metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C120⋊9C4, C6.4D40, C24⋊1Dic5, C8⋊1Dic15, C40⋊5Dic3, C30.25D8, C2.1D120, C10.4D24, C60.23Q8, C30.11Q16, C2.2Dic60, C4.5Dic30, C6.2Dic20, C22.9D60, C20.20Dic6, C10.2Dic12, C12.20Dic10, (C2×C40).5S3, (C2×C8).3D15, (C2×C24).5D5, C5⋊3(C24⋊1C4), C3⋊2(C40⋊5C4), (C2×C120).9C2, (C2×C4).72D30, (C2×C6).15D20, C15⋊10(C2.D8), C30.40(C4⋊C4), C60⋊5C4.3C2, C60.231(C2×C4), (C2×C20).386D6, (C2×C30).101D4, (C2×C10).15D12, C6.8(C4⋊Dic5), C4.7(C2×Dic15), C2.4(C60⋊5C4), (C2×C12).388D10, C12.36(C2×Dic5), C20.57(C2×Dic3), (C2×C60).473C22, C10.15(C4⋊Dic3), SmallGroup(480,178)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C120⋊9C4
G = < a,b | a120=b4=1, bab-1=a-1 >
Subgroups: 404 in 72 conjugacy classes, 47 normal (37 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, C2×C10, C24, C2×Dic3, C2×C12, C30, C2.D8, C40, C2×Dic5, C2×C20, C4⋊Dic3, C2×C24, Dic15, C60, C2×C30, C4⋊Dic5, C2×C40, C24⋊1C4, C120, C2×Dic15, C2×C60, C40⋊5C4, C60⋊5C4, C2×C120, C120⋊9C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, Dic3, D6, C4⋊C4, D8, Q16, Dic5, D10, Dic6, D12, C2×Dic3, D15, C2.D8, Dic10, D20, C2×Dic5, D24, Dic12, C4⋊Dic3, Dic15, D30, D40, Dic20, C4⋊Dic5, C24⋊1C4, Dic30, D60, C2×Dic15, C40⋊5C4, D120, Dic60, C60⋊5C4, C120⋊9C4
►Regular action on 480 points
Generators in S480
(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)
(1 367 136 274)(2 366 137 273)(3 365 138 272)(4 364 139 271)(5 363 140 270)(6 362 141 269)(7 361 142 268)(8 480 143 267)(9 479 144 266)(10 478 145 265)(11 477 146 264)(12 476 147 263)(13 475 148 262)(14 474 149 261)(15 473 150 260)(16 472 151 259)(17 471 152 258)(18 470 153 257)(19 469 154 256)(20 468 155 255)(21 467 156 254)(22 466 157 253)(23 465 158 252)(24 464 159 251)(25 463 160 250)(26 462 161 249)(27 461 162 248)(28 460 163 247)(29 459 164 246)(30 458 165 245)(31 457 166 244)(32 456 167 243)(33 455 168 242)(34 454 169 241)(35 453 170 360)(36 452 171 359)(37 451 172 358)(38 450 173 357)(39 449 174 356)(40 448 175 355)(41 447 176 354)(42 446 177 353)(43 445 178 352)(44 444 179 351)(45 443 180 350)(46 442 181 349)(47 441 182 348)(48 440 183 347)(49 439 184 346)(50 438 185 345)(51 437 186 344)(52 436 187 343)(53 435 188 342)(54 434 189 341)(55 433 190 340)(56 432 191 339)(57 431 192 338)(58 430 193 337)(59 429 194 336)(60 428 195 335)(61 427 196 334)(62 426 197 333)(63 425 198 332)(64 424 199 331)(65 423 200 330)(66 422 201 329)(67 421 202 328)(68 420 203 327)(69 419 204 326)(70 418 205 325)(71 417 206 324)(72 416 207 323)(73 415 208 322)(74 414 209 321)(75 413 210 320)(76 412 211 319)(77 411 212 318)(78 410 213 317)(79 409 214 316)(80 408 215 315)(81 407 216 314)(82 406 217 313)(83 405 218 312)(84 404 219 311)(85 403 220 310)(86 402 221 309)(87 401 222 308)(88 400 223 307)(89 399 224 306)(90 398 225 305)(91 397 226 304)(92 396 227 303)(93 395 228 302)(94 394 229 301)(95 393 230 300)(96 392 231 299)(97 391 232 298)(98 390 233 297)(99 389 234 296)(100 388 235 295)(101 387 236 294)(102 386 237 293)(103 385 238 292)(104 384 239 291)(105 383 240 290)(106 382 121 289)(107 381 122 288)(108 380 123 287)(109 379 124 286)(110 378 125 285)(111 377 126 284)(112 376 127 283)(113 375 128 282)(114 374 129 281)(115 373 130 280)(116 372 131 279)(117 371 132 278)(118 370 133 277)(119 369 134 276)(120 368 135 275)
G:=sub<Sym(480)| (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), (1,367,136,274)(2,366,137,273)(3,365,138,272)(4,364,139,271)(5,363,140,270)(6,362,141,269)(7,361,142,268)(8,480,143,267)(9,479,144,266)(10,478,145,265)(11,477,146,264)(12,476,147,263)(13,475,148,262)(14,474,149,261)(15,473,150,260)(16,472,151,259)(17,471,152,258)(18,470,153,257)(19,469,154,256)(20,468,155,255)(21,467,156,254)(22,466,157,253)(23,465,158,252)(24,464,159,251)(25,463,160,250)(26,462,161,249)(27,461,162,248)(28,460,163,247)(29,459,164,246)(30,458,165,245)(31,457,166,244)(32,456,167,243)(33,455,168,242)(34,454,169,241)(35,453,170,360)(36,452,171,359)(37,451,172,358)(38,450,173,357)(39,449,174,356)(40,448,175,355)(41,447,176,354)(42,446,177,353)(43,445,178,352)(44,444,179,351)(45,443,180,350)(46,442,181,349)(47,441,182,348)(48,440,183,347)(49,439,184,346)(50,438,185,345)(51,437,186,344)(52,436,187,343)(53,435,188,342)(54,434,189,341)(55,433,190,340)(56,432,191,339)(57,431,192,338)(58,430,193,337)(59,429,194,336)(60,428,195,335)(61,427,196,334)(62,426,197,333)(63,425,198,332)(64,424,199,331)(65,423,200,330)(66,422,201,329)(67,421,202,328)(68,420,203,327)(69,419,204,326)(70,418,205,325)(71,417,206,324)(72,416,207,323)(73,415,208,322)(74,414,209,321)(75,413,210,320)(76,412,211,319)(77,411,212,318)(78,410,213,317)(79,409,214,316)(80,408,215,315)(81,407,216,314)(82,406,217,313)(83,405,218,312)(84,404,219,311)(85,403,220,310)(86,402,221,309)(87,401,222,308)(88,400,223,307)(89,399,224,306)(90,398,225,305)(91,397,226,304)(92,396,227,303)(93,395,228,302)(94,394,229,301)(95,393,230,300)(96,392,231,299)(97,391,232,298)(98,390,233,297)(99,389,234,296)(100,388,235,295)(101,387,236,294)(102,386,237,293)(103,385,238,292)(104,384,239,291)(105,383,240,290)(106,382,121,289)(107,381,122,288)(108,380,123,287)(109,379,124,286)(110,378,125,285)(111,377,126,284)(112,376,127,283)(113,375,128,282)(114,374,129,281)(115,373,130,280)(116,372,131,279)(117,371,132,278)(118,370,133,277)(119,369,134,276)(120,368,135,275)>;
G:=Group( (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), (1,367,136,274)(2,366,137,273)(3,365,138,272)(4,364,139,271)(5,363,140,270)(6,362,141,269)(7,361,142,268)(8,480,143,267)(9,479,144,266)(10,478,145,265)(11,477,146,264)(12,476,147,263)(13,475,148,262)(14,474,149,261)(15,473,150,260)(16,472,151,259)(17,471,152,258)(18,470,153,257)(19,469,154,256)(20,468,155,255)(21,467,156,254)(22,466,157,253)(23,465,158,252)(24,464,159,251)(25,463,160,250)(26,462,161,249)(27,461,162,248)(28,460,163,247)(29,459,164,246)(30,458,165,245)(31,457,166,244)(32,456,167,243)(33,455,168,242)(34,454,169,241)(35,453,170,360)(36,452,171,359)(37,451,172,358)(38,450,173,357)(39,449,174,356)(40,448,175,355)(41,447,176,354)(42,446,177,353)(43,445,178,352)(44,444,179,351)(45,443,180,350)(46,442,181,349)(47,441,182,348)(48,440,183,347)(49,439,184,346)(50,438,185,345)(51,437,186,344)(52,436,187,343)(53,435,188,342)(54,434,189,341)(55,433,190,340)(56,432,191,339)(57,431,192,338)(58,430,193,337)(59,429,194,336)(60,428,195,335)(61,427,196,334)(62,426,197,333)(63,425,198,332)(64,424,199,331)(65,423,200,330)(66,422,201,329)(67,421,202,328)(68,420,203,327)(69,419,204,326)(70,418,205,325)(71,417,206,324)(72,416,207,323)(73,415,208,322)(74,414,209,321)(75,413,210,320)(76,412,211,319)(77,411,212,318)(78,410,213,317)(79,409,214,316)(80,408,215,315)(81,407,216,314)(82,406,217,313)(83,405,218,312)(84,404,219,311)(85,403,220,310)(86,402,221,309)(87,401,222,308)(88,400,223,307)(89,399,224,306)(90,398,225,305)(91,397,226,304)(92,396,227,303)(93,395,228,302)(94,394,229,301)(95,393,230,300)(96,392,231,299)(97,391,232,298)(98,390,233,297)(99,389,234,296)(100,388,235,295)(101,387,236,294)(102,386,237,293)(103,385,238,292)(104,384,239,291)(105,383,240,290)(106,382,121,289)(107,381,122,288)(108,380,123,287)(109,379,124,286)(110,378,125,285)(111,377,126,284)(112,376,127,283)(113,375,128,282)(114,374,129,281)(115,373,130,280)(116,372,131,279)(117,371,132,278)(118,370,133,277)(119,369,134,276)(120,368,135,275) );
G=PermutationGroup([[(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)], [(1,367,136,274),(2,366,137,273),(3,365,138,272),(4,364,139,271),(5,363,140,270),(6,362,141,269),(7,361,142,268),(8,480,143,267),(9,479,144,266),(10,478,145,265),(11,477,146,264),(12,476,147,263),(13,475,148,262),(14,474,149,261),(15,473,150,260),(16,472,151,259),(17,471,152,258),(18,470,153,257),(19,469,154,256),(20,468,155,255),(21,467,156,254),(22,466,157,253),(23,465,158,252),(24,464,159,251),(25,463,160,250),(26,462,161,249),(27,461,162,248),(28,460,163,247),(29,459,164,246),(30,458,165,245),(31,457,166,244),(32,456,167,243),(33,455,168,242),(34,454,169,241),(35,453,170,360),(36,452,171,359),(37,451,172,358),(38,450,173,357),(39,449,174,356),(40,448,175,355),(41,447,176,354),(42,446,177,353),(43,445,178,352),(44,444,179,351),(45,443,180,350),(46,442,181,349),(47,441,182,348),(48,440,183,347),(49,439,184,346),(50,438,185,345),(51,437,186,344),(52,436,187,343),(53,435,188,342),(54,434,189,341),(55,433,190,340),(56,432,191,339),(57,431,192,338),(58,430,193,337),(59,429,194,336),(60,428,195,335),(61,427,196,334),(62,426,197,333),(63,425,198,332),(64,424,199,331),(65,423,200,330),(66,422,201,329),(67,421,202,328),(68,420,203,327),(69,419,204,326),(70,418,205,325),(71,417,206,324),(72,416,207,323),(73,415,208,322),(74,414,209,321),(75,413,210,320),(76,412,211,319),(77,411,212,318),(78,410,213,317),(79,409,214,316),(80,408,215,315),(81,407,216,314),(82,406,217,313),(83,405,218,312),(84,404,219,311),(85,403,220,310),(86,402,221,309),(87,401,222,308),(88,400,223,307),(89,399,224,306),(90,398,225,305),(91,397,226,304),(92,396,227,303),(93,395,228,302),(94,394,229,301),(95,393,230,300),(96,392,231,299),(97,391,232,298),(98,390,233,297),(99,389,234,296),(100,388,235,295),(101,387,236,294),(102,386,237,293),(103,385,238,292),(104,384,239,291),(105,383,240,290),(106,382,121,289),(107,381,122,288),(108,380,123,287),(109,379,124,286),(110,378,125,285),(111,377,126,284),(112,376,127,283),(113,375,128,282),(114,374,129,281),(115,373,130,280),(116,372,131,279),(117,371,132,278),(118,370,133,277),(119,369,134,276),(120,368,135,275)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 60 | 60 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | + | - | - | + | - | + | + | - | + | + | - | - | + | + | - | - | + | + | - | |
| image | C1 | C2 | C2 | C4 | S3 | Q8 | D4 | D5 | Dic3 | D6 | D8 | Q16 | Dic5 | D10 | Dic6 | D12 | D15 | Dic10 | D20 | D24 | Dic12 | Dic15 | D30 | D40 | Dic20 | Dic30 | D60 | D120 | Dic60 |
| kernel | C120⋊9C4 | C60⋊5C4 | C2×C120 | C120 | C2×C40 | C60 | C2×C30 | C2×C24 | C40 | C2×C20 | C30 | C30 | C24 | C2×C12 | C20 | C2×C10 | C2×C8 | C12 | C2×C6 | C10 | C10 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of C120⋊9C4 ►in GL5(𝔽241)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 173 | 63 | 0 | 0 |
| 0 | 178 | 30 | 0 | 0 |
| 0 | 0 | 0 | 20 | 228 |
| 0 | 0 | 0 | 207 | 227 |
| 64 | 0 | 0 | 0 | 0 |
| 0 | 211 | 64 | 0 | 0 |
| 0 | 178 | 30 | 0 | 0 |
| 0 | 0 | 0 | 192 | 49 |
| 0 | 0 | 0 | 133 | 49 |
G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,173,178,0,0,0,63,30,0,0,0,0,0,20,207,0,0,0,228,227],[64,0,0,0,0,0,211,178,0,0,0,64,30,0,0,0,0,0,192,133,0,0,0,49,49] >;
C120⋊9C4 in GAP, Magma, Sage, TeX
C_{120}\rtimes_9C_4 % in TeX
G:=Group("C120:9C4"); // GroupNames label
G:=SmallGroup(480,178);
// by ID
G=gap.SmallGroup(480,178);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,176,675,80,2693,18822]);
// Polycyclic
G:=Group<a,b|a^120=b^4=1,b*a*b^-1=a^-1>;
// generators/relations