,124),(46,107,62,123),(47,106,63,122),(48,105,64,121),(65,145,81,129),(66,144,82,160),(67,143,83,159),(68,142,84,158),(69,141,85,157),(70,140,86,156),(71,139,87,155),(72,138,88,154),(73,137,89,153),(74,136,90,152),(75,135,91,151),(76,134,92,150),(77,133,93,149),(78,132,94,148),(79,131,95,147),(80,130,96,146),(161,209,177,193),(162,208,178,224),(163,207,179,223),(164,206,180,222),(165,205,181,221),(166,204,182,220),(167,203,183,219),(168,202,184,218),(169,201,185,217),(170,200,186,216),(171,199,187,215),(172,198,188,214),(173,197,189,213),(174,196,190,212),(175,195,191,211),(176,194,192,210),(225,277,241,261),(226,276,242,260),(227,275,243,259),(228,274,244,258),(229,273,245,257),(230,272,246,288),(231,271,247,287),(232,270,248,286),(233,269,249,285),(234,268,250,284),(235,267,251,283),(236,266,252,282),(237,265,253,281),(238,264,254,280),(239,263,255,279),(240,262,256,278)]])
41 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 10A | 10B | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 32A | ··· | 32H | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 32 | ··· | 32 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 16 | 80 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 16 | 16 | 16 | 16 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | D4 | D5 | D8 | D10 | D16 | C5⋊D4 | Q64 | D4⋊D5 | C5⋊D16 | C5⋊Q64 |
| kernel | C5⋊Q64 | C5⋊2C32 | Dic40 | C5×Q32 | C40 | Q32 | C20 | C16 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 4 | 8 |
Matrix representation of C5⋊Q64 ►in GL4(𝔽641) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 640 | 1 |
| 0 | 0 | 277 | 363 |
| 371 | 440 | 0 | 0 |
| 625 | 527 | 0 | 0 |
| 0 | 0 | 411 | 218 |
| 0 | 0 | 119 | 230 |
| 262 | 85 | 0 | 0 |
| 399 | 379 | 0 | 0 |
| 0 | 0 | 180 | 478 |
| 0 | 0 | 360 | 461 |
G:=sub<GL(4,GF(641))| [1,0,0,0,0,1,0,0,0,0,640,277,0,0,1,363],[371,625,0,0,440,527,0,0,0,0,411,119,0,0,218,230],[262,399,0,0,85,379,0,0,0,0,180,360,0,0,478,461] >;
C5⋊Q64 in GAP, Magma, Sage, TeX
C_5\rtimes Q_{64} % in TeX
G:=Group("C5:Q64"); // GroupNames label
G:=SmallGroup(320,80);
// by ID
G=gap.SmallGroup(320,80);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,85,232,254,135,142,675,346,192,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^5=b^32=1,c^2=b^16,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export