metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5⋊C64, C10.C32, C40.2C8, C80.3C4, C16.4F5, C20.2C16, C2.(C5⋊C32), C8.4(C5⋊C8), C4.2(C5⋊C16), C5⋊2C32.2C2, SmallGroup(320,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C5⋊C64 |
Generators and relations for C5⋊C64
G = < a,b | a5=b64=1, bab-1=a3 >
Smallest permutation representation of C5⋊C64
►Regular action on 320 points
Generators in S320
(1 139 205 113 315)(2 114 140 316 206)(3 317 115 207 141)(4 208 318 142 116)(5 143 209 117 319)(6 118 144 320 210)(7 257 119 211 145)(8 212 258 146 120)(9 147 213 121 259)(10 122 148 260 214)(11 261 123 215 149)(12 216 262 150 124)(13 151 217 125 263)(14 126 152 264 218)(15 265 127 219 153)(16 220 266 154 128)(17 155 221 65 267)(18 66 156 268 222)(19 269 67 223 157)(20 224 270 158 68)(21 159 225 69 271)(22 70 160 272 226)(23 273 71 227 161)(24 228 274 162 72)(25 163 229 73 275)(26 74 164 276 230)(27 277 75 231 165)(28 232 278 166 76)(29 167 233 77 279)(30 78 168 280 234)(31 281 79 235 169)(32 236 282 170 80)(33 171 237 81 283)(34 82 172 284 238)(35 285 83 239 173)(36 240 286 174 84)(37 175 241 85 287)(38 86 176 288 242)(39 289 87 243 177)(40 244 290 178 88)(41 179 245 89 291)(42 90 180 292 246)(43 293 91 247 181)(44 248 294 182 92)(45 183 249 93 295)(46 94 184 296 250)(47 297 95 251 185)(48 252 298 186 96)(49 187 253 97 299)(50 98 188 300 254)(51 301 99 255 189)(52 256 302 190 100)(53 191 193 101 303)(54 102 192 304 194)(55 305 103 195 129)(56 196 306 130 104)(57 131 197 105 307)(58 106 132 308 198)(59 309 107 199 133)(60 200 310 134 108)(61 135 201 109 311)(62 110 136 312 202)(63 313 111 203 137)(64 204 314 138 112)
(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)
G:=sub<Sym(320)| (1,139,205,113,315)(2,114,140,316,206)(3,317,115,207,141)(4,208,318,142,116)(5,143,209,117,319)(6,118,144,320,210)(7,257,119,211,145)(8,212,258,146,120)(9,147,213,121,259)(10,122,148,260,214)(11,261,123,215,149)(12,216,262,150,124)(13,151,217,125,263)(14,126,152,264,218)(15,265,127,219,153)(16,220,266,154,128)(17,155,221,65,267)(18,66,156,268,222)(19,269,67,223,157)(20,224,270,158,68)(21,159,225,69,271)(22,70,160,272,226)(23,273,71,227,161)(24,228,274,162,72)(25,163,229,73,275)(26,74,164,276,230)(27,277,75,231,165)(28,232,278,166,76)(29,167,233,77,279)(30,78,168,280,234)(31,281,79,235,169)(32,236,282,170,80)(33,171,237,81,283)(34,82,172,284,238)(35,285,83,239,173)(36,240,286,174,84)(37,175,241,85,287)(38,86,176,288,242)(39,289,87,243,177)(40,244,290,178,88)(41,179,245,89,291)(42,90,180,292,246)(43,293,91,247,181)(44,248,294,182,92)(45,183,249,93,295)(46,94,184,296,250)(47,297,95,251,185)(48,252,298,186,96)(49,187,253,97,299)(50,98,188,300,254)(51,301,99,255,189)(52,256,302,190,100)(53,191,193,101,303)(54,102,192,304,194)(55,305,103,195,129)(56,196,306,130,104)(57,131,197,105,307)(58,106,132,308,198)(59,309,107,199,133)(60,200,310,134,108)(61,135,201,109,311)(62,110,136,312,202)(63,313,111,203,137)(64,204,314,138,112), (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)>;
G:=Group( (1,139,205,113,315)(2,114,140,316,206)(3,317,115,207,141)(4,208,318,142,116)(5,143,209,117,319)(6,118,144,320,210)(7,257,119,211,145)(8,212,258,146,120)(9,147,213,121,259)(10,122,148,260,214)(11,261,123,215,149)(12,216,262,150,124)(13,151,217,125,263)(14,126,152,264,218)(15,265,127,219,153)(16,220,266,154,128)(17,155,221,65,267)(18,66,156,268,222)(19,269,67,223,157)(20,224,270,158,68)(21,159,225,69,271)(22,70,160,272,226)(23,273,71,227,161)(24,228,274,162,72)(25,163,229,73,275)(26,74,164,276,230)(27,277,75,231,165)(28,232,278,166,76)(29,167,233,77,279)(30,78,168,280,234)(31,281,79,235,169)(32,236,282,170,80)(33,171,237,81,283)(34,82,172,284,238)(35,285,83,239,173)(36,240,286,174,84)(37,175,241,85,287)(38,86,176,288,242)(39,289,87,243,177)(40,244,290,178,88)(41,179,245,89,291)(42,90,180,292,246)(43,293,91,247,181)(44,248,294,182,92)(45,183,249,93,295)(46,94,184,296,250)(47,297,95,251,185)(48,252,298,186,96)(49,187,253,97,299)(50,98,188,300,254)(51,301,99,255,189)(52,256,302,190,100)(53,191,193,101,303)(54,102,192,304,194)(55,305,103,195,129)(56,196,306,130,104)(57,131,197,105,307)(58,106,132,308,198)(59,309,107,199,133)(60,200,310,134,108)(61,135,201,109,311)(62,110,136,312,202)(63,313,111,203,137)(64,204,314,138,112), (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) );
G=PermutationGroup([[(1,139,205,113,315),(2,114,140,316,206),(3,317,115,207,141),(4,208,318,142,116),(5,143,209,117,319),(6,118,144,320,210),(7,257,119,211,145),(8,212,258,146,120),(9,147,213,121,259),(10,122,148,260,214),(11,261,123,215,149),(12,216,262,150,124),(13,151,217,125,263),(14,126,152,264,218),(15,265,127,219,153),(16,220,266,154,128),(17,155,221,65,267),(18,66,156,268,222),(19,269,67,223,157),(20,224,270,158,68),(21,159,225,69,271),(22,70,160,272,226),(23,273,71,227,161),(24,228,274,162,72),(25,163,229,73,275),(26,74,164,276,230),(27,277,75,231,165),(28,232,278,166,76),(29,167,233,77,279),(30,78,168,280,234),(31,281,79,235,169),(32,236,282,170,80),(33,171,237,81,283),(34,82,172,284,238),(35,285,83,239,173),(36,240,286,174,84),(37,175,241,85,287),(38,86,176,288,242),(39,289,87,243,177),(40,244,290,178,88),(41,179,245,89,291),(42,90,180,292,246),(43,293,91,247,181),(44,248,294,182,92),(45,183,249,93,295),(46,94,184,296,250),(47,297,95,251,185),(48,252,298,186,96),(49,187,253,97,299),(50,98,188,300,254),(51,301,99,255,189),(52,256,302,190,100),(53,191,193,101,303),(54,102,192,304,194),(55,305,103,195,129),(56,196,306,130,104),(57,131,197,105,307),(58,106,132,308,198),(59,309,107,199,133),(60,200,310,134,108),(61,135,201,109,311),(62,110,136,312,202),(63,313,111,203,137),(64,204,314,138,112)], [(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)]])
80 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5 | 8A | 8B | 8C | 8D | 10 | 16A | ··· | 16H | 20A | 20B | 32A | ··· | 32P | 40A | 40B | 40C | 40D | 64A | ··· | 64AF | 80A | ··· | 80H |
| order | 1 | 2 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 10 | 16 | ··· | 16 | 20 | 20 | 32 | ··· | 32 | 40 | 40 | 40 | 40 | 64 | ··· | 64 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 4 | 1 | ··· | 1 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 5 | ··· | 5 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | C64 | F5 | C5⋊C8 | C5⋊C16 | C5⋊C32 | C5⋊C64 |
| kernel | C5⋊C64 | C5⋊2C32 | C80 | C40 | C20 | C10 | C5 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 32 | 1 | 1 | 2 | 4 | 8 |
Matrix representation of C5⋊C64 ►in GL4(𝔽641) generated by
| 640 | 1 | 0 | 0 |
| 640 | 0 | 1 | 0 |
| 640 | 0 | 0 | 1 |
| 640 | 0 | 0 | 0 |
| 636 | 292 | 365 | 504 |
| 360 | 155 | 491 | 499 |
| 486 | 150 | 142 | 223 |
| 137 | 515 | 5 | 349 |
G:=sub<GL(4,GF(641))| [640,640,640,640,1,0,0,0,0,1,0,0,0,0,1,0],[636,360,486,137,292,155,150,515,365,491,142,5,504,499,223,349] >;
C5⋊C64 in GAP, Magma, Sage, TeX
C_5\rtimes C_{64} % in TeX
G:=Group("C5:C64"); // GroupNames label
G:=SmallGroup(320,3);
// by ID
G=gap.SmallGroup(320,3);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,14,36,58,80,102,6278,6285]);
// Polycyclic
G:=Group<a,b|a^5=b^64=1,b*a*b^-1=a^3>;
// generators/relations
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