metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5⋊2C64, C80.7C4, C40.8C8, C32.2D5, C20.5C16, C10.2C32, C160.3C2, C16.3Dic5, C2.(C5⋊2C32), C8.4(C5⋊2C8), C4.2(C5⋊2C16), SmallGroup(320,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C5⋊2C64 |
Generators and relations for C5⋊2C64
G = < a,b | a5=b64=1, bab-1=a-1 >
Smallest permutation representation of C5⋊2C64
►Regular action on 320 points
Generators in S320
(1 207 147 123 259)(2 260 124 148 208)(3 209 149 125 261)(4 262 126 150 210)(5 211 151 127 263)(6 264 128 152 212)(7 213 153 65 265)(8 266 66 154 214)(9 215 155 67 267)(10 268 68 156 216)(11 217 157 69 269)(12 270 70 158 218)(13 219 159 71 271)(14 272 72 160 220)(15 221 161 73 273)(16 274 74 162 222)(17 223 163 75 275)(18 276 76 164 224)(19 225 165 77 277)(20 278 78 166 226)(21 227 167 79 279)(22 280 80 168 228)(23 229 169 81 281)(24 282 82 170 230)(25 231 171 83 283)(26 284 84 172 232)(27 233 173 85 285)(28 286 86 174 234)(29 235 175 87 287)(30 288 88 176 236)(31 237 177 89 289)(32 290 90 178 238)(33 239 179 91 291)(34 292 92 180 240)(35 241 181 93 293)(36 294 94 182 242)(37 243 183 95 295)(38 296 96 184 244)(39 245 185 97 297)(40 298 98 186 246)(41 247 187 99 299)(42 300 100 188 248)(43 249 189 101 301)(44 302 102 190 250)(45 251 191 103 303)(46 304 104 192 252)(47 253 129 105 305)(48 306 106 130 254)(49 255 131 107 307)(50 308 108 132 256)(51 193 133 109 309)(52 310 110 134 194)(53 195 135 111 311)(54 312 112 136 196)(55 197 137 113 313)(56 314 114 138 198)(57 199 139 115 315)(58 316 116 140 200)(59 201 141 117 317)(60 318 118 142 202)(61 203 143 119 319)(62 320 120 144 204)(63 205 145 121 257)(64 258 122 146 206)
(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)| (1,207,147,123,259)(2,260,124,148,208)(3,209,149,125,261)(4,262,126,150,210)(5,211,151,127,263)(6,264,128,152,212)(7,213,153,65,265)(8,266,66,154,214)(9,215,155,67,267)(10,268,68,156,216)(11,217,157,69,269)(12,270,70,158,218)(13,219,159,71,271)(14,272,72,160,220)(15,221,161,73,273)(16,274,74,162,222)(17,223,163,75,275)(18,276,76,164,224)(19,225,165,77,277)(20,278,78,166,226)(21,227,167,79,279)(22,280,80,168,228)(23,229,169,81,281)(24,282,82,170,230)(25,231,171,83,283)(26,284,84,172,232)(27,233,173,85,285)(28,286,86,174,234)(29,235,175,87,287)(30,288,88,176,236)(31,237,177,89,289)(32,290,90,178,238)(33,239,179,91,291)(34,292,92,180,240)(35,241,181,93,293)(36,294,94,182,242)(37,243,183,95,295)(38,296,96,184,244)(39,245,185,97,297)(40,298,98,186,246)(41,247,187,99,299)(42,300,100,188,248)(43,249,189,101,301)(44,302,102,190,250)(45,251,191,103,303)(46,304,104,192,252)(47,253,129,105,305)(48,306,106,130,254)(49,255,131,107,307)(50,308,108,132,256)(51,193,133,109,309)(52,310,110,134,194)(53,195,135,111,311)(54,312,112,136,196)(55,197,137,113,313)(56,314,114,138,198)(57,199,139,115,315)(58,316,116,140,200)(59,201,141,117,317)(60,318,118,142,202)(61,203,143,119,319)(62,320,120,144,204)(63,205,145,121,257)(64,258,122,146,206), (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( (1,207,147,123,259)(2,260,124,148,208)(3,209,149,125,261)(4,262,126,150,210)(5,211,151,127,263)(6,264,128,152,212)(7,213,153,65,265)(8,266,66,154,214)(9,215,155,67,267)(10,268,68,156,216)(11,217,157,69,269)(12,270,70,158,218)(13,219,159,71,271)(14,272,72,160,220)(15,221,161,73,273)(16,274,74,162,222)(17,223,163,75,275)(18,276,76,164,224)(19,225,165,77,277)(20,278,78,166,226)(21,227,167,79,279)(22,280,80,168,228)(23,229,169,81,281)(24,282,82,170,230)(25,231,171,83,283)(26,284,84,172,232)(27,233,173,85,285)(28,286,86,174,234)(29,235,175,87,287)(30,288,88,176,236)(31,237,177,89,289)(32,290,90,178,238)(33,239,179,91,291)(34,292,92,180,240)(35,241,181,93,293)(36,294,94,182,242)(37,243,183,95,295)(38,296,96,184,244)(39,245,185,97,297)(40,298,98,186,246)(41,247,187,99,299)(42,300,100,188,248)(43,249,189,101,301)(44,302,102,190,250)(45,251,191,103,303)(46,304,104,192,252)(47,253,129,105,305)(48,306,106,130,254)(49,255,131,107,307)(50,308,108,132,256)(51,193,133,109,309)(52,310,110,134,194)(53,195,135,111,311)(54,312,112,136,196)(55,197,137,113,313)(56,314,114,138,198)(57,199,139,115,315)(58,316,116,140,200)(59,201,141,117,317)(60,318,118,142,202)(61,203,143,119,319)(62,320,120,144,204)(63,205,145,121,257)(64,258,122,146,206), (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([[(1,207,147,123,259),(2,260,124,148,208),(3,209,149,125,261),(4,262,126,150,210),(5,211,151,127,263),(6,264,128,152,212),(7,213,153,65,265),(8,266,66,154,214),(9,215,155,67,267),(10,268,68,156,216),(11,217,157,69,269),(12,270,70,158,218),(13,219,159,71,271),(14,272,72,160,220),(15,221,161,73,273),(16,274,74,162,222),(17,223,163,75,275),(18,276,76,164,224),(19,225,165,77,277),(20,278,78,166,226),(21,227,167,79,279),(22,280,80,168,228),(23,229,169,81,281),(24,282,82,170,230),(25,231,171,83,283),(26,284,84,172,232),(27,233,173,85,285),(28,286,86,174,234),(29,235,175,87,287),(30,288,88,176,236),(31,237,177,89,289),(32,290,90,178,238),(33,239,179,91,291),(34,292,92,180,240),(35,241,181,93,293),(36,294,94,182,242),(37,243,183,95,295),(38,296,96,184,244),(39,245,185,97,297),(40,298,98,186,246),(41,247,187,99,299),(42,300,100,188,248),(43,249,189,101,301),(44,302,102,190,250),(45,251,191,103,303),(46,304,104,192,252),(47,253,129,105,305),(48,306,106,130,254),(49,255,131,107,307),(50,308,108,132,256),(51,193,133,109,309),(52,310,110,134,194),(53,195,135,111,311),(54,312,112,136,196),(55,197,137,113,313),(56,314,114,138,198),(57,199,139,115,315),(58,316,116,140,200),(59,201,141,117,317),(60,318,118,142,202),(61,203,143,119,319),(62,320,120,144,204),(63,205,145,121,257),(64,258,122,146,206)], [(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)]])
128 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 32A | ··· | 32P | 40A | ··· | 40H | 64A | ··· | 64AF | 80A | ··· | 80P | 160A | ··· | 160AF |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 32 | ··· | 32 | 40 | ··· | 40 | 64 | ··· | 64 | 80 | ··· | 80 | 160 | ··· | 160 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | C64 | D5 | Dic5 | C5⋊2C8 | C5⋊2C16 | C5⋊2C32 | C5⋊2C64 |
| kernel | C5⋊2C64 | C160 | C80 | C40 | C20 | C10 | C5 | C32 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 32 | 2 | 2 | 4 | 8 | 16 | 32 |
Matrix representation of C5⋊2C64 ►in GL2(𝔽641) generated by
| 0 | 1 |
| 640 | 362 |
| 152 | 591 |
| 489 | 489 |
G:=sub<GL(2,GF(641))| [0,640,1,362],[152,489,591,489] >;
C5⋊2C64 in GAP, Magma, Sage, TeX
C_5\rtimes_2C_{64} % in TeX
G:=Group("C5:2C64"); // GroupNames label
G:=SmallGroup(320,1);
// by ID
G=gap.SmallGroup(320,1);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,14,36,58,80,102,12550]);
// Polycyclic
G:=Group<a,b|a^5=b^64=1,b*a*b^-1=a^-1>;
// generators/relations
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