metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.8D20, C60.78D4, C30.1Q16, C30.7SD16, Dic10⋊6Dic3, C12.4(C4×D5), (C2×C30).17D4, (C2×C20).50D6, C4⋊Dic3.8D5, C4.7(D5×Dic3), C60.122(C2×C4), C15⋊3(Q8⋊C4), (C2×C12).51D10, C5⋊3(Q8⋊2Dic3), C3⋊3(C10.Q16), C6.5(D4.D5), C6.4(C5⋊Q16), (C3×Dic10)⋊10C4, C20.12(C3⋊D4), C4.22(C3⋊D20), (C6×Dic10).6C2, (C2×Dic10).7S3, C20.23(C2×Dic3), C10.4(C3⋊Q16), C2.1(C15⋊Q16), C30.53(C22⋊C4), (C2×C60).183C22, C10.5(Q8⋊2S3), C2.2(C20.D6), C6.28(D10⋊C4), C2.7(D10⋊Dic3), C22.14(C15⋊D4), C10.17(C6.D4), (C2×C4).186(S3×D5), (C5×C4⋊Dic3).7C2, (C2×C15⋊3C8).11C2, (C2×C6).45(C5⋊D4), (C2×C10).45(C3⋊D4), SmallGroup(480,46)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.Q16
G = < a,b,c | a30=b8=1, c2=a15b4, bab-1=a-1, cac-1=a11, cbc-1=a15b-1 >
Subgroups: 316 in 84 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, Q8, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8, C30, Q8⋊C4, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C2×C3⋊C8, C4⋊Dic3, C6×Q8, C5×Dic3, C3×Dic5, C60, C2×C30, C2×C5⋊2C8, C5×C4⋊C4, C2×Dic10, Q8⋊2Dic3, C15⋊3C8, C3×Dic10, C3×Dic10, C6×Dic5, C10×Dic3, C2×C60, C10.Q16, C5×C4⋊Dic3, C2×C15⋊3C8, C6×Dic10, C30.Q16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, SD16, Q16, D10, C2×Dic3, C3⋊D4, Q8⋊C4, C4×D5, D20, C5⋊D4, Q8⋊2S3, C3⋊Q16, C6.D4, S3×D5, D10⋊C4, D4.D5, C5⋊Q16, Q8⋊2Dic3, D5×Dic3, C15⋊D4, C3⋊D20, C10.Q16, C20.D6, C15⋊Q16, D10⋊Dic3, C30.Q16
►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)
(1 330 456 287 252 376 409 117)(2 329 457 286 253 375 410 116)(3 328 458 285 254 374 411 115)(4 327 459 284 255 373 412 114)(5 326 460 283 256 372 413 113)(6 325 461 282 257 371 414 112)(7 324 462 281 258 370 415 111)(8 323 463 280 259 369 416 110)(9 322 464 279 260 368 417 109)(10 321 465 278 261 367 418 108)(11 320 466 277 262 366 419 107)(12 319 467 276 263 365 420 106)(13 318 468 275 264 364 391 105)(14 317 469 274 265 363 392 104)(15 316 470 273 266 362 393 103)(16 315 471 272 267 361 394 102)(17 314 472 271 268 390 395 101)(18 313 473 300 269 389 396 100)(19 312 474 299 270 388 397 99)(20 311 475 298 241 387 398 98)(21 310 476 297 242 386 399 97)(22 309 477 296 243 385 400 96)(23 308 478 295 244 384 401 95)(24 307 479 294 245 383 402 94)(25 306 480 293 246 382 403 93)(26 305 451 292 247 381 404 92)(27 304 452 291 248 380 405 91)(28 303 453 290 249 379 406 120)(29 302 454 289 250 378 407 119)(30 301 455 288 251 377 408 118)(31 141 82 220 448 185 168 357)(32 140 83 219 449 184 169 356)(33 139 84 218 450 183 170 355)(34 138 85 217 421 182 171 354)(35 137 86 216 422 181 172 353)(36 136 87 215 423 210 173 352)(37 135 88 214 424 209 174 351)(38 134 89 213 425 208 175 350)(39 133 90 212 426 207 176 349)(40 132 61 211 427 206 177 348)(41 131 62 240 428 205 178 347)(42 130 63 239 429 204 179 346)(43 129 64 238 430 203 180 345)(44 128 65 237 431 202 151 344)(45 127 66 236 432 201 152 343)(46 126 67 235 433 200 153 342)(47 125 68 234 434 199 154 341)(48 124 69 233 435 198 155 340)(49 123 70 232 436 197 156 339)(50 122 71 231 437 196 157 338)(51 121 72 230 438 195 158 337)(52 150 73 229 439 194 159 336)(53 149 74 228 440 193 160 335)(54 148 75 227 441 192 161 334)(55 147 76 226 442 191 162 333)(56 146 77 225 443 190 163 332)(57 145 78 224 444 189 164 331)(58 144 79 223 445 188 165 360)(59 143 80 222 446 187 166 359)(60 142 81 221 447 186 167 358)
(1 73 267 174)(2 84 268 155)(3 65 269 166)(4 76 270 177)(5 87 241 158)(6 68 242 169)(7 79 243 180)(8 90 244 161)(9 71 245 172)(10 82 246 153)(11 63 247 164)(12 74 248 175)(13 85 249 156)(14 66 250 167)(15 77 251 178)(16 88 252 159)(17 69 253 170)(18 80 254 151)(19 61 255 162)(20 72 256 173)(21 83 257 154)(22 64 258 165)(23 75 259 176)(24 86 260 157)(25 67 261 168)(26 78 262 179)(27 89 263 160)(28 70 264 171)(29 81 265 152)(30 62 266 163)(31 403 433 465)(32 414 434 476)(33 395 435 457)(34 406 436 468)(35 417 437 479)(36 398 438 460)(37 409 439 471)(38 420 440 452)(39 401 441 463)(40 412 442 474)(41 393 443 455)(42 404 444 466)(43 415 445 477)(44 396 446 458)(45 407 447 469)(46 418 448 480)(47 399 449 461)(48 410 450 472)(49 391 421 453)(50 402 422 464)(51 413 423 475)(52 394 424 456)(53 405 425 467)(54 416 426 478)(55 397 427 459)(56 408 428 470)(57 419 429 451)(58 400 430 462)(59 411 431 473)(60 392 432 454)(91 228 276 350)(92 239 277 331)(93 220 278 342)(94 231 279 353)(95 212 280 334)(96 223 281 345)(97 234 282 356)(98 215 283 337)(99 226 284 348)(100 237 285 359)(101 218 286 340)(102 229 287 351)(103 240 288 332)(104 221 289 343)(105 232 290 354)(106 213 291 335)(107 224 292 346)(108 235 293 357)(109 216 294 338)(110 227 295 349)(111 238 296 360)(112 219 297 341)(113 230 298 352)(114 211 299 333)(115 222 300 344)(116 233 271 355)(117 214 272 336)(118 225 273 347)(119 236 274 358)(120 217 275 339)(121 387 210 326)(122 368 181 307)(123 379 182 318)(124 390 183 329)(125 371 184 310)(126 382 185 321)(127 363 186 302)(128 374 187 313)(129 385 188 324)(130 366 189 305)(131 377 190 316)(132 388 191 327)(133 369 192 308)(134 380 193 319)(135 361 194 330)(136 372 195 311)(137 383 196 322)(138 364 197 303)(139 375 198 314)(140 386 199 325)(141 367 200 306)(142 378 201 317)(143 389 202 328)(144 370 203 309)(145 381 204 320)(146 362 205 301)(147 373 206 312)(148 384 207 323)(149 365 208 304)(150 376 209 315)
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), (1,330,456,287,252,376,409,117)(2,329,457,286,253,375,410,116)(3,328,458,285,254,374,411,115)(4,327,459,284,255,373,412,114)(5,326,460,283,256,372,413,113)(6,325,461,282,257,371,414,112)(7,324,462,281,258,370,415,111)(8,323,463,280,259,369,416,110)(9,322,464,279,260,368,417,109)(10,321,465,278,261,367,418,108)(11,320,466,277,262,366,419,107)(12,319,467,276,263,365,420,106)(13,318,468,275,264,364,391,105)(14,317,469,274,265,363,392,104)(15,316,470,273,266,362,393,103)(16,315,471,272,267,361,394,102)(17,314,472,271,268,390,395,101)(18,313,473,300,269,389,396,100)(19,312,474,299,270,388,397,99)(20,311,475,298,241,387,398,98)(21,310,476,297,242,386,399,97)(22,309,477,296,243,385,400,96)(23,308,478,295,244,384,401,95)(24,307,479,294,245,383,402,94)(25,306,480,293,246,382,403,93)(26,305,451,292,247,381,404,92)(27,304,452,291,248,380,405,91)(28,303,453,290,249,379,406,120)(29,302,454,289,250,378,407,119)(30,301,455,288,251,377,408,118)(31,141,82,220,448,185,168,357)(32,140,83,219,449,184,169,356)(33,139,84,218,450,183,170,355)(34,138,85,217,421,182,171,354)(35,137,86,216,422,181,172,353)(36,136,87,215,423,210,173,352)(37,135,88,214,424,209,174,351)(38,134,89,213,425,208,175,350)(39,133,90,212,426,207,176,349)(40,132,61,211,427,206,177,348)(41,131,62,240,428,205,178,347)(42,130,63,239,429,204,179,346)(43,129,64,238,430,203,180,345)(44,128,65,237,431,202,151,344)(45,127,66,236,432,201,152,343)(46,126,67,235,433,200,153,342)(47,125,68,234,434,199,154,341)(48,124,69,233,435,198,155,340)(49,123,70,232,436,197,156,339)(50,122,71,231,437,196,157,338)(51,121,72,230,438,195,158,337)(52,150,73,229,439,194,159,336)(53,149,74,228,440,193,160,335)(54,148,75,227,441,192,161,334)(55,147,76,226,442,191,162,333)(56,146,77,225,443,190,163,332)(57,145,78,224,444,189,164,331)(58,144,79,223,445,188,165,360)(59,143,80,222,446,187,166,359)(60,142,81,221,447,186,167,358), (1,73,267,174)(2,84,268,155)(3,65,269,166)(4,76,270,177)(5,87,241,158)(6,68,242,169)(7,79,243,180)(8,90,244,161)(9,71,245,172)(10,82,246,153)(11,63,247,164)(12,74,248,175)(13,85,249,156)(14,66,250,167)(15,77,251,178)(16,88,252,159)(17,69,253,170)(18,80,254,151)(19,61,255,162)(20,72,256,173)(21,83,257,154)(22,64,258,165)(23,75,259,176)(24,86,260,157)(25,67,261,168)(26,78,262,179)(27,89,263,160)(28,70,264,171)(29,81,265,152)(30,62,266,163)(31,403,433,465)(32,414,434,476)(33,395,435,457)(34,406,436,468)(35,417,437,479)(36,398,438,460)(37,409,439,471)(38,420,440,452)(39,401,441,463)(40,412,442,474)(41,393,443,455)(42,404,444,466)(43,415,445,477)(44,396,446,458)(45,407,447,469)(46,418,448,480)(47,399,449,461)(48,410,450,472)(49,391,421,453)(50,402,422,464)(51,413,423,475)(52,394,424,456)(53,405,425,467)(54,416,426,478)(55,397,427,459)(56,408,428,470)(57,419,429,451)(58,400,430,462)(59,411,431,473)(60,392,432,454)(91,228,276,350)(92,239,277,331)(93,220,278,342)(94,231,279,353)(95,212,280,334)(96,223,281,345)(97,234,282,356)(98,215,283,337)(99,226,284,348)(100,237,285,359)(101,218,286,340)(102,229,287,351)(103,240,288,332)(104,221,289,343)(105,232,290,354)(106,213,291,335)(107,224,292,346)(108,235,293,357)(109,216,294,338)(110,227,295,349)(111,238,296,360)(112,219,297,341)(113,230,298,352)(114,211,299,333)(115,222,300,344)(116,233,271,355)(117,214,272,336)(118,225,273,347)(119,236,274,358)(120,217,275,339)(121,387,210,326)(122,368,181,307)(123,379,182,318)(124,390,183,329)(125,371,184,310)(126,382,185,321)(127,363,186,302)(128,374,187,313)(129,385,188,324)(130,366,189,305)(131,377,190,316)(132,388,191,327)(133,369,192,308)(134,380,193,319)(135,361,194,330)(136,372,195,311)(137,383,196,322)(138,364,197,303)(139,375,198,314)(140,386,199,325)(141,367,200,306)(142,378,201,317)(143,389,202,328)(144,370,203,309)(145,381,204,320)(146,362,205,301)(147,373,206,312)(148,384,207,323)(149,365,208,304)(150,376,209,315)>;
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), (1,330,456,287,252,376,409,117)(2,329,457,286,253,375,410,116)(3,328,458,285,254,374,411,115)(4,327,459,284,255,373,412,114)(5,326,460,283,256,372,413,113)(6,325,461,282,257,371,414,112)(7,324,462,281,258,370,415,111)(8,323,463,280,259,369,416,110)(9,322,464,279,260,368,417,109)(10,321,465,278,261,367,418,108)(11,320,466,277,262,366,419,107)(12,319,467,276,263,365,420,106)(13,318,468,275,264,364,391,105)(14,317,469,274,265,363,392,104)(15,316,470,273,266,362,393,103)(16,315,471,272,267,361,394,102)(17,314,472,271,268,390,395,101)(18,313,473,300,269,389,396,100)(19,312,474,299,270,388,397,99)(20,311,475,298,241,387,398,98)(21,310,476,297,242,386,399,97)(22,309,477,296,243,385,400,96)(23,308,478,295,244,384,401,95)(24,307,479,294,245,383,402,94)(25,306,480,293,246,382,403,93)(26,305,451,292,247,381,404,92)(27,304,452,291,248,380,405,91)(28,303,453,290,249,379,406,120)(29,302,454,289,250,378,407,119)(30,301,455,288,251,377,408,118)(31,141,82,220,448,185,168,357)(32,140,83,219,449,184,169,356)(33,139,84,218,450,183,170,355)(34,138,85,217,421,182,171,354)(35,137,86,216,422,181,172,353)(36,136,87,215,423,210,173,352)(37,135,88,214,424,209,174,351)(38,134,89,213,425,208,175,350)(39,133,90,212,426,207,176,349)(40,132,61,211,427,206,177,348)(41,131,62,240,428,205,178,347)(42,130,63,239,429,204,179,346)(43,129,64,238,430,203,180,345)(44,128,65,237,431,202,151,344)(45,127,66,236,432,201,152,343)(46,126,67,235,433,200,153,342)(47,125,68,234,434,199,154,341)(48,124,69,233,435,198,155,340)(49,123,70,232,436,197,156,339)(50,122,71,231,437,196,157,338)(51,121,72,230,438,195,158,337)(52,150,73,229,439,194,159,336)(53,149,74,228,440,193,160,335)(54,148,75,227,441,192,161,334)(55,147,76,226,442,191,162,333)(56,146,77,225,443,190,163,332)(57,145,78,224,444,189,164,331)(58,144,79,223,445,188,165,360)(59,143,80,222,446,187,166,359)(60,142,81,221,447,186,167,358), (1,73,267,174)(2,84,268,155)(3,65,269,166)(4,76,270,177)(5,87,241,158)(6,68,242,169)(7,79,243,180)(8,90,244,161)(9,71,245,172)(10,82,246,153)(11,63,247,164)(12,74,248,175)(13,85,249,156)(14,66,250,167)(15,77,251,178)(16,88,252,159)(17,69,253,170)(18,80,254,151)(19,61,255,162)(20,72,256,173)(21,83,257,154)(22,64,258,165)(23,75,259,176)(24,86,260,157)(25,67,261,168)(26,78,262,179)(27,89,263,160)(28,70,264,171)(29,81,265,152)(30,62,266,163)(31,403,433,465)(32,414,434,476)(33,395,435,457)(34,406,436,468)(35,417,437,479)(36,398,438,460)(37,409,439,471)(38,420,440,452)(39,401,441,463)(40,412,442,474)(41,393,443,455)(42,404,444,466)(43,415,445,477)(44,396,446,458)(45,407,447,469)(46,418,448,480)(47,399,449,461)(48,410,450,472)(49,391,421,453)(50,402,422,464)(51,413,423,475)(52,394,424,456)(53,405,425,467)(54,416,426,478)(55,397,427,459)(56,408,428,470)(57,419,429,451)(58,400,430,462)(59,411,431,473)(60,392,432,454)(91,228,276,350)(92,239,277,331)(93,220,278,342)(94,231,279,353)(95,212,280,334)(96,223,281,345)(97,234,282,356)(98,215,283,337)(99,226,284,348)(100,237,285,359)(101,218,286,340)(102,229,287,351)(103,240,288,332)(104,221,289,343)(105,232,290,354)(106,213,291,335)(107,224,292,346)(108,235,293,357)(109,216,294,338)(110,227,295,349)(111,238,296,360)(112,219,297,341)(113,230,298,352)(114,211,299,333)(115,222,300,344)(116,233,271,355)(117,214,272,336)(118,225,273,347)(119,236,274,358)(120,217,275,339)(121,387,210,326)(122,368,181,307)(123,379,182,318)(124,390,183,329)(125,371,184,310)(126,382,185,321)(127,363,186,302)(128,374,187,313)(129,385,188,324)(130,366,189,305)(131,377,190,316)(132,388,191,327)(133,369,192,308)(134,380,193,319)(135,361,194,330)(136,372,195,311)(137,383,196,322)(138,364,197,303)(139,375,198,314)(140,386,199,325)(141,367,200,306)(142,378,201,317)(143,389,202,328)(144,370,203,309)(145,381,204,320)(146,362,205,301)(147,373,206,312)(148,384,207,323)(149,365,208,304)(150,376,209,315) );
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)], [(1,330,456,287,252,376,409,117),(2,329,457,286,253,375,410,116),(3,328,458,285,254,374,411,115),(4,327,459,284,255,373,412,114),(5,326,460,283,256,372,413,113),(6,325,461,282,257,371,414,112),(7,324,462,281,258,370,415,111),(8,323,463,280,259,369,416,110),(9,322,464,279,260,368,417,109),(10,321,465,278,261,367,418,108),(11,320,466,277,262,366,419,107),(12,319,467,276,263,365,420,106),(13,318,468,275,264,364,391,105),(14,317,469,274,265,363,392,104),(15,316,470,273,266,362,393,103),(16,315,471,272,267,361,394,102),(17,314,472,271,268,390,395,101),(18,313,473,300,269,389,396,100),(19,312,474,299,270,388,397,99),(20,311,475,298,241,387,398,98),(21,310,476,297,242,386,399,97),(22,309,477,296,243,385,400,96),(23,308,478,295,244,384,401,95),(24,307,479,294,245,383,402,94),(25,306,480,293,246,382,403,93),(26,305,451,292,247,381,404,92),(27,304,452,291,248,380,405,91),(28,303,453,290,249,379,406,120),(29,302,454,289,250,378,407,119),(30,301,455,288,251,377,408,118),(31,141,82,220,448,185,168,357),(32,140,83,219,449,184,169,356),(33,139,84,218,450,183,170,355),(34,138,85,217,421,182,171,354),(35,137,86,216,422,181,172,353),(36,136,87,215,423,210,173,352),(37,135,88,214,424,209,174,351),(38,134,89,213,425,208,175,350),(39,133,90,212,426,207,176,349),(40,132,61,211,427,206,177,348),(41,131,62,240,428,205,178,347),(42,130,63,239,429,204,179,346),(43,129,64,238,430,203,180,345),(44,128,65,237,431,202,151,344),(45,127,66,236,432,201,152,343),(46,126,67,235,433,200,153,342),(47,125,68,234,434,199,154,341),(48,124,69,233,435,198,155,340),(49,123,70,232,436,197,156,339),(50,122,71,231,437,196,157,338),(51,121,72,230,438,195,158,337),(52,150,73,229,439,194,159,336),(53,149,74,228,440,193,160,335),(54,148,75,227,441,192,161,334),(55,147,76,226,442,191,162,333),(56,146,77,225,443,190,163,332),(57,145,78,224,444,189,164,331),(58,144,79,223,445,188,165,360),(59,143,80,222,446,187,166,359),(60,142,81,221,447,186,167,358)], [(1,73,267,174),(2,84,268,155),(3,65,269,166),(4,76,270,177),(5,87,241,158),(6,68,242,169),(7,79,243,180),(8,90,244,161),(9,71,245,172),(10,82,246,153),(11,63,247,164),(12,74,248,175),(13,85,249,156),(14,66,250,167),(15,77,251,178),(16,88,252,159),(17,69,253,170),(18,80,254,151),(19,61,255,162),(20,72,256,173),(21,83,257,154),(22,64,258,165),(23,75,259,176),(24,86,260,157),(25,67,261,168),(26,78,262,179),(27,89,263,160),(28,70,264,171),(29,81,265,152),(30,62,266,163),(31,403,433,465),(32,414,434,476),(33,395,435,457),(34,406,436,468),(35,417,437,479),(36,398,438,460),(37,409,439,471),(38,420,440,452),(39,401,441,463),(40,412,442,474),(41,393,443,455),(42,404,444,466),(43,415,445,477),(44,396,446,458),(45,407,447,469),(46,418,448,480),(47,399,449,461),(48,410,450,472),(49,391,421,453),(50,402,422,464),(51,413,423,475),(52,394,424,456),(53,405,425,467),(54,416,426,478),(55,397,427,459),(56,408,428,470),(57,419,429,451),(58,400,430,462),(59,411,431,473),(60,392,432,454),(91,228,276,350),(92,239,277,331),(93,220,278,342),(94,231,279,353),(95,212,280,334),(96,223,281,345),(97,234,282,356),(98,215,283,337),(99,226,284,348),(100,237,285,359),(101,218,286,340),(102,229,287,351),(103,240,288,332),(104,221,289,343),(105,232,290,354),(106,213,291,335),(107,224,292,346),(108,235,293,357),(109,216,294,338),(110,227,295,349),(111,238,296,360),(112,219,297,341),(113,230,298,352),(114,211,299,333),(115,222,300,344),(116,233,271,355),(117,214,272,336),(118,225,273,347),(119,236,274,358),(120,217,275,339),(121,387,210,326),(122,368,181,307),(123,379,182,318),(124,390,183,329),(125,371,184,310),(126,382,185,321),(127,363,186,302),(128,374,187,313),(129,385,188,324),(130,366,189,305),(131,377,190,316),(132,388,191,327),(133,369,192,308),(134,380,193,319),(135,361,194,330),(136,372,195,311),(137,383,196,322),(138,364,197,303),(139,375,198,314),(140,386,199,325),(141,367,200,306),(142,378,201,317),(143,389,202,328),(144,370,203,309),(145,381,204,320),(146,362,205,301),(147,373,206,312),(148,384,207,323),(149,365,208,304),(150,376,209,315)]])
60 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 | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| 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 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | + | + | - | + | - | - | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D5 | Dic3 | D6 | SD16 | Q16 | D10 | C3⋊D4 | C3⋊D4 | C4×D5 | D20 | C5⋊D4 | Q8⋊2S3 | C3⋊Q16 | S3×D5 | D4.D5 | C5⋊Q16 | D5×Dic3 | C3⋊D20 | C15⋊D4 | C20.D6 | C15⋊Q16 |
| kernel | C30.Q16 | C5×C4⋊Dic3 | C2×C15⋊3C8 | C6×Dic10 | C3×Dic10 | C2×Dic10 | C60 | C2×C30 | C4⋊Dic3 | Dic10 | C2×C20 | C30 | C30 | C2×C12 | C20 | C2×C10 | C12 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C30.Q16 ►in GL6(𝔽241)
| 0 | 240 | 0 | 0 | 0 | 0 |
| 1 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 208 | 160 | 0 | 0 | 0 | 0 |
| 189 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 172 | 146 | 0 | 0 |
| 0 | 0 | 215 | 69 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 184 |
| 0 | 0 | 0 | 0 | 93 | 38 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 172 | 146 | 0 | 0 |
| 0 | 0 | 215 | 69 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 107 |
| 0 | 0 | 0 | 0 | 68 | 227 |
G:=sub<GL(6,GF(241))| [0,1,0,0,0,0,240,52,0,0,0,0,0,0,240,1,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[208,189,0,0,0,0,160,33,0,0,0,0,0,0,172,215,0,0,0,0,146,69,0,0,0,0,0,0,0,93,0,0,0,0,184,38],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,172,215,0,0,0,0,146,69,0,0,0,0,0,0,14,68,0,0,0,0,107,227] >;
C30.Q16 in GAP, Magma, Sage, TeX
C_{30}.Q_{16} % in TeX
G:=Group("C30.Q16"); // GroupNames label
G:=SmallGroup(480,46);
// by ID
G=gap.SmallGroup(480,46);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,36,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^30=b^8=1,c^2=a^15*b^4,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b^-1>;
// generators/relations