metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C120⋊10C4, C24⋊2Dic5, C40⋊6Dic3, C8⋊2Dic15, C60.22Q8, C4.4Dic30, C22.8D60, C20.19Dic6, C30.17SD16, C12.19Dic10, (C2×C40).8S3, (C2×C8).6D15, (C2×C24).8D5, C3⋊2(C40⋊6C4), C5⋊3(C8⋊Dic3), (C2×C4).71D30, (C2×C6).14D20, C15⋊10(C4.Q8), C30.39(C4⋊C4), C60⋊5C4.2C2, (C2×C120).12C2, C60.230(C2×C4), (C2×C20).385D6, (C2×C10).14D12, (C2×C30).100D4, C6.2(C40⋊C2), C6.7(C4⋊Dic5), C4.6(C2×Dic15), C2.3(C60⋊5C4), C2.2(C24⋊D5), C10.2(C24⋊C2), (C2×C12).387D10, C20.56(C2×Dic3), C12.35(C2×Dic5), (C2×C60).472C22, C10.14(C4⋊Dic3), SmallGroup(480,177)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C120⋊10C4
G = < a,b | a120=b4=1, bab-1=a59 >
Subgroups: 404 in 72 conjugacy classes, 47 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, Dic3, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, C2×C10, C24, C2×Dic3, C2×C12, C30, C30, C4.Q8, C40, C2×Dic5, C2×C20, C4⋊Dic3, C2×C24, Dic15, C60, C2×C30, C4⋊Dic5, C2×C40, C8⋊Dic3, C120, C2×Dic15, C2×C60, C40⋊6C4, C60⋊5C4, C2×C120, C120⋊10C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, Dic3, D6, C4⋊C4, SD16, Dic5, D10, Dic6, D12, C2×Dic3, D15, C4.Q8, Dic10, D20, C2×Dic5, C24⋊C2, C4⋊Dic3, Dic15, D30, C40⋊C2, C4⋊Dic5, C8⋊Dic3, Dic30, D60, C2×Dic15, C40⋊6C4, C24⋊D5, C60⋊5C4, C120⋊10C4
►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)
(1 181 358 470)(2 240 359 409)(3 179 360 468)(4 238 241 407)(5 177 242 466)(6 236 243 405)(7 175 244 464)(8 234 245 403)(9 173 246 462)(10 232 247 401)(11 171 248 460)(12 230 249 399)(13 169 250 458)(14 228 251 397)(15 167 252 456)(16 226 253 395)(17 165 254 454)(18 224 255 393)(19 163 256 452)(20 222 257 391)(21 161 258 450)(22 220 259 389)(23 159 260 448)(24 218 261 387)(25 157 262 446)(26 216 263 385)(27 155 264 444)(28 214 265 383)(29 153 266 442)(30 212 267 381)(31 151 268 440)(32 210 269 379)(33 149 270 438)(34 208 271 377)(35 147 272 436)(36 206 273 375)(37 145 274 434)(38 204 275 373)(39 143 276 432)(40 202 277 371)(41 141 278 430)(42 200 279 369)(43 139 280 428)(44 198 281 367)(45 137 282 426)(46 196 283 365)(47 135 284 424)(48 194 285 363)(49 133 286 422)(50 192 287 361)(51 131 288 420)(52 190 289 479)(53 129 290 418)(54 188 291 477)(55 127 292 416)(56 186 293 475)(57 125 294 414)(58 184 295 473)(59 123 296 412)(60 182 297 471)(61 121 298 410)(62 180 299 469)(63 239 300 408)(64 178 301 467)(65 237 302 406)(66 176 303 465)(67 235 304 404)(68 174 305 463)(69 233 306 402)(70 172 307 461)(71 231 308 400)(72 170 309 459)(73 229 310 398)(74 168 311 457)(75 227 312 396)(76 166 313 455)(77 225 314 394)(78 164 315 453)(79 223 316 392)(80 162 317 451)(81 221 318 390)(82 160 319 449)(83 219 320 388)(84 158 321 447)(85 217 322 386)(86 156 323 445)(87 215 324 384)(88 154 325 443)(89 213 326 382)(90 152 327 441)(91 211 328 380)(92 150 329 439)(93 209 330 378)(94 148 331 437)(95 207 332 376)(96 146 333 435)(97 205 334 374)(98 144 335 433)(99 203 336 372)(100 142 337 431)(101 201 338 370)(102 140 339 429)(103 199 340 368)(104 138 341 427)(105 197 342 366)(106 136 343 425)(107 195 344 364)(108 134 345 423)(109 193 346 362)(110 132 347 421)(111 191 348 480)(112 130 349 419)(113 189 350 478)(114 128 351 417)(115 187 352 476)(116 126 353 415)(117 185 354 474)(118 124 355 413)(119 183 356 472)(120 122 357 411)
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), (1,181,358,470)(2,240,359,409)(3,179,360,468)(4,238,241,407)(5,177,242,466)(6,236,243,405)(7,175,244,464)(8,234,245,403)(9,173,246,462)(10,232,247,401)(11,171,248,460)(12,230,249,399)(13,169,250,458)(14,228,251,397)(15,167,252,456)(16,226,253,395)(17,165,254,454)(18,224,255,393)(19,163,256,452)(20,222,257,391)(21,161,258,450)(22,220,259,389)(23,159,260,448)(24,218,261,387)(25,157,262,446)(26,216,263,385)(27,155,264,444)(28,214,265,383)(29,153,266,442)(30,212,267,381)(31,151,268,440)(32,210,269,379)(33,149,270,438)(34,208,271,377)(35,147,272,436)(36,206,273,375)(37,145,274,434)(38,204,275,373)(39,143,276,432)(40,202,277,371)(41,141,278,430)(42,200,279,369)(43,139,280,428)(44,198,281,367)(45,137,282,426)(46,196,283,365)(47,135,284,424)(48,194,285,363)(49,133,286,422)(50,192,287,361)(51,131,288,420)(52,190,289,479)(53,129,290,418)(54,188,291,477)(55,127,292,416)(56,186,293,475)(57,125,294,414)(58,184,295,473)(59,123,296,412)(60,182,297,471)(61,121,298,410)(62,180,299,469)(63,239,300,408)(64,178,301,467)(65,237,302,406)(66,176,303,465)(67,235,304,404)(68,174,305,463)(69,233,306,402)(70,172,307,461)(71,231,308,400)(72,170,309,459)(73,229,310,398)(74,168,311,457)(75,227,312,396)(76,166,313,455)(77,225,314,394)(78,164,315,453)(79,223,316,392)(80,162,317,451)(81,221,318,390)(82,160,319,449)(83,219,320,388)(84,158,321,447)(85,217,322,386)(86,156,323,445)(87,215,324,384)(88,154,325,443)(89,213,326,382)(90,152,327,441)(91,211,328,380)(92,150,329,439)(93,209,330,378)(94,148,331,437)(95,207,332,376)(96,146,333,435)(97,205,334,374)(98,144,335,433)(99,203,336,372)(100,142,337,431)(101,201,338,370)(102,140,339,429)(103,199,340,368)(104,138,341,427)(105,197,342,366)(106,136,343,425)(107,195,344,364)(108,134,345,423)(109,193,346,362)(110,132,347,421)(111,191,348,480)(112,130,349,419)(113,189,350,478)(114,128,351,417)(115,187,352,476)(116,126,353,415)(117,185,354,474)(118,124,355,413)(119,183,356,472)(120,122,357,411)>;
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), (1,181,358,470)(2,240,359,409)(3,179,360,468)(4,238,241,407)(5,177,242,466)(6,236,243,405)(7,175,244,464)(8,234,245,403)(9,173,246,462)(10,232,247,401)(11,171,248,460)(12,230,249,399)(13,169,250,458)(14,228,251,397)(15,167,252,456)(16,226,253,395)(17,165,254,454)(18,224,255,393)(19,163,256,452)(20,222,257,391)(21,161,258,450)(22,220,259,389)(23,159,260,448)(24,218,261,387)(25,157,262,446)(26,216,263,385)(27,155,264,444)(28,214,265,383)(29,153,266,442)(30,212,267,381)(31,151,268,440)(32,210,269,379)(33,149,270,438)(34,208,271,377)(35,147,272,436)(36,206,273,375)(37,145,274,434)(38,204,275,373)(39,143,276,432)(40,202,277,371)(41,141,278,430)(42,200,279,369)(43,139,280,428)(44,198,281,367)(45,137,282,426)(46,196,283,365)(47,135,284,424)(48,194,285,363)(49,133,286,422)(50,192,287,361)(51,131,288,420)(52,190,289,479)(53,129,290,418)(54,188,291,477)(55,127,292,416)(56,186,293,475)(57,125,294,414)(58,184,295,473)(59,123,296,412)(60,182,297,471)(61,121,298,410)(62,180,299,469)(63,239,300,408)(64,178,301,467)(65,237,302,406)(66,176,303,465)(67,235,304,404)(68,174,305,463)(69,233,306,402)(70,172,307,461)(71,231,308,400)(72,170,309,459)(73,229,310,398)(74,168,311,457)(75,227,312,396)(76,166,313,455)(77,225,314,394)(78,164,315,453)(79,223,316,392)(80,162,317,451)(81,221,318,390)(82,160,319,449)(83,219,320,388)(84,158,321,447)(85,217,322,386)(86,156,323,445)(87,215,324,384)(88,154,325,443)(89,213,326,382)(90,152,327,441)(91,211,328,380)(92,150,329,439)(93,209,330,378)(94,148,331,437)(95,207,332,376)(96,146,333,435)(97,205,334,374)(98,144,335,433)(99,203,336,372)(100,142,337,431)(101,201,338,370)(102,140,339,429)(103,199,340,368)(104,138,341,427)(105,197,342,366)(106,136,343,425)(107,195,344,364)(108,134,345,423)(109,193,346,362)(110,132,347,421)(111,191,348,480)(112,130,349,419)(113,189,350,478)(114,128,351,417)(115,187,352,476)(116,126,353,415)(117,185,354,474)(118,124,355,413)(119,183,356,472)(120,122,357,411) );
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)], [(1,181,358,470),(2,240,359,409),(3,179,360,468),(4,238,241,407),(5,177,242,466),(6,236,243,405),(7,175,244,464),(8,234,245,403),(9,173,246,462),(10,232,247,401),(11,171,248,460),(12,230,249,399),(13,169,250,458),(14,228,251,397),(15,167,252,456),(16,226,253,395),(17,165,254,454),(18,224,255,393),(19,163,256,452),(20,222,257,391),(21,161,258,450),(22,220,259,389),(23,159,260,448),(24,218,261,387),(25,157,262,446),(26,216,263,385),(27,155,264,444),(28,214,265,383),(29,153,266,442),(30,212,267,381),(31,151,268,440),(32,210,269,379),(33,149,270,438),(34,208,271,377),(35,147,272,436),(36,206,273,375),(37,145,274,434),(38,204,275,373),(39,143,276,432),(40,202,277,371),(41,141,278,430),(42,200,279,369),(43,139,280,428),(44,198,281,367),(45,137,282,426),(46,196,283,365),(47,135,284,424),(48,194,285,363),(49,133,286,422),(50,192,287,361),(51,131,288,420),(52,190,289,479),(53,129,290,418),(54,188,291,477),(55,127,292,416),(56,186,293,475),(57,125,294,414),(58,184,295,473),(59,123,296,412),(60,182,297,471),(61,121,298,410),(62,180,299,469),(63,239,300,408),(64,178,301,467),(65,237,302,406),(66,176,303,465),(67,235,304,404),(68,174,305,463),(69,233,306,402),(70,172,307,461),(71,231,308,400),(72,170,309,459),(73,229,310,398),(74,168,311,457),(75,227,312,396),(76,166,313,455),(77,225,314,394),(78,164,315,453),(79,223,316,392),(80,162,317,451),(81,221,318,390),(82,160,319,449),(83,219,320,388),(84,158,321,447),(85,217,322,386),(86,156,323,445),(87,215,324,384),(88,154,325,443),(89,213,326,382),(90,152,327,441),(91,211,328,380),(92,150,329,439),(93,209,330,378),(94,148,331,437),(95,207,332,376),(96,146,333,435),(97,205,334,374),(98,144,335,433),(99,203,336,372),(100,142,337,431),(101,201,338,370),(102,140,339,429),(103,199,340,368),(104,138,341,427),(105,197,342,366),(106,136,343,425),(107,195,344,364),(108,134,345,423),(109,193,346,362),(110,132,347,421),(111,191,348,480),(112,130,349,419),(113,189,350,478),(114,128,351,417),(115,187,352,476),(116,126,353,415),(117,185,354,474),(118,124,355,413),(119,183,356,472),(120,122,357,411)]])
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 |
| type | + | + | + | + | - | + | + | - | + | - | + | - | + | + | - | + | - | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | S3 | Q8 | D4 | D5 | Dic3 | D6 | SD16 | Dic5 | D10 | Dic6 | D12 | D15 | Dic10 | D20 | C24⋊C2 | Dic15 | D30 | C40⋊C2 | Dic30 | D60 | C24⋊D5 |
| kernel | C120⋊10C4 | C60⋊5C4 | C2×C120 | C120 | C2×C40 | C60 | C2×C30 | C2×C24 | C40 | C2×C20 | C30 | C24 | C2×C12 | C20 | C2×C10 | C2×C8 | C12 | C2×C6 | C10 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of C120⋊10C4 ►in GL5(𝔽241)
| 240 | 0 | 0 | 0 | 0 |
| 0 | 57 | 6 | 0 | 0 |
| 0 | 235 | 122 | 0 | 0 |
| 0 | 0 | 0 | 94 | 166 |
| 0 | 0 | 0 | 106 | 200 |
| 177 | 0 | 0 | 0 | 0 |
| 0 | 54 | 89 | 0 | 0 |
| 0 | 192 | 187 | 0 | 0 |
| 0 | 0 | 0 | 152 | 236 |
| 0 | 0 | 0 | 138 | 89 |
G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,57,235,0,0,0,6,122,0,0,0,0,0,94,106,0,0,0,166,200],[177,0,0,0,0,0,54,192,0,0,0,89,187,0,0,0,0,0,152,138,0,0,0,236,89] >;
C120⋊10C4 in GAP, Magma, Sage, TeX
C_{120}\rtimes_{10}C_4 % in TeX
G:=Group("C120:10C4"); // GroupNames label
G:=SmallGroup(480,177);
// by ID
G=gap.SmallGroup(480,177);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,675,80,2693,18822]);
// Polycyclic
G:=Group<a,b|a^120=b^4=1,b*a*b^-1=a^59>;
// generators/relations