direct product, abelian, monomial, 5-elementary
Aliases: C5×C80, SmallGroup(400,51)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C80 |
| C1 — C5×C80 |
| C1 — C5×C80 |
Generators and relations for C5×C80
G = < a,b | a5=b80=1, ab=ba >
Smallest permutation representation of C5×C80
►Regular action on 400 points
Generators in S400
(1 153 395 196 319)(2 154 396 197 320)(3 155 397 198 241)(4 156 398 199 242)(5 157 399 200 243)(6 158 400 201 244)(7 159 321 202 245)(8 160 322 203 246)(9 81 323 204 247)(10 82 324 205 248)(11 83 325 206 249)(12 84 326 207 250)(13 85 327 208 251)(14 86 328 209 252)(15 87 329 210 253)(16 88 330 211 254)(17 89 331 212 255)(18 90 332 213 256)(19 91 333 214 257)(20 92 334 215 258)(21 93 335 216 259)(22 94 336 217 260)(23 95 337 218 261)(24 96 338 219 262)(25 97 339 220 263)(26 98 340 221 264)(27 99 341 222 265)(28 100 342 223 266)(29 101 343 224 267)(30 102 344 225 268)(31 103 345 226 269)(32 104 346 227 270)(33 105 347 228 271)(34 106 348 229 272)(35 107 349 230 273)(36 108 350 231 274)(37 109 351 232 275)(38 110 352 233 276)(39 111 353 234 277)(40 112 354 235 278)(41 113 355 236 279)(42 114 356 237 280)(43 115 357 238 281)(44 116 358 239 282)(45 117 359 240 283)(46 118 360 161 284)(47 119 361 162 285)(48 120 362 163 286)(49 121 363 164 287)(50 122 364 165 288)(51 123 365 166 289)(52 124 366 167 290)(53 125 367 168 291)(54 126 368 169 292)(55 127 369 170 293)(56 128 370 171 294)(57 129 371 172 295)(58 130 372 173 296)(59 131 373 174 297)(60 132 374 175 298)(61 133 375 176 299)(62 134 376 177 300)(63 135 377 178 301)(64 136 378 179 302)(65 137 379 180 303)(66 138 380 181 304)(67 139 381 182 305)(68 140 382 183 306)(69 141 383 184 307)(70 142 384 185 308)(71 143 385 186 309)(72 144 386 187 310)(73 145 387 188 311)(74 146 388 189 312)(75 147 389 190 313)(76 148 390 191 314)(77 149 391 192 315)(78 150 392 193 316)(79 151 393 194 317)(80 152 394 195 318)
(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)(321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400)
G:=sub<Sym(400)| (1,153,395,196,319)(2,154,396,197,320)(3,155,397,198,241)(4,156,398,199,242)(5,157,399,200,243)(6,158,400,201,244)(7,159,321,202,245)(8,160,322,203,246)(9,81,323,204,247)(10,82,324,205,248)(11,83,325,206,249)(12,84,326,207,250)(13,85,327,208,251)(14,86,328,209,252)(15,87,329,210,253)(16,88,330,211,254)(17,89,331,212,255)(18,90,332,213,256)(19,91,333,214,257)(20,92,334,215,258)(21,93,335,216,259)(22,94,336,217,260)(23,95,337,218,261)(24,96,338,219,262)(25,97,339,220,263)(26,98,340,221,264)(27,99,341,222,265)(28,100,342,223,266)(29,101,343,224,267)(30,102,344,225,268)(31,103,345,226,269)(32,104,346,227,270)(33,105,347,228,271)(34,106,348,229,272)(35,107,349,230,273)(36,108,350,231,274)(37,109,351,232,275)(38,110,352,233,276)(39,111,353,234,277)(40,112,354,235,278)(41,113,355,236,279)(42,114,356,237,280)(43,115,357,238,281)(44,116,358,239,282)(45,117,359,240,283)(46,118,360,161,284)(47,119,361,162,285)(48,120,362,163,286)(49,121,363,164,287)(50,122,364,165,288)(51,123,365,166,289)(52,124,366,167,290)(53,125,367,168,291)(54,126,368,169,292)(55,127,369,170,293)(56,128,370,171,294)(57,129,371,172,295)(58,130,372,173,296)(59,131,373,174,297)(60,132,374,175,298)(61,133,375,176,299)(62,134,376,177,300)(63,135,377,178,301)(64,136,378,179,302)(65,137,379,180,303)(66,138,380,181,304)(67,139,381,182,305)(68,140,382,183,306)(69,141,383,184,307)(70,142,384,185,308)(71,143,385,186,309)(72,144,386,187,310)(73,145,387,188,311)(74,146,388,189,312)(75,147,389,190,313)(76,148,390,191,314)(77,149,391,192,315)(78,150,392,193,316)(79,151,393,194,317)(80,152,394,195,318), (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)(321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400)>;
G:=Group( (1,153,395,196,319)(2,154,396,197,320)(3,155,397,198,241)(4,156,398,199,242)(5,157,399,200,243)(6,158,400,201,244)(7,159,321,202,245)(8,160,322,203,246)(9,81,323,204,247)(10,82,324,205,248)(11,83,325,206,249)(12,84,326,207,250)(13,85,327,208,251)(14,86,328,209,252)(15,87,329,210,253)(16,88,330,211,254)(17,89,331,212,255)(18,90,332,213,256)(19,91,333,214,257)(20,92,334,215,258)(21,93,335,216,259)(22,94,336,217,260)(23,95,337,218,261)(24,96,338,219,262)(25,97,339,220,263)(26,98,340,221,264)(27,99,341,222,265)(28,100,342,223,266)(29,101,343,224,267)(30,102,344,225,268)(31,103,345,226,269)(32,104,346,227,270)(33,105,347,228,271)(34,106,348,229,272)(35,107,349,230,273)(36,108,350,231,274)(37,109,351,232,275)(38,110,352,233,276)(39,111,353,234,277)(40,112,354,235,278)(41,113,355,236,279)(42,114,356,237,280)(43,115,357,238,281)(44,116,358,239,282)(45,117,359,240,283)(46,118,360,161,284)(47,119,361,162,285)(48,120,362,163,286)(49,121,363,164,287)(50,122,364,165,288)(51,123,365,166,289)(52,124,366,167,290)(53,125,367,168,291)(54,126,368,169,292)(55,127,369,170,293)(56,128,370,171,294)(57,129,371,172,295)(58,130,372,173,296)(59,131,373,174,297)(60,132,374,175,298)(61,133,375,176,299)(62,134,376,177,300)(63,135,377,178,301)(64,136,378,179,302)(65,137,379,180,303)(66,138,380,181,304)(67,139,381,182,305)(68,140,382,183,306)(69,141,383,184,307)(70,142,384,185,308)(71,143,385,186,309)(72,144,386,187,310)(73,145,387,188,311)(74,146,388,189,312)(75,147,389,190,313)(76,148,390,191,314)(77,149,391,192,315)(78,150,392,193,316)(79,151,393,194,317)(80,152,394,195,318), (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)(321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400) );
G=PermutationGroup([[(1,153,395,196,319),(2,154,396,197,320),(3,155,397,198,241),(4,156,398,199,242),(5,157,399,200,243),(6,158,400,201,244),(7,159,321,202,245),(8,160,322,203,246),(9,81,323,204,247),(10,82,324,205,248),(11,83,325,206,249),(12,84,326,207,250),(13,85,327,208,251),(14,86,328,209,252),(15,87,329,210,253),(16,88,330,211,254),(17,89,331,212,255),(18,90,332,213,256),(19,91,333,214,257),(20,92,334,215,258),(21,93,335,216,259),(22,94,336,217,260),(23,95,337,218,261),(24,96,338,219,262),(25,97,339,220,263),(26,98,340,221,264),(27,99,341,222,265),(28,100,342,223,266),(29,101,343,224,267),(30,102,344,225,268),(31,103,345,226,269),(32,104,346,227,270),(33,105,347,228,271),(34,106,348,229,272),(35,107,349,230,273),(36,108,350,231,274),(37,109,351,232,275),(38,110,352,233,276),(39,111,353,234,277),(40,112,354,235,278),(41,113,355,236,279),(42,114,356,237,280),(43,115,357,238,281),(44,116,358,239,282),(45,117,359,240,283),(46,118,360,161,284),(47,119,361,162,285),(48,120,362,163,286),(49,121,363,164,287),(50,122,364,165,288),(51,123,365,166,289),(52,124,366,167,290),(53,125,367,168,291),(54,126,368,169,292),(55,127,369,170,293),(56,128,370,171,294),(57,129,371,172,295),(58,130,372,173,296),(59,131,373,174,297),(60,132,374,175,298),(61,133,375,176,299),(62,134,376,177,300),(63,135,377,178,301),(64,136,378,179,302),(65,137,379,180,303),(66,138,380,181,304),(67,139,381,182,305),(68,140,382,183,306),(69,141,383,184,307),(70,142,384,185,308),(71,143,385,186,309),(72,144,386,187,310),(73,145,387,188,311),(74,146,388,189,312),(75,147,389,190,313),(76,148,390,191,314),(77,149,391,192,315),(78,150,392,193,316),(79,151,393,194,317),(80,152,394,195,318)], [(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),(321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400)]])
400 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | ··· | 5X | 8A | 8B | 8C | 8D | 10A | ··· | 10X | 16A | ··· | 16H | 20A | ··· | 20AV | 40A | ··· | 40CR | 80A | ··· | 80GJ |
| order | 1 | 2 | 4 | 4 | 5 | ··· | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
400 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||
| image | C1 | C2 | C4 | C5 | C8 | C10 | C16 | C20 | C40 | C80 |
| kernel | C5×C80 | C5×C40 | C5×C20 | C80 | C5×C10 | C40 | C52 | C20 | C10 | C5 |
| # reps | 1 | 1 | 2 | 24 | 4 | 24 | 8 | 48 | 96 | 192 |
Matrix representation of C5×C80 ►in GL2(𝔽241) generated by
| 205 | 0 |
| 0 | 87 |
| 240 | 0 |
| 0 | 102 |
G:=sub<GL(2,GF(241))| [205,0,0,87],[240,0,0,102] >;
C5×C80 in GAP, Magma, Sage, TeX
C_5\times C_{80} % in TeX
G:=Group("C5xC80"); // GroupNames label
G:=SmallGroup(400,51);
// by ID
G=gap.SmallGroup(400,51);
# by ID
G:=PCGroup([6,-2,-5,-5,-2,-2,-2,300,69,88]);
// Polycyclic
G:=Group<a,b|a^5=b^80=1,a*b=b*a>;
// generators/relations
Export