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G = C5×C80order 400 = 24·52

Abelian group of type [5,80]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C80, SmallGroup(400,51)

Series: Derived Chief Lower central Upper central

C1 — C5×C80
C1C2C4C8C40C5×C40 — 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
(1 222 99 398 306)(2 223 100 399 307)(3 224 101 400 308)(4 225 102 321 309)(5 226 103 322 310)(6 227 104 323 311)(7 228 105 324 312)(8 229 106 325 313)(9 230 107 326 314)(10 231 108 327 315)(11 232 109 328 316)(12 233 110 329 317)(13 234 111 330 318)(14 235 112 331 319)(15 236 113 332 320)(16 237 114 333 241)(17 238 115 334 242)(18 239 116 335 243)(19 240 117 336 244)(20 161 118 337 245)(21 162 119 338 246)(22 163 120 339 247)(23 164 121 340 248)(24 165 122 341 249)(25 166 123 342 250)(26 167 124 343 251)(27 168 125 344 252)(28 169 126 345 253)(29 170 127 346 254)(30 171 128 347 255)(31 172 129 348 256)(32 173 130 349 257)(33 174 131 350 258)(34 175 132 351 259)(35 176 133 352 260)(36 177 134 353 261)(37 178 135 354 262)(38 179 136 355 263)(39 180 137 356 264)(40 181 138 357 265)(41 182 139 358 266)(42 183 140 359 267)(43 184 141 360 268)(44 185 142 361 269)(45 186 143 362 270)(46 187 144 363 271)(47 188 145 364 272)(48 189 146 365 273)(49 190 147 366 274)(50 191 148 367 275)(51 192 149 368 276)(52 193 150 369 277)(53 194 151 370 278)(54 195 152 371 279)(55 196 153 372 280)(56 197 154 373 281)(57 198 155 374 282)(58 199 156 375 283)(59 200 157 376 284)(60 201 158 377 285)(61 202 159 378 286)(62 203 160 379 287)(63 204 81 380 288)(64 205 82 381 289)(65 206 83 382 290)(66 207 84 383 291)(67 208 85 384 292)(68 209 86 385 293)(69 210 87 386 294)(70 211 88 387 295)(71 212 89 388 296)(72 213 90 389 297)(73 214 91 390 298)(74 215 92 391 299)(75 216 93 392 300)(76 217 94 393 301)(77 218 95 394 302)(78 219 96 395 303)(79 220 97 396 304)(80 221 98 397 305)
(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)| (1,222,99,398,306)(2,223,100,399,307)(3,224,101,400,308)(4,225,102,321,309)(5,226,103,322,310)(6,227,104,323,311)(7,228,105,324,312)(8,229,106,325,313)(9,230,107,326,314)(10,231,108,327,315)(11,232,109,328,316)(12,233,110,329,317)(13,234,111,330,318)(14,235,112,331,319)(15,236,113,332,320)(16,237,114,333,241)(17,238,115,334,242)(18,239,116,335,243)(19,240,117,336,244)(20,161,118,337,245)(21,162,119,338,246)(22,163,120,339,247)(23,164,121,340,248)(24,165,122,341,249)(25,166,123,342,250)(26,167,124,343,251)(27,168,125,344,252)(28,169,126,345,253)(29,170,127,346,254)(30,171,128,347,255)(31,172,129,348,256)(32,173,130,349,257)(33,174,131,350,258)(34,175,132,351,259)(35,176,133,352,260)(36,177,134,353,261)(37,178,135,354,262)(38,179,136,355,263)(39,180,137,356,264)(40,181,138,357,265)(41,182,139,358,266)(42,183,140,359,267)(43,184,141,360,268)(44,185,142,361,269)(45,186,143,362,270)(46,187,144,363,271)(47,188,145,364,272)(48,189,146,365,273)(49,190,147,366,274)(50,191,148,367,275)(51,192,149,368,276)(52,193,150,369,277)(53,194,151,370,278)(54,195,152,371,279)(55,196,153,372,280)(56,197,154,373,281)(57,198,155,374,282)(58,199,156,375,283)(59,200,157,376,284)(60,201,158,377,285)(61,202,159,378,286)(62,203,160,379,287)(63,204,81,380,288)(64,205,82,381,289)(65,206,83,382,290)(66,207,84,383,291)(67,208,85,384,292)(68,209,86,385,293)(69,210,87,386,294)(70,211,88,387,295)(71,212,89,388,296)(72,213,90,389,297)(73,214,91,390,298)(74,215,92,391,299)(75,216,93,392,300)(76,217,94,393,301)(77,218,95,394,302)(78,219,96,395,303)(79,220,97,396,304)(80,221,98,397,305), (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( (1,222,99,398,306)(2,223,100,399,307)(3,224,101,400,308)(4,225,102,321,309)(5,226,103,322,310)(6,227,104,323,311)(7,228,105,324,312)(8,229,106,325,313)(9,230,107,326,314)(10,231,108,327,315)(11,232,109,328,316)(12,233,110,329,317)(13,234,111,330,318)(14,235,112,331,319)(15,236,113,332,320)(16,237,114,333,241)(17,238,115,334,242)(18,239,116,335,243)(19,240,117,336,244)(20,161,118,337,245)(21,162,119,338,246)(22,163,120,339,247)(23,164,121,340,248)(24,165,122,341,249)(25,166,123,342,250)(26,167,124,343,251)(27,168,125,344,252)(28,169,126,345,253)(29,170,127,346,254)(30,171,128,347,255)(31,172,129,348,256)(32,173,130,349,257)(33,174,131,350,258)(34,175,132,351,259)(35,176,133,352,260)(36,177,134,353,261)(37,178,135,354,262)(38,179,136,355,263)(39,180,137,356,264)(40,181,138,357,265)(41,182,139,358,266)(42,183,140,359,267)(43,184,141,360,268)(44,185,142,361,269)(45,186,143,362,270)(46,187,144,363,271)(47,188,145,364,272)(48,189,146,365,273)(49,190,147,366,274)(50,191,148,367,275)(51,192,149,368,276)(52,193,150,369,277)(53,194,151,370,278)(54,195,152,371,279)(55,196,153,372,280)(56,197,154,373,281)(57,198,155,374,282)(58,199,156,375,283)(59,200,157,376,284)(60,201,158,377,285)(61,202,159,378,286)(62,203,160,379,287)(63,204,81,380,288)(64,205,82,381,289)(65,206,83,382,290)(66,207,84,383,291)(67,208,85,384,292)(68,209,86,385,293)(69,210,87,386,294)(70,211,88,387,295)(71,212,89,388,296)(72,213,90,389,297)(73,214,91,390,298)(74,215,92,391,299)(75,216,93,392,300)(76,217,94,393,301)(77,218,95,394,302)(78,219,96,395,303)(79,220,97,396,304)(80,221,98,397,305), (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([(1,222,99,398,306),(2,223,100,399,307),(3,224,101,400,308),(4,225,102,321,309),(5,226,103,322,310),(6,227,104,323,311),(7,228,105,324,312),(8,229,106,325,313),(9,230,107,326,314),(10,231,108,327,315),(11,232,109,328,316),(12,233,110,329,317),(13,234,111,330,318),(14,235,112,331,319),(15,236,113,332,320),(16,237,114,333,241),(17,238,115,334,242),(18,239,116,335,243),(19,240,117,336,244),(20,161,118,337,245),(21,162,119,338,246),(22,163,120,339,247),(23,164,121,340,248),(24,165,122,341,249),(25,166,123,342,250),(26,167,124,343,251),(27,168,125,344,252),(28,169,126,345,253),(29,170,127,346,254),(30,171,128,347,255),(31,172,129,348,256),(32,173,130,349,257),(33,174,131,350,258),(34,175,132,351,259),(35,176,133,352,260),(36,177,134,353,261),(37,178,135,354,262),(38,179,136,355,263),(39,180,137,356,264),(40,181,138,357,265),(41,182,139,358,266),(42,183,140,359,267),(43,184,141,360,268),(44,185,142,361,269),(45,186,143,362,270),(46,187,144,363,271),(47,188,145,364,272),(48,189,146,365,273),(49,190,147,366,274),(50,191,148,367,275),(51,192,149,368,276),(52,193,150,369,277),(53,194,151,370,278),(54,195,152,371,279),(55,196,153,372,280),(56,197,154,373,281),(57,198,155,374,282),(58,199,156,375,283),(59,200,157,376,284),(60,201,158,377,285),(61,202,159,378,286),(62,203,160,379,287),(63,204,81,380,288),(64,205,82,381,289),(65,206,83,382,290),(66,207,84,383,291),(67,208,85,384,292),(68,209,86,385,293),(69,210,87,386,294),(70,211,88,387,295),(71,212,89,388,296),(72,213,90,389,297),(73,214,91,390,298),(74,215,92,391,299),(75,216,93,392,300),(76,217,94,393,301),(77,218,95,394,302),(78,219,96,395,303),(79,220,97,396,304),(80,221,98,397,305)], [(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 4A4B5A···5X8A8B8C8D10A···10X16A···16H20A···20AV40A···40CR80A···80GJ
order12445···5888810···1016···1620···2040···4080···80
size11111···111111···11···11···11···11···1

400 irreducible representations

dim1111111111
type++
imageC1C2C4C5C8C10C16C20C40C80
kernelC5×C80C5×C40C5×C20C80C5×C10C40C52C20C10C5
# reps1122442484896192

Matrix representation of C5×C80 in GL2(𝔽241) generated by

2050
087
,
2400
0102
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

Subgroup lattice of C5×C80 in TeX

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