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G = C4×C120order 480 = 25·3·5

Abelian group of type [4,120]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C120, SmallGroup(480,199)

Series: Derived Chief Lower central Upper central

C1 — C4×C120
C1C2C22C2×C4C2×C20C2×C60C2×C120 — C4×C120
C1 — C4×C120
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 [×2], C3, C4 [×6], C22, C5, C6, C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C10, C10 [×2], C12 [×6], C2×C6, C15, C42, C2×C8 [×2], C20 [×6], C2×C10, C24 [×4], C2×C12, C2×C12 [×2], C30, C30 [×2], C4×C8, C40 [×4], C2×C20, C2×C20 [×2], C4×C12, C2×C24 [×2], C60 [×6], C2×C30, C4×C20, C2×C40 [×2], C4×C24, C120 [×4], C2×C60, C2×C60 [×2], C4×C40, C4×C60, C2×C120 [×2], C4×C120
Quotients: C1, C2 [×3], C3, C4 [×6], C22, C5, C6 [×3], C8 [×4], C2×C4 [×3], C10 [×3], C12 [×6], C2×C6, C15, C42, C2×C8 [×2], C20 [×6], C2×C10, C24 [×4], C2×C12 [×3], C30 [×3], C4×C8, C40 [×4], C2×C20 [×3], C4×C12, C2×C24 [×2], C60 [×6], C2×C30, C4×C20, C2×C40 [×2], C4×C24, C120 [×4], C2×C60 [×3], C4×C40, C4×C60, C2×C120 [×2], C4×C120

Smallest permutation representation of C4×C120
Regular action on 480 points
Generators in S480
(1 444 335 234)(2 445 336 235)(3 446 337 236)(4 447 338 237)(5 448 339 238)(6 449 340 239)(7 450 341 240)(8 451 342 121)(9 452 343 122)(10 453 344 123)(11 454 345 124)(12 455 346 125)(13 456 347 126)(14 457 348 127)(15 458 349 128)(16 459 350 129)(17 460 351 130)(18 461 352 131)(19 462 353 132)(20 463 354 133)(21 464 355 134)(22 465 356 135)(23 466 357 136)(24 467 358 137)(25 468 359 138)(26 469 360 139)(27 470 241 140)(28 471 242 141)(29 472 243 142)(30 473 244 143)(31 474 245 144)(32 475 246 145)(33 476 247 146)(34 477 248 147)(35 478 249 148)(36 479 250 149)(37 480 251 150)(38 361 252 151)(39 362 253 152)(40 363 254 153)(41 364 255 154)(42 365 256 155)(43 366 257 156)(44 367 258 157)(45 368 259 158)(46 369 260 159)(47 370 261 160)(48 371 262 161)(49 372 263 162)(50 373 264 163)(51 374 265 164)(52 375 266 165)(53 376 267 166)(54 377 268 167)(55 378 269 168)(56 379 270 169)(57 380 271 170)(58 381 272 171)(59 382 273 172)(60 383 274 173)(61 384 275 174)(62 385 276 175)(63 386 277 176)(64 387 278 177)(65 388 279 178)(66 389 280 179)(67 390 281 180)(68 391 282 181)(69 392 283 182)(70 393 284 183)(71 394 285 184)(72 395 286 185)(73 396 287 186)(74 397 288 187)(75 398 289 188)(76 399 290 189)(77 400 291 190)(78 401 292 191)(79 402 293 192)(80 403 294 193)(81 404 295 194)(82 405 296 195)(83 406 297 196)(84 407 298 197)(85 408 299 198)(86 409 300 199)(87 410 301 200)(88 411 302 201)(89 412 303 202)(90 413 304 203)(91 414 305 204)(92 415 306 205)(93 416 307 206)(94 417 308 207)(95 418 309 208)(96 419 310 209)(97 420 311 210)(98 421 312 211)(99 422 313 212)(100 423 314 213)(101 424 315 214)(102 425 316 215)(103 426 317 216)(104 427 318 217)(105 428 319 218)(106 429 320 219)(107 430 321 220)(108 431 322 221)(109 432 323 222)(110 433 324 223)(111 434 325 224)(112 435 326 225)(113 436 327 226)(114 437 328 227)(115 438 329 228)(116 439 330 229)(117 440 331 230)(118 441 332 231)(119 442 333 232)(120 443 334 233)
(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)| (1,444,335,234)(2,445,336,235)(3,446,337,236)(4,447,338,237)(5,448,339,238)(6,449,340,239)(7,450,341,240)(8,451,342,121)(9,452,343,122)(10,453,344,123)(11,454,345,124)(12,455,346,125)(13,456,347,126)(14,457,348,127)(15,458,349,128)(16,459,350,129)(17,460,351,130)(18,461,352,131)(19,462,353,132)(20,463,354,133)(21,464,355,134)(22,465,356,135)(23,466,357,136)(24,467,358,137)(25,468,359,138)(26,469,360,139)(27,470,241,140)(28,471,242,141)(29,472,243,142)(30,473,244,143)(31,474,245,144)(32,475,246,145)(33,476,247,146)(34,477,248,147)(35,478,249,148)(36,479,250,149)(37,480,251,150)(38,361,252,151)(39,362,253,152)(40,363,254,153)(41,364,255,154)(42,365,256,155)(43,366,257,156)(44,367,258,157)(45,368,259,158)(46,369,260,159)(47,370,261,160)(48,371,262,161)(49,372,263,162)(50,373,264,163)(51,374,265,164)(52,375,266,165)(53,376,267,166)(54,377,268,167)(55,378,269,168)(56,379,270,169)(57,380,271,170)(58,381,272,171)(59,382,273,172)(60,383,274,173)(61,384,275,174)(62,385,276,175)(63,386,277,176)(64,387,278,177)(65,388,279,178)(66,389,280,179)(67,390,281,180)(68,391,282,181)(69,392,283,182)(70,393,284,183)(71,394,285,184)(72,395,286,185)(73,396,287,186)(74,397,288,187)(75,398,289,188)(76,399,290,189)(77,400,291,190)(78,401,292,191)(79,402,293,192)(80,403,294,193)(81,404,295,194)(82,405,296,195)(83,406,297,196)(84,407,298,197)(85,408,299,198)(86,409,300,199)(87,410,301,200)(88,411,302,201)(89,412,303,202)(90,413,304,203)(91,414,305,204)(92,415,306,205)(93,416,307,206)(94,417,308,207)(95,418,309,208)(96,419,310,209)(97,420,311,210)(98,421,312,211)(99,422,313,212)(100,423,314,213)(101,424,315,214)(102,425,316,215)(103,426,317,216)(104,427,318,217)(105,428,319,218)(106,429,320,219)(107,430,321,220)(108,431,322,221)(109,432,323,222)(110,433,324,223)(111,434,325,224)(112,435,326,225)(113,436,327,226)(114,437,328,227)(115,438,329,228)(116,439,330,229)(117,440,331,230)(118,441,332,231)(119,442,333,232)(120,443,334,233), (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( (1,444,335,234)(2,445,336,235)(3,446,337,236)(4,447,338,237)(5,448,339,238)(6,449,340,239)(7,450,341,240)(8,451,342,121)(9,452,343,122)(10,453,344,123)(11,454,345,124)(12,455,346,125)(13,456,347,126)(14,457,348,127)(15,458,349,128)(16,459,350,129)(17,460,351,130)(18,461,352,131)(19,462,353,132)(20,463,354,133)(21,464,355,134)(22,465,356,135)(23,466,357,136)(24,467,358,137)(25,468,359,138)(26,469,360,139)(27,470,241,140)(28,471,242,141)(29,472,243,142)(30,473,244,143)(31,474,245,144)(32,475,246,145)(33,476,247,146)(34,477,248,147)(35,478,249,148)(36,479,250,149)(37,480,251,150)(38,361,252,151)(39,362,253,152)(40,363,254,153)(41,364,255,154)(42,365,256,155)(43,366,257,156)(44,367,258,157)(45,368,259,158)(46,369,260,159)(47,370,261,160)(48,371,262,161)(49,372,263,162)(50,373,264,163)(51,374,265,164)(52,375,266,165)(53,376,267,166)(54,377,268,167)(55,378,269,168)(56,379,270,169)(57,380,271,170)(58,381,272,171)(59,382,273,172)(60,383,274,173)(61,384,275,174)(62,385,276,175)(63,386,277,176)(64,387,278,177)(65,388,279,178)(66,389,280,179)(67,390,281,180)(68,391,282,181)(69,392,283,182)(70,393,284,183)(71,394,285,184)(72,395,286,185)(73,396,287,186)(74,397,288,187)(75,398,289,188)(76,399,290,189)(77,400,291,190)(78,401,292,191)(79,402,293,192)(80,403,294,193)(81,404,295,194)(82,405,296,195)(83,406,297,196)(84,407,298,197)(85,408,299,198)(86,409,300,199)(87,410,301,200)(88,411,302,201)(89,412,303,202)(90,413,304,203)(91,414,305,204)(92,415,306,205)(93,416,307,206)(94,417,308,207)(95,418,309,208)(96,419,310,209)(97,420,311,210)(98,421,312,211)(99,422,313,212)(100,423,314,213)(101,424,315,214)(102,425,316,215)(103,426,317,216)(104,427,318,217)(105,428,319,218)(106,429,320,219)(107,430,321,220)(108,431,322,221)(109,432,323,222)(110,433,324,223)(111,434,325,224)(112,435,326,225)(113,436,327,226)(114,437,328,227)(115,438,329,228)(116,439,330,229)(117,440,331,230)(118,441,332,231)(119,442,333,232)(120,443,334,233), (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([(1,444,335,234),(2,445,336,235),(3,446,337,236),(4,447,338,237),(5,448,339,238),(6,449,340,239),(7,450,341,240),(8,451,342,121),(9,452,343,122),(10,453,344,123),(11,454,345,124),(12,455,346,125),(13,456,347,126),(14,457,348,127),(15,458,349,128),(16,459,350,129),(17,460,351,130),(18,461,352,131),(19,462,353,132),(20,463,354,133),(21,464,355,134),(22,465,356,135),(23,466,357,136),(24,467,358,137),(25,468,359,138),(26,469,360,139),(27,470,241,140),(28,471,242,141),(29,472,243,142),(30,473,244,143),(31,474,245,144),(32,475,246,145),(33,476,247,146),(34,477,248,147),(35,478,249,148),(36,479,250,149),(37,480,251,150),(38,361,252,151),(39,362,253,152),(40,363,254,153),(41,364,255,154),(42,365,256,155),(43,366,257,156),(44,367,258,157),(45,368,259,158),(46,369,260,159),(47,370,261,160),(48,371,262,161),(49,372,263,162),(50,373,264,163),(51,374,265,164),(52,375,266,165),(53,376,267,166),(54,377,268,167),(55,378,269,168),(56,379,270,169),(57,380,271,170),(58,381,272,171),(59,382,273,172),(60,383,274,173),(61,384,275,174),(62,385,276,175),(63,386,277,176),(64,387,278,177),(65,388,279,178),(66,389,280,179),(67,390,281,180),(68,391,282,181),(69,392,283,182),(70,393,284,183),(71,394,285,184),(72,395,286,185),(73,396,287,186),(74,397,288,187),(75,398,289,188),(76,399,290,189),(77,400,291,190),(78,401,292,191),(79,402,293,192),(80,403,294,193),(81,404,295,194),(82,405,296,195),(83,406,297,196),(84,407,298,197),(85,408,299,198),(86,409,300,199),(87,410,301,200),(88,411,302,201),(89,412,303,202),(90,413,304,203),(91,414,305,204),(92,415,306,205),(93,416,307,206),(94,417,308,207),(95,418,309,208),(96,419,310,209),(97,420,311,210),(98,421,312,211),(99,422,313,212),(100,423,314,213),(101,424,315,214),(102,425,316,215),(103,426,317,216),(104,427,318,217),(105,428,319,218),(106,429,320,219),(107,430,321,220),(108,431,322,221),(109,432,323,222),(110,433,324,223),(111,434,325,224),(112,435,326,225),(113,436,327,226),(114,437,328,227),(115,438,329,228),(116,439,330,229),(117,440,331,230),(118,441,332,231),(119,442,333,232),(120,443,334,233)], [(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 2A2B2C3A3B4A···4L5A5B5C5D6A···6F8A···8P10A···10L12A···12X15A···15H20A···20AV24A···24AF30A···30X40A···40BL60A···60CR120A···120DX
order1222334···455556···68···810···1012···1215···1520···2024···2430···3040···4060···60120···120
size1111111···111111···11···11···11···11···11···11···11···11···11···11···1

480 irreducible representations

dim111111111111111111111111
type+++
imageC1C2C2C3C4C4C5C6C6C8C10C10C12C12C15C20C20C24C30C30C40C60C60C120
kernelC4×C120C4×C60C2×C120C4×C40C120C2×C60C4×C24C4×C20C2×C40C60C4×C12C2×C24C40C2×C20C4×C8C24C2×C12C20C42C2×C8C12C8C2×C4C4
# reps11228442416481688321632816646432128

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

1770
0240
,
1740
0180
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

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