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## G = C4×C120order 480 = 25·3·5

### Abelian group of type [4,120]

Aliases: C4×C120, SmallGroup(480,199)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C120
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — C2×C60 — C2×C120 — C4×C120
 Lower central C1 — C4×C120
 Upper central C1 — C4×C120

Generators and relations for C4×C120
G = < a,b | a4=b120=1, ab=ba >

Subgroups: 88, all normal (24 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, C12, C2×C6, C15, C42, C2×C8, C20, C2×C10, C24, C2×C12, C2×C12, C30, C30, C4×C8, C40, C2×C20, C2×C20, C4×C12, C2×C24, C60, C2×C30, C4×C20, C2×C40, C4×C24, C120, C2×C60, C2×C60, C4×C40, C4×C60, C2×C120, C4×C120
Quotients: C1, C2, C3, C4, C22, C5, C6, C8, C2×C4, C10, C12, C2×C6, C15, C42, C2×C8, C20, C2×C10, C24, C2×C12, C30, C4×C8, C40, C2×C20, C4×C12, C2×C24, C60, C2×C30, C4×C20, C2×C40, C4×C24, C120, C2×C60, C4×C40, C4×C60, C2×C120, C4×C120

Smallest permutation representation of C4×C120
Regular action on 480 points
Generators in S480
(1 406 147 351)(2 407 148 352)(3 408 149 353)(4 409 150 354)(5 410 151 355)(6 411 152 356)(7 412 153 357)(8 413 154 358)(9 414 155 359)(10 415 156 360)(11 416 157 241)(12 417 158 242)(13 418 159 243)(14 419 160 244)(15 420 161 245)(16 421 162 246)(17 422 163 247)(18 423 164 248)(19 424 165 249)(20 425 166 250)(21 426 167 251)(22 427 168 252)(23 428 169 253)(24 429 170 254)(25 430 171 255)(26 431 172 256)(27 432 173 257)(28 433 174 258)(29 434 175 259)(30 435 176 260)(31 436 177 261)(32 437 178 262)(33 438 179 263)(34 439 180 264)(35 440 181 265)(36 441 182 266)(37 442 183 267)(38 443 184 268)(39 444 185 269)(40 445 186 270)(41 446 187 271)(42 447 188 272)(43 448 189 273)(44 449 190 274)(45 450 191 275)(46 451 192 276)(47 452 193 277)(48 453 194 278)(49 454 195 279)(50 455 196 280)(51 456 197 281)(52 457 198 282)(53 458 199 283)(54 459 200 284)(55 460 201 285)(56 461 202 286)(57 462 203 287)(58 463 204 288)(59 464 205 289)(60 465 206 290)(61 466 207 291)(62 467 208 292)(63 468 209 293)(64 469 210 294)(65 470 211 295)(66 471 212 296)(67 472 213 297)(68 473 214 298)(69 474 215 299)(70 475 216 300)(71 476 217 301)(72 477 218 302)(73 478 219 303)(74 479 220 304)(75 480 221 305)(76 361 222 306)(77 362 223 307)(78 363 224 308)(79 364 225 309)(80 365 226 310)(81 366 227 311)(82 367 228 312)(83 368 229 313)(84 369 230 314)(85 370 231 315)(86 371 232 316)(87 372 233 317)(88 373 234 318)(89 374 235 319)(90 375 236 320)(91 376 237 321)(92 377 238 322)(93 378 239 323)(94 379 240 324)(95 380 121 325)(96 381 122 326)(97 382 123 327)(98 383 124 328)(99 384 125 329)(100 385 126 330)(101 386 127 331)(102 387 128 332)(103 388 129 333)(104 389 130 334)(105 390 131 335)(106 391 132 336)(107 392 133 337)(108 393 134 338)(109 394 135 339)(110 395 136 340)(111 396 137 341)(112 397 138 342)(113 398 139 343)(114 399 140 344)(115 400 141 345)(116 401 142 346)(117 402 143 347)(118 403 144 348)(119 404 145 349)(120 405 146 350)
(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)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480)

G:=sub<Sym(480)| (1,406,147,351)(2,407,148,352)(3,408,149,353)(4,409,150,354)(5,410,151,355)(6,411,152,356)(7,412,153,357)(8,413,154,358)(9,414,155,359)(10,415,156,360)(11,416,157,241)(12,417,158,242)(13,418,159,243)(14,419,160,244)(15,420,161,245)(16,421,162,246)(17,422,163,247)(18,423,164,248)(19,424,165,249)(20,425,166,250)(21,426,167,251)(22,427,168,252)(23,428,169,253)(24,429,170,254)(25,430,171,255)(26,431,172,256)(27,432,173,257)(28,433,174,258)(29,434,175,259)(30,435,176,260)(31,436,177,261)(32,437,178,262)(33,438,179,263)(34,439,180,264)(35,440,181,265)(36,441,182,266)(37,442,183,267)(38,443,184,268)(39,444,185,269)(40,445,186,270)(41,446,187,271)(42,447,188,272)(43,448,189,273)(44,449,190,274)(45,450,191,275)(46,451,192,276)(47,452,193,277)(48,453,194,278)(49,454,195,279)(50,455,196,280)(51,456,197,281)(52,457,198,282)(53,458,199,283)(54,459,200,284)(55,460,201,285)(56,461,202,286)(57,462,203,287)(58,463,204,288)(59,464,205,289)(60,465,206,290)(61,466,207,291)(62,467,208,292)(63,468,209,293)(64,469,210,294)(65,470,211,295)(66,471,212,296)(67,472,213,297)(68,473,214,298)(69,474,215,299)(70,475,216,300)(71,476,217,301)(72,477,218,302)(73,478,219,303)(74,479,220,304)(75,480,221,305)(76,361,222,306)(77,362,223,307)(78,363,224,308)(79,364,225,309)(80,365,226,310)(81,366,227,311)(82,367,228,312)(83,368,229,313)(84,369,230,314)(85,370,231,315)(86,371,232,316)(87,372,233,317)(88,373,234,318)(89,374,235,319)(90,375,236,320)(91,376,237,321)(92,377,238,322)(93,378,239,323)(94,379,240,324)(95,380,121,325)(96,381,122,326)(97,382,123,327)(98,383,124,328)(99,384,125,329)(100,385,126,330)(101,386,127,331)(102,387,128,332)(103,388,129,333)(104,389,130,334)(105,390,131,335)(106,391,132,336)(107,392,133,337)(108,393,134,338)(109,394,135,339)(110,395,136,340)(111,396,137,341)(112,397,138,342)(113,398,139,343)(114,399,140,344)(115,400,141,345)(116,401,142,346)(117,402,143,347)(118,403,144,348)(119,404,145,349)(120,405,146,350), (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)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480)>;

G:=Group( (1,406,147,351)(2,407,148,352)(3,408,149,353)(4,409,150,354)(5,410,151,355)(6,411,152,356)(7,412,153,357)(8,413,154,358)(9,414,155,359)(10,415,156,360)(11,416,157,241)(12,417,158,242)(13,418,159,243)(14,419,160,244)(15,420,161,245)(16,421,162,246)(17,422,163,247)(18,423,164,248)(19,424,165,249)(20,425,166,250)(21,426,167,251)(22,427,168,252)(23,428,169,253)(24,429,170,254)(25,430,171,255)(26,431,172,256)(27,432,173,257)(28,433,174,258)(29,434,175,259)(30,435,176,260)(31,436,177,261)(32,437,178,262)(33,438,179,263)(34,439,180,264)(35,440,181,265)(36,441,182,266)(37,442,183,267)(38,443,184,268)(39,444,185,269)(40,445,186,270)(41,446,187,271)(42,447,188,272)(43,448,189,273)(44,449,190,274)(45,450,191,275)(46,451,192,276)(47,452,193,277)(48,453,194,278)(49,454,195,279)(50,455,196,280)(51,456,197,281)(52,457,198,282)(53,458,199,283)(54,459,200,284)(55,460,201,285)(56,461,202,286)(57,462,203,287)(58,463,204,288)(59,464,205,289)(60,465,206,290)(61,466,207,291)(62,467,208,292)(63,468,209,293)(64,469,210,294)(65,470,211,295)(66,471,212,296)(67,472,213,297)(68,473,214,298)(69,474,215,299)(70,475,216,300)(71,476,217,301)(72,477,218,302)(73,478,219,303)(74,479,220,304)(75,480,221,305)(76,361,222,306)(77,362,223,307)(78,363,224,308)(79,364,225,309)(80,365,226,310)(81,366,227,311)(82,367,228,312)(83,368,229,313)(84,369,230,314)(85,370,231,315)(86,371,232,316)(87,372,233,317)(88,373,234,318)(89,374,235,319)(90,375,236,320)(91,376,237,321)(92,377,238,322)(93,378,239,323)(94,379,240,324)(95,380,121,325)(96,381,122,326)(97,382,123,327)(98,383,124,328)(99,384,125,329)(100,385,126,330)(101,386,127,331)(102,387,128,332)(103,388,129,333)(104,389,130,334)(105,390,131,335)(106,391,132,336)(107,392,133,337)(108,393,134,338)(109,394,135,339)(110,395,136,340)(111,396,137,341)(112,397,138,342)(113,398,139,343)(114,399,140,344)(115,400,141,345)(116,401,142,346)(117,402,143,347)(118,403,144,348)(119,404,145,349)(120,405,146,350), (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)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480) );

G=PermutationGroup([[(1,406,147,351),(2,407,148,352),(3,408,149,353),(4,409,150,354),(5,410,151,355),(6,411,152,356),(7,412,153,357),(8,413,154,358),(9,414,155,359),(10,415,156,360),(11,416,157,241),(12,417,158,242),(13,418,159,243),(14,419,160,244),(15,420,161,245),(16,421,162,246),(17,422,163,247),(18,423,164,248),(19,424,165,249),(20,425,166,250),(21,426,167,251),(22,427,168,252),(23,428,169,253),(24,429,170,254),(25,430,171,255),(26,431,172,256),(27,432,173,257),(28,433,174,258),(29,434,175,259),(30,435,176,260),(31,436,177,261),(32,437,178,262),(33,438,179,263),(34,439,180,264),(35,440,181,265),(36,441,182,266),(37,442,183,267),(38,443,184,268),(39,444,185,269),(40,445,186,270),(41,446,187,271),(42,447,188,272),(43,448,189,273),(44,449,190,274),(45,450,191,275),(46,451,192,276),(47,452,193,277),(48,453,194,278),(49,454,195,279),(50,455,196,280),(51,456,197,281),(52,457,198,282),(53,458,199,283),(54,459,200,284),(55,460,201,285),(56,461,202,286),(57,462,203,287),(58,463,204,288),(59,464,205,289),(60,465,206,290),(61,466,207,291),(62,467,208,292),(63,468,209,293),(64,469,210,294),(65,470,211,295),(66,471,212,296),(67,472,213,297),(68,473,214,298),(69,474,215,299),(70,475,216,300),(71,476,217,301),(72,477,218,302),(73,478,219,303),(74,479,220,304),(75,480,221,305),(76,361,222,306),(77,362,223,307),(78,363,224,308),(79,364,225,309),(80,365,226,310),(81,366,227,311),(82,367,228,312),(83,368,229,313),(84,369,230,314),(85,370,231,315),(86,371,232,316),(87,372,233,317),(88,373,234,318),(89,374,235,319),(90,375,236,320),(91,376,237,321),(92,377,238,322),(93,378,239,323),(94,379,240,324),(95,380,121,325),(96,381,122,326),(97,382,123,327),(98,383,124,328),(99,384,125,329),(100,385,126,330),(101,386,127,331),(102,387,128,332),(103,388,129,333),(104,389,130,334),(105,390,131,335),(106,391,132,336),(107,392,133,337),(108,393,134,338),(109,394,135,339),(110,395,136,340),(111,396,137,341),(112,397,138,342),(113,398,139,343),(114,399,140,344),(115,400,141,345),(116,401,142,346),(117,402,143,347),(118,403,144,348),(119,404,145,349),(120,405,146,350)], [(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),(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480)]])

480 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A ··· 4L 5A 5B 5C 5D 6A ··· 6F 8A ··· 8P 10A ··· 10L 12A ··· 12X 15A ··· 15H 20A ··· 20AV 24A ··· 24AF 30A ··· 30X 40A ··· 40BL 60A ··· 60CR 120A ··· 120DX order 1 2 2 2 3 3 4 ··· 4 5 5 5 5 6 ··· 6 8 ··· 8 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 1 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

480 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C4 C5 C6 C6 C8 C10 C10 C12 C12 C15 C20 C20 C24 C30 C30 C40 C60 C60 C120 kernel C4×C120 C4×C60 C2×C120 C4×C40 C120 C2×C60 C4×C24 C4×C20 C2×C40 C60 C4×C12 C2×C24 C40 C2×C20 C4×C8 C24 C2×C12 C20 C42 C2×C8 C12 C8 C2×C4 C4 # reps 1 1 2 2 8 4 4 2 4 16 4 8 16 8 8 32 16 32 8 16 64 64 32 128

Matrix representation of C4×C120 in GL2(𝔽241) generated by

 177 0 0 240
,
 174 0 0 180
G:=sub<GL(2,GF(241))| [177,0,0,240],[174,0,0,180] >;

C4×C120 in GAP, Magma, Sage, TeX

C_4\times C_{120}
% in TeX

G:=Group("C4xC120");
// GroupNames label

G:=SmallGroup(480,199);
// by ID

G=gap.SmallGroup(480,199);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,420,848,172]);
// Polycyclic

G:=Group<a,b|a^4=b^120=1,a*b=b*a>;
// generators/relations

׿
×
𝔽