direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4⋊C4×C30, (C2×C4)⋊3C60, C4⋊2(C2×C60), (C2×C12)⋊8C20, C60⋊45(C2×C4), C12⋊9(C2×C20), (C2×C60)⋊28C4, C2.2(D4×C30), C20⋊12(C2×C12), (C2×C20)⋊13C12, C2.1(Q8×C30), C10.65(C6×D4), C6.65(D4×C10), (C2×C30).25Q8, C6.18(Q8×C10), C10.18(C6×Q8), (C2×C30).194D4, C30.448(C2×D4), (C22×C20).9C6, C2.2(C22×C60), (C22×C4).5C30, C30.116(C2×Q8), C22.3(Q8×C15), (C22×C60).33C2, C22.10(C2×C60), C23.16(C2×C30), C6.30(C22×C20), (C22×C12).5C10, C22.13(D4×C15), (C2×C60).575C22, (C2×C30).451C23, C30.237(C22×C4), C10.43(C22×C12), C22.5(C22×C30), (C22×C30).187C22, (C2×C6).8(C5×Q8), (C2×C6).51(C5×D4), (C2×C10).8(C3×Q8), (C2×C6).40(C2×C20), (C2×C4).10(C2×C30), (C2×C10).51(C3×D4), (C2×C10).60(C2×C12), (C2×C30).208(C2×C4), (C2×C20).123(C2×C6), (C2×C12).77(C2×C10), (C2×C6).71(C22×C10), (C22×C10).57(C2×C6), (C22×C6).49(C2×C10), (C2×C10).71(C22×C6), SmallGroup(480,921)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4×C30
G = < a,b,c | a30=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 216 in 184 conjugacy classes, 152 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, C12, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×C12, C2×C12, C22×C6, C30, C30, C2×C4⋊C4, C2×C20, C2×C20, C22×C10, C3×C4⋊C4, C22×C12, C22×C12, C60, C60, C2×C30, C2×C30, C5×C4⋊C4, C22×C20, C22×C20, C6×C4⋊C4, C2×C60, C2×C60, C22×C30, C10×C4⋊C4, C15×C4⋊C4, C22×C60, C22×C60, C4⋊C4×C30
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, Q8, C23, C10, C12, C2×C6, C15, C4⋊C4, C22×C4, C2×D4, C2×Q8, C20, C2×C10, C2×C12, C3×D4, C3×Q8, C22×C6, C30, C2×C4⋊C4, C2×C20, C5×D4, C5×Q8, C22×C10, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C60, C2×C30, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C6×C4⋊C4, C2×C60, D4×C15, Q8×C15, C22×C30, C10×C4⋊C4, C15×C4⋊C4, C22×C60, D4×C30, Q8×C30, C4⋊C4×C30
►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)
(1 136 380 67)(2 137 381 68)(3 138 382 69)(4 139 383 70)(5 140 384 71)(6 141 385 72)(7 142 386 73)(8 143 387 74)(9 144 388 75)(10 145 389 76)(11 146 390 77)(12 147 361 78)(13 148 362 79)(14 149 363 80)(15 150 364 81)(16 121 365 82)(17 122 366 83)(18 123 367 84)(19 124 368 85)(20 125 369 86)(21 126 370 87)(22 127 371 88)(23 128 372 89)(24 129 373 90)(25 130 374 61)(26 131 375 62)(27 132 376 63)(28 133 377 64)(29 134 378 65)(30 135 379 66)(31 279 351 201)(32 280 352 202)(33 281 353 203)(34 282 354 204)(35 283 355 205)(36 284 356 206)(37 285 357 207)(38 286 358 208)(39 287 359 209)(40 288 360 210)(41 289 331 181)(42 290 332 182)(43 291 333 183)(44 292 334 184)(45 293 335 185)(46 294 336 186)(47 295 337 187)(48 296 338 188)(49 297 339 189)(50 298 340 190)(51 299 341 191)(52 300 342 192)(53 271 343 193)(54 272 344 194)(55 273 345 195)(56 274 346 196)(57 275 347 197)(58 276 348 198)(59 277 349 199)(60 278 350 200)(91 408 480 219)(92 409 451 220)(93 410 452 221)(94 411 453 222)(95 412 454 223)(96 413 455 224)(97 414 456 225)(98 415 457 226)(99 416 458 227)(100 417 459 228)(101 418 460 229)(102 419 461 230)(103 420 462 231)(104 391 463 232)(105 392 464 233)(106 393 465 234)(107 394 466 235)(108 395 467 236)(109 396 468 237)(110 397 469 238)(111 398 470 239)(112 399 471 240)(113 400 472 211)(114 401 473 212)(115 402 474 213)(116 403 475 214)(117 404 476 215)(118 405 477 216)(119 406 478 217)(120 407 479 218)(151 445 322 260)(152 446 323 261)(153 447 324 262)(154 448 325 263)(155 449 326 264)(156 450 327 265)(157 421 328 266)(158 422 329 267)(159 423 330 268)(160 424 301 269)(161 425 302 270)(162 426 303 241)(163 427 304 242)(164 428 305 243)(165 429 306 244)(166 430 307 245)(167 431 308 246)(168 432 309 247)(169 433 310 248)(170 434 311 249)(171 435 312 250)(172 436 313 251)(173 437 314 252)(174 438 315 253)(175 439 316 254)(176 440 317 255)(177 441 318 256)(178 442 319 257)(179 443 320 258)(180 444 321 259)
(1 266 228 287)(2 267 229 288)(3 268 230 289)(4 269 231 290)(5 270 232 291)(6 241 233 292)(7 242 234 293)(8 243 235 294)(9 244 236 295)(10 245 237 296)(11 246 238 297)(12 247 239 298)(13 248 240 299)(14 249 211 300)(15 250 212 271)(16 251 213 272)(17 252 214 273)(18 253 215 274)(19 254 216 275)(20 255 217 276)(21 256 218 277)(22 257 219 278)(23 258 220 279)(24 259 221 280)(25 260 222 281)(26 261 223 282)(27 262 224 283)(28 263 225 284)(29 264 226 285)(30 265 227 286)(31 128 320 92)(32 129 321 93)(33 130 322 94)(34 131 323 95)(35 132 324 96)(36 133 325 97)(37 134 326 98)(38 135 327 99)(39 136 328 100)(40 137 329 101)(41 138 330 102)(42 139 301 103)(43 140 302 104)(44 141 303 105)(45 142 304 106)(46 143 305 107)(47 144 306 108)(48 145 307 109)(49 146 308 110)(50 147 309 111)(51 148 310 112)(52 149 311 113)(53 150 312 114)(54 121 313 115)(55 122 314 116)(56 123 315 117)(57 124 316 118)(58 125 317 119)(59 126 318 120)(60 127 319 91)(61 151 453 353)(62 152 454 354)(63 153 455 355)(64 154 456 356)(65 155 457 357)(66 156 458 358)(67 157 459 359)(68 158 460 360)(69 159 461 331)(70 160 462 332)(71 161 463 333)(72 162 464 334)(73 163 465 335)(74 164 466 336)(75 165 467 337)(76 166 468 338)(77 167 469 339)(78 168 470 340)(79 169 471 341)(80 170 472 342)(81 171 473 343)(82 172 474 344)(83 173 475 345)(84 174 476 346)(85 175 477 347)(86 176 478 348)(87 177 479 349)(88 178 480 350)(89 179 451 351)(90 180 452 352)(181 382 423 419)(182 383 424 420)(183 384 425 391)(184 385 426 392)(185 386 427 393)(186 387 428 394)(187 388 429 395)(188 389 430 396)(189 390 431 397)(190 361 432 398)(191 362 433 399)(192 363 434 400)(193 364 435 401)(194 365 436 402)(195 366 437 403)(196 367 438 404)(197 368 439 405)(198 369 440 406)(199 370 441 407)(200 371 442 408)(201 372 443 409)(202 373 444 410)(203 374 445 411)(204 375 446 412)(205 376 447 413)(206 377 448 414)(207 378 449 415)(208 379 450 416)(209 380 421 417)(210 381 422 418)
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), (1,136,380,67)(2,137,381,68)(3,138,382,69)(4,139,383,70)(5,140,384,71)(6,141,385,72)(7,142,386,73)(8,143,387,74)(9,144,388,75)(10,145,389,76)(11,146,390,77)(12,147,361,78)(13,148,362,79)(14,149,363,80)(15,150,364,81)(16,121,365,82)(17,122,366,83)(18,123,367,84)(19,124,368,85)(20,125,369,86)(21,126,370,87)(22,127,371,88)(23,128,372,89)(24,129,373,90)(25,130,374,61)(26,131,375,62)(27,132,376,63)(28,133,377,64)(29,134,378,65)(30,135,379,66)(31,279,351,201)(32,280,352,202)(33,281,353,203)(34,282,354,204)(35,283,355,205)(36,284,356,206)(37,285,357,207)(38,286,358,208)(39,287,359,209)(40,288,360,210)(41,289,331,181)(42,290,332,182)(43,291,333,183)(44,292,334,184)(45,293,335,185)(46,294,336,186)(47,295,337,187)(48,296,338,188)(49,297,339,189)(50,298,340,190)(51,299,341,191)(52,300,342,192)(53,271,343,193)(54,272,344,194)(55,273,345,195)(56,274,346,196)(57,275,347,197)(58,276,348,198)(59,277,349,199)(60,278,350,200)(91,408,480,219)(92,409,451,220)(93,410,452,221)(94,411,453,222)(95,412,454,223)(96,413,455,224)(97,414,456,225)(98,415,457,226)(99,416,458,227)(100,417,459,228)(101,418,460,229)(102,419,461,230)(103,420,462,231)(104,391,463,232)(105,392,464,233)(106,393,465,234)(107,394,466,235)(108,395,467,236)(109,396,468,237)(110,397,469,238)(111,398,470,239)(112,399,471,240)(113,400,472,211)(114,401,473,212)(115,402,474,213)(116,403,475,214)(117,404,476,215)(118,405,477,216)(119,406,478,217)(120,407,479,218)(151,445,322,260)(152,446,323,261)(153,447,324,262)(154,448,325,263)(155,449,326,264)(156,450,327,265)(157,421,328,266)(158,422,329,267)(159,423,330,268)(160,424,301,269)(161,425,302,270)(162,426,303,241)(163,427,304,242)(164,428,305,243)(165,429,306,244)(166,430,307,245)(167,431,308,246)(168,432,309,247)(169,433,310,248)(170,434,311,249)(171,435,312,250)(172,436,313,251)(173,437,314,252)(174,438,315,253)(175,439,316,254)(176,440,317,255)(177,441,318,256)(178,442,319,257)(179,443,320,258)(180,444,321,259), (1,266,228,287)(2,267,229,288)(3,268,230,289)(4,269,231,290)(5,270,232,291)(6,241,233,292)(7,242,234,293)(8,243,235,294)(9,244,236,295)(10,245,237,296)(11,246,238,297)(12,247,239,298)(13,248,240,299)(14,249,211,300)(15,250,212,271)(16,251,213,272)(17,252,214,273)(18,253,215,274)(19,254,216,275)(20,255,217,276)(21,256,218,277)(22,257,219,278)(23,258,220,279)(24,259,221,280)(25,260,222,281)(26,261,223,282)(27,262,224,283)(28,263,225,284)(29,264,226,285)(30,265,227,286)(31,128,320,92)(32,129,321,93)(33,130,322,94)(34,131,323,95)(35,132,324,96)(36,133,325,97)(37,134,326,98)(38,135,327,99)(39,136,328,100)(40,137,329,101)(41,138,330,102)(42,139,301,103)(43,140,302,104)(44,141,303,105)(45,142,304,106)(46,143,305,107)(47,144,306,108)(48,145,307,109)(49,146,308,110)(50,147,309,111)(51,148,310,112)(52,149,311,113)(53,150,312,114)(54,121,313,115)(55,122,314,116)(56,123,315,117)(57,124,316,118)(58,125,317,119)(59,126,318,120)(60,127,319,91)(61,151,453,353)(62,152,454,354)(63,153,455,355)(64,154,456,356)(65,155,457,357)(66,156,458,358)(67,157,459,359)(68,158,460,360)(69,159,461,331)(70,160,462,332)(71,161,463,333)(72,162,464,334)(73,163,465,335)(74,164,466,336)(75,165,467,337)(76,166,468,338)(77,167,469,339)(78,168,470,340)(79,169,471,341)(80,170,472,342)(81,171,473,343)(82,172,474,344)(83,173,475,345)(84,174,476,346)(85,175,477,347)(86,176,478,348)(87,177,479,349)(88,178,480,350)(89,179,451,351)(90,180,452,352)(181,382,423,419)(182,383,424,420)(183,384,425,391)(184,385,426,392)(185,386,427,393)(186,387,428,394)(187,388,429,395)(188,389,430,396)(189,390,431,397)(190,361,432,398)(191,362,433,399)(192,363,434,400)(193,364,435,401)(194,365,436,402)(195,366,437,403)(196,367,438,404)(197,368,439,405)(198,369,440,406)(199,370,441,407)(200,371,442,408)(201,372,443,409)(202,373,444,410)(203,374,445,411)(204,375,446,412)(205,376,447,413)(206,377,448,414)(207,378,449,415)(208,379,450,416)(209,380,421,417)(210,381,422,418)>;
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), (1,136,380,67)(2,137,381,68)(3,138,382,69)(4,139,383,70)(5,140,384,71)(6,141,385,72)(7,142,386,73)(8,143,387,74)(9,144,388,75)(10,145,389,76)(11,146,390,77)(12,147,361,78)(13,148,362,79)(14,149,363,80)(15,150,364,81)(16,121,365,82)(17,122,366,83)(18,123,367,84)(19,124,368,85)(20,125,369,86)(21,126,370,87)(22,127,371,88)(23,128,372,89)(24,129,373,90)(25,130,374,61)(26,131,375,62)(27,132,376,63)(28,133,377,64)(29,134,378,65)(30,135,379,66)(31,279,351,201)(32,280,352,202)(33,281,353,203)(34,282,354,204)(35,283,355,205)(36,284,356,206)(37,285,357,207)(38,286,358,208)(39,287,359,209)(40,288,360,210)(41,289,331,181)(42,290,332,182)(43,291,333,183)(44,292,334,184)(45,293,335,185)(46,294,336,186)(47,295,337,187)(48,296,338,188)(49,297,339,189)(50,298,340,190)(51,299,341,191)(52,300,342,192)(53,271,343,193)(54,272,344,194)(55,273,345,195)(56,274,346,196)(57,275,347,197)(58,276,348,198)(59,277,349,199)(60,278,350,200)(91,408,480,219)(92,409,451,220)(93,410,452,221)(94,411,453,222)(95,412,454,223)(96,413,455,224)(97,414,456,225)(98,415,457,226)(99,416,458,227)(100,417,459,228)(101,418,460,229)(102,419,461,230)(103,420,462,231)(104,391,463,232)(105,392,464,233)(106,393,465,234)(107,394,466,235)(108,395,467,236)(109,396,468,237)(110,397,469,238)(111,398,470,239)(112,399,471,240)(113,400,472,211)(114,401,473,212)(115,402,474,213)(116,403,475,214)(117,404,476,215)(118,405,477,216)(119,406,478,217)(120,407,479,218)(151,445,322,260)(152,446,323,261)(153,447,324,262)(154,448,325,263)(155,449,326,264)(156,450,327,265)(157,421,328,266)(158,422,329,267)(159,423,330,268)(160,424,301,269)(161,425,302,270)(162,426,303,241)(163,427,304,242)(164,428,305,243)(165,429,306,244)(166,430,307,245)(167,431,308,246)(168,432,309,247)(169,433,310,248)(170,434,311,249)(171,435,312,250)(172,436,313,251)(173,437,314,252)(174,438,315,253)(175,439,316,254)(176,440,317,255)(177,441,318,256)(178,442,319,257)(179,443,320,258)(180,444,321,259), (1,266,228,287)(2,267,229,288)(3,268,230,289)(4,269,231,290)(5,270,232,291)(6,241,233,292)(7,242,234,293)(8,243,235,294)(9,244,236,295)(10,245,237,296)(11,246,238,297)(12,247,239,298)(13,248,240,299)(14,249,211,300)(15,250,212,271)(16,251,213,272)(17,252,214,273)(18,253,215,274)(19,254,216,275)(20,255,217,276)(21,256,218,277)(22,257,219,278)(23,258,220,279)(24,259,221,280)(25,260,222,281)(26,261,223,282)(27,262,224,283)(28,263,225,284)(29,264,226,285)(30,265,227,286)(31,128,320,92)(32,129,321,93)(33,130,322,94)(34,131,323,95)(35,132,324,96)(36,133,325,97)(37,134,326,98)(38,135,327,99)(39,136,328,100)(40,137,329,101)(41,138,330,102)(42,139,301,103)(43,140,302,104)(44,141,303,105)(45,142,304,106)(46,143,305,107)(47,144,306,108)(48,145,307,109)(49,146,308,110)(50,147,309,111)(51,148,310,112)(52,149,311,113)(53,150,312,114)(54,121,313,115)(55,122,314,116)(56,123,315,117)(57,124,316,118)(58,125,317,119)(59,126,318,120)(60,127,319,91)(61,151,453,353)(62,152,454,354)(63,153,455,355)(64,154,456,356)(65,155,457,357)(66,156,458,358)(67,157,459,359)(68,158,460,360)(69,159,461,331)(70,160,462,332)(71,161,463,333)(72,162,464,334)(73,163,465,335)(74,164,466,336)(75,165,467,337)(76,166,468,338)(77,167,469,339)(78,168,470,340)(79,169,471,341)(80,170,472,342)(81,171,473,343)(82,172,474,344)(83,173,475,345)(84,174,476,346)(85,175,477,347)(86,176,478,348)(87,177,479,349)(88,178,480,350)(89,179,451,351)(90,180,452,352)(181,382,423,419)(182,383,424,420)(183,384,425,391)(184,385,426,392)(185,386,427,393)(186,387,428,394)(187,388,429,395)(188,389,430,396)(189,390,431,397)(190,361,432,398)(191,362,433,399)(192,363,434,400)(193,364,435,401)(194,365,436,402)(195,366,437,403)(196,367,438,404)(197,368,439,405)(198,369,440,406)(199,370,441,407)(200,371,442,408)(201,372,443,409)(202,373,444,410)(203,374,445,411)(204,375,446,412)(205,376,447,413)(206,377,448,414)(207,378,449,415)(208,379,450,416)(209,380,421,417)(210,381,422,418) );
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)], [(1,136,380,67),(2,137,381,68),(3,138,382,69),(4,139,383,70),(5,140,384,71),(6,141,385,72),(7,142,386,73),(8,143,387,74),(9,144,388,75),(10,145,389,76),(11,146,390,77),(12,147,361,78),(13,148,362,79),(14,149,363,80),(15,150,364,81),(16,121,365,82),(17,122,366,83),(18,123,367,84),(19,124,368,85),(20,125,369,86),(21,126,370,87),(22,127,371,88),(23,128,372,89),(24,129,373,90),(25,130,374,61),(26,131,375,62),(27,132,376,63),(28,133,377,64),(29,134,378,65),(30,135,379,66),(31,279,351,201),(32,280,352,202),(33,281,353,203),(34,282,354,204),(35,283,355,205),(36,284,356,206),(37,285,357,207),(38,286,358,208),(39,287,359,209),(40,288,360,210),(41,289,331,181),(42,290,332,182),(43,291,333,183),(44,292,334,184),(45,293,335,185),(46,294,336,186),(47,295,337,187),(48,296,338,188),(49,297,339,189),(50,298,340,190),(51,299,341,191),(52,300,342,192),(53,271,343,193),(54,272,344,194),(55,273,345,195),(56,274,346,196),(57,275,347,197),(58,276,348,198),(59,277,349,199),(60,278,350,200),(91,408,480,219),(92,409,451,220),(93,410,452,221),(94,411,453,222),(95,412,454,223),(96,413,455,224),(97,414,456,225),(98,415,457,226),(99,416,458,227),(100,417,459,228),(101,418,460,229),(102,419,461,230),(103,420,462,231),(104,391,463,232),(105,392,464,233),(106,393,465,234),(107,394,466,235),(108,395,467,236),(109,396,468,237),(110,397,469,238),(111,398,470,239),(112,399,471,240),(113,400,472,211),(114,401,473,212),(115,402,474,213),(116,403,475,214),(117,404,476,215),(118,405,477,216),(119,406,478,217),(120,407,479,218),(151,445,322,260),(152,446,323,261),(153,447,324,262),(154,448,325,263),(155,449,326,264),(156,450,327,265),(157,421,328,266),(158,422,329,267),(159,423,330,268),(160,424,301,269),(161,425,302,270),(162,426,303,241),(163,427,304,242),(164,428,305,243),(165,429,306,244),(166,430,307,245),(167,431,308,246),(168,432,309,247),(169,433,310,248),(170,434,311,249),(171,435,312,250),(172,436,313,251),(173,437,314,252),(174,438,315,253),(175,439,316,254),(176,440,317,255),(177,441,318,256),(178,442,319,257),(179,443,320,258),(180,444,321,259)], [(1,266,228,287),(2,267,229,288),(3,268,230,289),(4,269,231,290),(5,270,232,291),(6,241,233,292),(7,242,234,293),(8,243,235,294),(9,244,236,295),(10,245,237,296),(11,246,238,297),(12,247,239,298),(13,248,240,299),(14,249,211,300),(15,250,212,271),(16,251,213,272),(17,252,214,273),(18,253,215,274),(19,254,216,275),(20,255,217,276),(21,256,218,277),(22,257,219,278),(23,258,220,279),(24,259,221,280),(25,260,222,281),(26,261,223,282),(27,262,224,283),(28,263,225,284),(29,264,226,285),(30,265,227,286),(31,128,320,92),(32,129,321,93),(33,130,322,94),(34,131,323,95),(35,132,324,96),(36,133,325,97),(37,134,326,98),(38,135,327,99),(39,136,328,100),(40,137,329,101),(41,138,330,102),(42,139,301,103),(43,140,302,104),(44,141,303,105),(45,142,304,106),(46,143,305,107),(47,144,306,108),(48,145,307,109),(49,146,308,110),(50,147,309,111),(51,148,310,112),(52,149,311,113),(53,150,312,114),(54,121,313,115),(55,122,314,116),(56,123,315,117),(57,124,316,118),(58,125,317,119),(59,126,318,120),(60,127,319,91),(61,151,453,353),(62,152,454,354),(63,153,455,355),(64,154,456,356),(65,155,457,357),(66,156,458,358),(67,157,459,359),(68,158,460,360),(69,159,461,331),(70,160,462,332),(71,161,463,333),(72,162,464,334),(73,163,465,335),(74,164,466,336),(75,165,467,337),(76,166,468,338),(77,167,469,339),(78,168,470,340),(79,169,471,341),(80,170,472,342),(81,171,473,343),(82,172,474,344),(83,173,475,345),(84,174,476,346),(85,175,477,347),(86,176,478,348),(87,177,479,349),(88,178,480,350),(89,179,451,351),(90,180,452,352),(181,382,423,419),(182,383,424,420),(183,384,425,391),(184,385,426,392),(185,386,427,393),(186,387,428,394),(187,388,429,395),(188,389,430,396),(189,390,431,397),(190,361,432,398),(191,362,433,399),(192,363,434,400),(193,364,435,401),(194,365,436,402),(195,366,437,403),(196,367,438,404),(197,368,439,405),(198,369,440,406),(199,370,441,407),(200,371,442,408),(201,372,443,409),(202,373,444,410),(203,374,445,411),(204,375,446,412),(205,376,447,413),(206,377,448,414),(207,378,449,415),(208,379,450,416),(209,380,421,417),(210,381,422,418)]])
300 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4L | 5A | 5B | 5C | 5D | 6A | ··· | 6N | 10A | ··· | 10AB | 12A | ··· | 12X | 15A | ··· | 15H | 20A | ··· | 20AV | 30A | ··· | 30BD | 60A | ··· | 60CR |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
300 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C60 | D4 | Q8 | C3×D4 | C3×Q8 | C5×D4 | C5×Q8 | D4×C15 | Q8×C15 |
| kernel | C4⋊C4×C30 | C15×C4⋊C4 | C22×C60 | C10×C4⋊C4 | C2×C60 | C6×C4⋊C4 | C5×C4⋊C4 | C22×C20 | C3×C4⋊C4 | C22×C12 | C2×C20 | C2×C4⋊C4 | C2×C12 | C4⋊C4 | C22×C4 | C2×C4 | C2×C30 | C2×C30 | C2×C10 | C2×C10 | C2×C6 | C2×C6 | C22 | C22 |
| # reps | 1 | 4 | 3 | 2 | 8 | 4 | 8 | 6 | 16 | 12 | 16 | 8 | 32 | 32 | 24 | 64 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 16 |
Matrix representation of C4⋊C4×C30 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 27 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 60 | 0 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 59 | 42 |
| 0 | 0 | 0 | 42 | 2 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,60,0,0,0,0,0,47,0,0,0,0,0,27,0,0,0,0,0,27],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,1,0],[11,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,59,42,0,0,0,42,2] >;
C4⋊C4×C30 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times C_{30} % in TeX
G:=Group("C4:C4xC30"); // GroupNames label
G:=SmallGroup(480,921);
// by ID
G=gap.SmallGroup(480,921);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1680,1709,848]);
// Polycyclic
G:=Group<a,b,c|a^30=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations