metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C120⋊13C4, C8⋊3Dic15, C24⋊5Dic5, C40⋊8Dic3, C30.28C42, C30.26M4(2), (C2×C8).8D15, C15⋊3C8⋊10C4, C5⋊6(C24⋊C4), C3⋊3(C40⋊8C4), C20.91(C4×S3), (C2×C40).12S3, (C2×C24).15D5, (C2×C4).93D30, C12.59(C4×D5), C4.21(C4×D15), C15⋊14(C8⋊C4), (C2×C120).19C2, C60.237(C2×C4), (C2×C20).407D6, C6.5(C8⋊D5), C2.4(C4×Dic15), C6.11(C4×Dic5), C2.2(C40⋊S3), (C2×C12).411D10, (C4×Dic15).6C2, C10.23(C4×Dic3), C12.42(C2×Dic5), C4.12(C2×Dic15), C20.63(C2×Dic3), C22.10(C4×D15), C10.10(C8⋊S3), (C2×C60).493C22, (C2×Dic15).13C4, (C2×C6).28(C4×D5), (C2×C10).53(C4×S3), (C2×C15⋊3C8).10C2, (C2×C30).135(C2×C4), SmallGroup(480,175)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C120⋊13C4
G = < a,b | a120=b4=1, bab-1=a29 >
Subgroups: 308 in 80 conjugacy classes, 51 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, C2×C4, C10, C10, Dic3, C12, C2×C6, C15, C42, C2×C8, C2×C8, Dic5, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×C12, C30, C30, C8⋊C4, C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, Dic15, C60, C2×C30, C2×C5⋊2C8, C4×Dic5, C2×C40, C24⋊C4, C15⋊3C8, C120, C2×Dic15, C2×C60, C40⋊8C4, C2×C15⋊3C8, C4×Dic15, C2×C120, C120⋊13C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, C42, M4(2), Dic5, D10, C4×S3, C2×Dic3, D15, C8⋊C4, C4×D5, C2×Dic5, C8⋊S3, C4×Dic3, Dic15, D30, C8⋊D5, C4×Dic5, C24⋊C4, C4×D15, C2×Dic15, C40⋊8C4, C40⋊S3, C4×Dic15, C120⋊13C4
►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)
(1 380 246 167)(2 409 247 196)(3 438 248 225)(4 467 249 134)(5 376 250 163)(6 405 251 192)(7 434 252 221)(8 463 253 130)(9 372 254 159)(10 401 255 188)(11 430 256 217)(12 459 257 126)(13 368 258 155)(14 397 259 184)(15 426 260 213)(16 455 261 122)(17 364 262 151)(18 393 263 180)(19 422 264 209)(20 451 265 238)(21 480 266 147)(22 389 267 176)(23 418 268 205)(24 447 269 234)(25 476 270 143)(26 385 271 172)(27 414 272 201)(28 443 273 230)(29 472 274 139)(30 381 275 168)(31 410 276 197)(32 439 277 226)(33 468 278 135)(34 377 279 164)(35 406 280 193)(36 435 281 222)(37 464 282 131)(38 373 283 160)(39 402 284 189)(40 431 285 218)(41 460 286 127)(42 369 287 156)(43 398 288 185)(44 427 289 214)(45 456 290 123)(46 365 291 152)(47 394 292 181)(48 423 293 210)(49 452 294 239)(50 361 295 148)(51 390 296 177)(52 419 297 206)(53 448 298 235)(54 477 299 144)(55 386 300 173)(56 415 301 202)(57 444 302 231)(58 473 303 140)(59 382 304 169)(60 411 305 198)(61 440 306 227)(62 469 307 136)(63 378 308 165)(64 407 309 194)(65 436 310 223)(66 465 311 132)(67 374 312 161)(68 403 313 190)(69 432 314 219)(70 461 315 128)(71 370 316 157)(72 399 317 186)(73 428 318 215)(74 457 319 124)(75 366 320 153)(76 395 321 182)(77 424 322 211)(78 453 323 240)(79 362 324 149)(80 391 325 178)(81 420 326 207)(82 449 327 236)(83 478 328 145)(84 387 329 174)(85 416 330 203)(86 445 331 232)(87 474 332 141)(88 383 333 170)(89 412 334 199)(90 441 335 228)(91 470 336 137)(92 379 337 166)(93 408 338 195)(94 437 339 224)(95 466 340 133)(96 375 341 162)(97 404 342 191)(98 433 343 220)(99 462 344 129)(100 371 345 158)(101 400 346 187)(102 429 347 216)(103 458 348 125)(104 367 349 154)(105 396 350 183)(106 425 351 212)(107 454 352 121)(108 363 353 150)(109 392 354 179)(110 421 355 208)(111 450 356 237)(112 479 357 146)(113 388 358 175)(114 417 359 204)(115 446 360 233)(116 475 241 142)(117 384 242 171)(118 413 243 200)(119 442 244 229)(120 471 245 138)
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), (1,380,246,167)(2,409,247,196)(3,438,248,225)(4,467,249,134)(5,376,250,163)(6,405,251,192)(7,434,252,221)(8,463,253,130)(9,372,254,159)(10,401,255,188)(11,430,256,217)(12,459,257,126)(13,368,258,155)(14,397,259,184)(15,426,260,213)(16,455,261,122)(17,364,262,151)(18,393,263,180)(19,422,264,209)(20,451,265,238)(21,480,266,147)(22,389,267,176)(23,418,268,205)(24,447,269,234)(25,476,270,143)(26,385,271,172)(27,414,272,201)(28,443,273,230)(29,472,274,139)(30,381,275,168)(31,410,276,197)(32,439,277,226)(33,468,278,135)(34,377,279,164)(35,406,280,193)(36,435,281,222)(37,464,282,131)(38,373,283,160)(39,402,284,189)(40,431,285,218)(41,460,286,127)(42,369,287,156)(43,398,288,185)(44,427,289,214)(45,456,290,123)(46,365,291,152)(47,394,292,181)(48,423,293,210)(49,452,294,239)(50,361,295,148)(51,390,296,177)(52,419,297,206)(53,448,298,235)(54,477,299,144)(55,386,300,173)(56,415,301,202)(57,444,302,231)(58,473,303,140)(59,382,304,169)(60,411,305,198)(61,440,306,227)(62,469,307,136)(63,378,308,165)(64,407,309,194)(65,436,310,223)(66,465,311,132)(67,374,312,161)(68,403,313,190)(69,432,314,219)(70,461,315,128)(71,370,316,157)(72,399,317,186)(73,428,318,215)(74,457,319,124)(75,366,320,153)(76,395,321,182)(77,424,322,211)(78,453,323,240)(79,362,324,149)(80,391,325,178)(81,420,326,207)(82,449,327,236)(83,478,328,145)(84,387,329,174)(85,416,330,203)(86,445,331,232)(87,474,332,141)(88,383,333,170)(89,412,334,199)(90,441,335,228)(91,470,336,137)(92,379,337,166)(93,408,338,195)(94,437,339,224)(95,466,340,133)(96,375,341,162)(97,404,342,191)(98,433,343,220)(99,462,344,129)(100,371,345,158)(101,400,346,187)(102,429,347,216)(103,458,348,125)(104,367,349,154)(105,396,350,183)(106,425,351,212)(107,454,352,121)(108,363,353,150)(109,392,354,179)(110,421,355,208)(111,450,356,237)(112,479,357,146)(113,388,358,175)(114,417,359,204)(115,446,360,233)(116,475,241,142)(117,384,242,171)(118,413,243,200)(119,442,244,229)(120,471,245,138)>;
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), (1,380,246,167)(2,409,247,196)(3,438,248,225)(4,467,249,134)(5,376,250,163)(6,405,251,192)(7,434,252,221)(8,463,253,130)(9,372,254,159)(10,401,255,188)(11,430,256,217)(12,459,257,126)(13,368,258,155)(14,397,259,184)(15,426,260,213)(16,455,261,122)(17,364,262,151)(18,393,263,180)(19,422,264,209)(20,451,265,238)(21,480,266,147)(22,389,267,176)(23,418,268,205)(24,447,269,234)(25,476,270,143)(26,385,271,172)(27,414,272,201)(28,443,273,230)(29,472,274,139)(30,381,275,168)(31,410,276,197)(32,439,277,226)(33,468,278,135)(34,377,279,164)(35,406,280,193)(36,435,281,222)(37,464,282,131)(38,373,283,160)(39,402,284,189)(40,431,285,218)(41,460,286,127)(42,369,287,156)(43,398,288,185)(44,427,289,214)(45,456,290,123)(46,365,291,152)(47,394,292,181)(48,423,293,210)(49,452,294,239)(50,361,295,148)(51,390,296,177)(52,419,297,206)(53,448,298,235)(54,477,299,144)(55,386,300,173)(56,415,301,202)(57,444,302,231)(58,473,303,140)(59,382,304,169)(60,411,305,198)(61,440,306,227)(62,469,307,136)(63,378,308,165)(64,407,309,194)(65,436,310,223)(66,465,311,132)(67,374,312,161)(68,403,313,190)(69,432,314,219)(70,461,315,128)(71,370,316,157)(72,399,317,186)(73,428,318,215)(74,457,319,124)(75,366,320,153)(76,395,321,182)(77,424,322,211)(78,453,323,240)(79,362,324,149)(80,391,325,178)(81,420,326,207)(82,449,327,236)(83,478,328,145)(84,387,329,174)(85,416,330,203)(86,445,331,232)(87,474,332,141)(88,383,333,170)(89,412,334,199)(90,441,335,228)(91,470,336,137)(92,379,337,166)(93,408,338,195)(94,437,339,224)(95,466,340,133)(96,375,341,162)(97,404,342,191)(98,433,343,220)(99,462,344,129)(100,371,345,158)(101,400,346,187)(102,429,347,216)(103,458,348,125)(104,367,349,154)(105,396,350,183)(106,425,351,212)(107,454,352,121)(108,363,353,150)(109,392,354,179)(110,421,355,208)(111,450,356,237)(112,479,357,146)(113,388,358,175)(114,417,359,204)(115,446,360,233)(116,475,241,142)(117,384,242,171)(118,413,243,200)(119,442,244,229)(120,471,245,138) );
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)], [(1,380,246,167),(2,409,247,196),(3,438,248,225),(4,467,249,134),(5,376,250,163),(6,405,251,192),(7,434,252,221),(8,463,253,130),(9,372,254,159),(10,401,255,188),(11,430,256,217),(12,459,257,126),(13,368,258,155),(14,397,259,184),(15,426,260,213),(16,455,261,122),(17,364,262,151),(18,393,263,180),(19,422,264,209),(20,451,265,238),(21,480,266,147),(22,389,267,176),(23,418,268,205),(24,447,269,234),(25,476,270,143),(26,385,271,172),(27,414,272,201),(28,443,273,230),(29,472,274,139),(30,381,275,168),(31,410,276,197),(32,439,277,226),(33,468,278,135),(34,377,279,164),(35,406,280,193),(36,435,281,222),(37,464,282,131),(38,373,283,160),(39,402,284,189),(40,431,285,218),(41,460,286,127),(42,369,287,156),(43,398,288,185),(44,427,289,214),(45,456,290,123),(46,365,291,152),(47,394,292,181),(48,423,293,210),(49,452,294,239),(50,361,295,148),(51,390,296,177),(52,419,297,206),(53,448,298,235),(54,477,299,144),(55,386,300,173),(56,415,301,202),(57,444,302,231),(58,473,303,140),(59,382,304,169),(60,411,305,198),(61,440,306,227),(62,469,307,136),(63,378,308,165),(64,407,309,194),(65,436,310,223),(66,465,311,132),(67,374,312,161),(68,403,313,190),(69,432,314,219),(70,461,315,128),(71,370,316,157),(72,399,317,186),(73,428,318,215),(74,457,319,124),(75,366,320,153),(76,395,321,182),(77,424,322,211),(78,453,323,240),(79,362,324,149),(80,391,325,178),(81,420,326,207),(82,449,327,236),(83,478,328,145),(84,387,329,174),(85,416,330,203),(86,445,331,232),(87,474,332,141),(88,383,333,170),(89,412,334,199),(90,441,335,228),(91,470,336,137),(92,379,337,166),(93,408,338,195),(94,437,339,224),(95,466,340,133),(96,375,341,162),(97,404,342,191),(98,433,343,220),(99,462,344,129),(100,371,345,158),(101,400,346,187),(102,429,347,216),(103,458,348,125),(104,367,349,154),(105,396,350,183),(106,425,351,212),(107,454,352,121),(108,363,353,150),(109,392,354,179),(110,421,355,208),(111,450,356,237),(112,479,357,146),(113,388,358,175),(114,417,359,204),(115,446,360,233),(116,475,241,142),(117,384,242,171),(118,413,243,200),(119,442,244,229),(120,471,245,138)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 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 | 1 | 1 | 1 | 1 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | Dic3 | D6 | M4(2) | Dic5 | D10 | C4×S3 | C4×S3 | D15 | C4×D5 | C4×D5 | C8⋊S3 | Dic15 | D30 | C8⋊D5 | C4×D15 | C4×D15 | C40⋊S3 |
| kernel | C120⋊13C4 | C2×C15⋊3C8 | C4×Dic15 | C2×C120 | C15⋊3C8 | C120 | C2×Dic15 | C2×C40 | 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 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of C120⋊13C4 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 218 | 61 |
| 0 | 180 | 196 |
| 177 | 0 | 0 |
| 0 | 184 | 109 |
| 0 | 94 | 57 |
G:=sub<GL(3,GF(241))| [1,0,0,0,218,180,0,61,196],[177,0,0,0,184,94,0,109,57] >;
C120⋊13C4 in GAP, Magma, Sage, TeX
C_{120}\rtimes_{13}C_4 % in TeX
G:=Group("C120:13C4"); // GroupNames label
G:=SmallGroup(480,175);
// by ID
G=gap.SmallGroup(480,175);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,80,2693,18822]);
// Polycyclic
G:=Group<a,b|a^120=b^4=1,b*a*b^-1=a^29>;
// generators/relations