direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C5⋊Q16, C30⋊8Q16, C60.130D4, C60.205C23, C5⋊3(C6×Q16), C10⋊2(C3×Q16), C15⋊17(C2×Q16), (C6×Q8).8D5, C20.20(C3×D4), C10.55(C6×D4), (Q8×C30).8C2, Q8.12(C6×D5), (Q8×C10).7C6, (C2×C30).168D4, C30.412(C2×D4), (C3×Q8).39D10, (C2×C12).366D10, C12.77(C5⋊D4), C20.16(C22×C6), (C2×Dic10).9C6, (C2×C60).298C22, (C6×Dic10).20C2, Dic10.10(C2×C6), C12.205(C22×D5), (Q8×C15).44C22, (C3×Dic10).52C22, C4.16(D5×C2×C6), C4.9(C3×C5⋊D4), (C2×C5⋊2C8).6C6, C5⋊2C8.9(C2×C6), (C2×C4).53(C6×D5), C2.19(C6×C5⋊D4), (C2×Q8).5(C3×D5), (C6×C5⋊2C8).18C2, (C2×C20).35(C2×C6), (C2×C10).43(C3×D4), C6.140(C2×C5⋊D4), (C5×Q8).15(C2×C6), (C2×C6).96(C5⋊D4), C22.24(C3×C5⋊D4), (C3×C5⋊2C8).49C22, SmallGroup(480,736)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C5⋊Q16
G = < a,b,c,d | a6=b5=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 304 in 120 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, C10, C10, C12, C12, C2×C6, C15, C2×C8, Q16, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, C30, C30, C2×Q16, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C2×C24, C3×Q16, C6×Q8, C6×Q8, C3×Dic5, C60, C60, C2×C30, C2×C5⋊2C8, C5⋊Q16, C2×Dic10, Q8×C10, C6×Q16, C3×C5⋊2C8, C3×Dic10, C3×Dic10, C6×Dic5, C2×C60, C2×C60, Q8×C15, Q8×C15, C2×C5⋊Q16, C6×C5⋊2C8, C3×C5⋊Q16, C6×Dic10, Q8×C30, C6×C5⋊Q16
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, Q16, C2×D4, D10, C3×D4, C22×C6, C3×D5, C2×Q16, C5⋊D4, C22×D5, C3×Q16, C6×D4, C6×D5, C5⋊Q16, C2×C5⋊D4, C6×Q16, C3×C5⋊D4, D5×C2×C6, C2×C5⋊Q16, C3×C5⋊Q16, C6×C5⋊D4, C6×C5⋊Q16
►Regular action on 480 points
Generators in S480
(1 304 87 323 128 47)(2 297 88 324 121 48)(3 298 81 325 122 41)(4 299 82 326 123 42)(5 300 83 327 124 43)(6 301 84 328 125 44)(7 302 85 321 126 45)(8 303 86 322 127 46)(9 234 28 431 471 73)(10 235 29 432 472 74)(11 236 30 425 465 75)(12 237 31 426 466 76)(13 238 32 427 467 77)(14 239 25 428 468 78)(15 240 26 429 469 79)(16 233 27 430 470 80)(17 419 338 248 477 258)(18 420 339 241 478 259)(19 421 340 242 479 260)(20 422 341 243 480 261)(21 423 342 244 473 262)(22 424 343 245 474 263)(23 417 344 246 475 264)(24 418 337 247 476 257)(33 133 173 68 314 229)(34 134 174 69 315 230)(35 135 175 70 316 231)(36 136 176 71 317 232)(37 129 169 72 318 225)(38 130 170 65 319 226)(39 131 171 66 320 227)(40 132 172 67 313 228)(49 214 185 89 155 105)(50 215 186 90 156 106)(51 216 187 91 157 107)(52 209 188 92 158 108)(53 210 189 93 159 109)(54 211 190 94 160 110)(55 212 191 95 153 111)(56 213 192 96 154 112)(57 394 373 277 450 293)(58 395 374 278 451 294)(59 396 375 279 452 295)(60 397 376 280 453 296)(61 398 369 273 454 289)(62 399 370 274 455 290)(63 400 371 275 456 291)(64 393 372 276 449 292)(97 463 383 356 222 142)(98 464 384 357 223 143)(99 457 377 358 224 144)(100 458 378 359 217 137)(101 459 379 360 218 138)(102 460 380 353 219 139)(103 461 381 354 220 140)(104 462 382 355 221 141)(113 350 392 193 267 145)(114 351 385 194 268 146)(115 352 386 195 269 147)(116 345 387 196 270 148)(117 346 388 197 271 149)(118 347 389 198 272 150)(119 348 390 199 265 151)(120 349 391 200 266 152)(161 201 365 402 445 285)(162 202 366 403 446 286)(163 203 367 404 447 287)(164 204 368 405 448 288)(165 205 361 406 441 281)(166 206 362 407 442 282)(167 207 363 408 443 283)(168 208 364 401 444 284)(177 252 308 413 332 437)(178 253 309 414 333 438)(179 254 310 415 334 439)(180 255 311 416 335 440)(181 256 312 409 336 433)(182 249 305 410 329 434)(183 250 306 411 330 435)(184 251 307 412 331 436)
(1 230 248 237 411)(2 412 238 241 231)(3 232 242 239 413)(4 414 240 243 225)(5 226 244 233 415)(6 416 234 245 227)(7 228 246 235 409)(8 410 236 247 229)(9 343 320 44 311)(10 312 45 313 344)(11 337 314 46 305)(12 306 47 315 338)(13 339 316 48 307)(14 308 41 317 340)(15 341 318 42 309)(16 310 43 319 342)(17 466 183 323 174)(18 175 324 184 467)(19 468 177 325 176)(20 169 326 178 469)(21 470 179 327 170)(22 171 328 180 471)(23 472 181 321 172)(24 173 322 182 465)(25 332 298 36 479)(26 480 37 299 333)(27 334 300 38 473)(28 474 39 301 335)(29 336 302 40 475)(30 476 33 303 329)(31 330 304 34 477)(32 478 35 297 331)(49 142 345 58 444)(50 445 59 346 143)(51 144 347 60 446)(52 447 61 348 137)(53 138 349 62 448)(54 441 63 350 139)(55 140 351 64 442)(56 443 57 352 141)(65 423 80 254 124)(66 125 255 73 424)(67 417 74 256 126)(68 127 249 75 418)(69 419 76 250 128)(70 121 251 77 420)(71 421 78 252 122)(72 123 253 79 422)(81 136 260 428 437)(82 438 429 261 129)(83 130 262 430 439)(84 440 431 263 131)(85 132 264 432 433)(86 434 425 257 133)(87 134 258 426 435)(88 436 427 259 135)(89 383 270 278 208)(90 201 279 271 384)(91 377 272 280 202)(92 203 273 265 378)(93 379 266 274 204)(94 205 275 267 380)(95 381 268 276 206)(96 207 277 269 382)(97 387 395 284 214)(98 215 285 396 388)(99 389 397 286 216)(100 209 287 398 390)(101 391 399 288 210)(102 211 281 400 392)(103 385 393 282 212)(104 213 283 394 386)(105 222 116 294 401)(106 402 295 117 223)(107 224 118 296 403)(108 404 289 119 217)(109 218 120 290 405)(110 406 291 113 219)(111 220 114 292 407)(112 408 293 115 221)(145 353 160 361 456)(146 449 362 153 354)(147 355 154 363 450)(148 451 364 155 356)(149 357 156 365 452)(150 453 366 157 358)(151 359 158 367 454)(152 455 368 159 360)(161 375 197 464 186)(162 187 457 198 376)(163 369 199 458 188)(164 189 459 200 370)(165 371 193 460 190)(166 191 461 194 372)(167 373 195 462 192)(168 185 463 196 374)
(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)
(1 448 5 444)(2 447 6 443)(3 446 7 442)(4 445 8 441)(9 115 13 119)(10 114 14 118)(11 113 15 117)(12 120 16 116)(17 379 21 383)(18 378 22 382)(19 377 23 381)(20 384 24 380)(25 389 29 385)(26 388 30 392)(27 387 31 391)(28 386 32 390)(33 211 37 215)(34 210 38 214)(35 209 39 213)(36 216 40 212)(41 403 45 407)(42 402 46 406)(43 401 47 405)(44 408 48 404)(49 230 53 226)(50 229 54 225)(51 228 55 232)(52 227 56 231)(57 412 61 416)(58 411 62 415)(59 410 63 414)(60 409 64 413)(65 155 69 159)(66 154 70 158)(67 153 71 157)(68 160 72 156)(73 147 77 151)(74 146 78 150)(75 145 79 149)(76 152 80 148)(81 162 85 166)(82 161 86 165)(83 168 87 164)(84 167 88 163)(89 174 93 170)(90 173 94 169)(91 172 95 176)(92 171 96 175)(97 477 101 473)(98 476 102 480)(99 475 103 479)(100 474 104 478)(105 315 109 319)(106 314 110 318)(107 313 111 317)(108 320 112 316)(121 367 125 363)(122 366 126 362)(123 365 127 361)(124 364 128 368)(129 186 133 190)(130 185 134 189)(131 192 135 188)(132 191 136 187)(137 245 141 241)(138 244 142 248)(139 243 143 247)(140 242 144 246)(177 280 181 276)(178 279 182 275)(179 278 183 274)(180 277 184 273)(193 429 197 425)(194 428 198 432)(195 427 199 431)(196 426 200 430)(201 322 205 326)(202 321 206 325)(203 328 207 324)(204 327 208 323)(217 343 221 339)(218 342 222 338)(219 341 223 337)(220 340 224 344)(233 345 237 349)(234 352 238 348)(235 351 239 347)(236 350 240 346)(249 456 253 452)(250 455 254 451)(251 454 255 450)(252 453 256 449)(257 460 261 464)(258 459 262 463)(259 458 263 462)(260 457 264 461)(265 471 269 467)(266 470 270 466)(267 469 271 465)(268 468 272 472)(281 299 285 303)(282 298 286 302)(283 297 287 301)(284 304 288 300)(289 311 293 307)(290 310 294 306)(291 309 295 305)(292 308 296 312)(329 400 333 396)(330 399 334 395)(331 398 335 394)(332 397 336 393)(353 422 357 418)(354 421 358 417)(355 420 359 424)(356 419 360 423)(369 440 373 436)(370 439 374 435)(371 438 375 434)(372 437 376 433)
G:=sub<Sym(480)| (1,304,87,323,128,47)(2,297,88,324,121,48)(3,298,81,325,122,41)(4,299,82,326,123,42)(5,300,83,327,124,43)(6,301,84,328,125,44)(7,302,85,321,126,45)(8,303,86,322,127,46)(9,234,28,431,471,73)(10,235,29,432,472,74)(11,236,30,425,465,75)(12,237,31,426,466,76)(13,238,32,427,467,77)(14,239,25,428,468,78)(15,240,26,429,469,79)(16,233,27,430,470,80)(17,419,338,248,477,258)(18,420,339,241,478,259)(19,421,340,242,479,260)(20,422,341,243,480,261)(21,423,342,244,473,262)(22,424,343,245,474,263)(23,417,344,246,475,264)(24,418,337,247,476,257)(33,133,173,68,314,229)(34,134,174,69,315,230)(35,135,175,70,316,231)(36,136,176,71,317,232)(37,129,169,72,318,225)(38,130,170,65,319,226)(39,131,171,66,320,227)(40,132,172,67,313,228)(49,214,185,89,155,105)(50,215,186,90,156,106)(51,216,187,91,157,107)(52,209,188,92,158,108)(53,210,189,93,159,109)(54,211,190,94,160,110)(55,212,191,95,153,111)(56,213,192,96,154,112)(57,394,373,277,450,293)(58,395,374,278,451,294)(59,396,375,279,452,295)(60,397,376,280,453,296)(61,398,369,273,454,289)(62,399,370,274,455,290)(63,400,371,275,456,291)(64,393,372,276,449,292)(97,463,383,356,222,142)(98,464,384,357,223,143)(99,457,377,358,224,144)(100,458,378,359,217,137)(101,459,379,360,218,138)(102,460,380,353,219,139)(103,461,381,354,220,140)(104,462,382,355,221,141)(113,350,392,193,267,145)(114,351,385,194,268,146)(115,352,386,195,269,147)(116,345,387,196,270,148)(117,346,388,197,271,149)(118,347,389,198,272,150)(119,348,390,199,265,151)(120,349,391,200,266,152)(161,201,365,402,445,285)(162,202,366,403,446,286)(163,203,367,404,447,287)(164,204,368,405,448,288)(165,205,361,406,441,281)(166,206,362,407,442,282)(167,207,363,408,443,283)(168,208,364,401,444,284)(177,252,308,413,332,437)(178,253,309,414,333,438)(179,254,310,415,334,439)(180,255,311,416,335,440)(181,256,312,409,336,433)(182,249,305,410,329,434)(183,250,306,411,330,435)(184,251,307,412,331,436), (1,230,248,237,411)(2,412,238,241,231)(3,232,242,239,413)(4,414,240,243,225)(5,226,244,233,415)(6,416,234,245,227)(7,228,246,235,409)(8,410,236,247,229)(9,343,320,44,311)(10,312,45,313,344)(11,337,314,46,305)(12,306,47,315,338)(13,339,316,48,307)(14,308,41,317,340)(15,341,318,42,309)(16,310,43,319,342)(17,466,183,323,174)(18,175,324,184,467)(19,468,177,325,176)(20,169,326,178,469)(21,470,179,327,170)(22,171,328,180,471)(23,472,181,321,172)(24,173,322,182,465)(25,332,298,36,479)(26,480,37,299,333)(27,334,300,38,473)(28,474,39,301,335)(29,336,302,40,475)(30,476,33,303,329)(31,330,304,34,477)(32,478,35,297,331)(49,142,345,58,444)(50,445,59,346,143)(51,144,347,60,446)(52,447,61,348,137)(53,138,349,62,448)(54,441,63,350,139)(55,140,351,64,442)(56,443,57,352,141)(65,423,80,254,124)(66,125,255,73,424)(67,417,74,256,126)(68,127,249,75,418)(69,419,76,250,128)(70,121,251,77,420)(71,421,78,252,122)(72,123,253,79,422)(81,136,260,428,437)(82,438,429,261,129)(83,130,262,430,439)(84,440,431,263,131)(85,132,264,432,433)(86,434,425,257,133)(87,134,258,426,435)(88,436,427,259,135)(89,383,270,278,208)(90,201,279,271,384)(91,377,272,280,202)(92,203,273,265,378)(93,379,266,274,204)(94,205,275,267,380)(95,381,268,276,206)(96,207,277,269,382)(97,387,395,284,214)(98,215,285,396,388)(99,389,397,286,216)(100,209,287,398,390)(101,391,399,288,210)(102,211,281,400,392)(103,385,393,282,212)(104,213,283,394,386)(105,222,116,294,401)(106,402,295,117,223)(107,224,118,296,403)(108,404,289,119,217)(109,218,120,290,405)(110,406,291,113,219)(111,220,114,292,407)(112,408,293,115,221)(145,353,160,361,456)(146,449,362,153,354)(147,355,154,363,450)(148,451,364,155,356)(149,357,156,365,452)(150,453,366,157,358)(151,359,158,367,454)(152,455,368,159,360)(161,375,197,464,186)(162,187,457,198,376)(163,369,199,458,188)(164,189,459,200,370)(165,371,193,460,190)(166,191,461,194,372)(167,373,195,462,192)(168,185,463,196,374), (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), (1,448,5,444)(2,447,6,443)(3,446,7,442)(4,445,8,441)(9,115,13,119)(10,114,14,118)(11,113,15,117)(12,120,16,116)(17,379,21,383)(18,378,22,382)(19,377,23,381)(20,384,24,380)(25,389,29,385)(26,388,30,392)(27,387,31,391)(28,386,32,390)(33,211,37,215)(34,210,38,214)(35,209,39,213)(36,216,40,212)(41,403,45,407)(42,402,46,406)(43,401,47,405)(44,408,48,404)(49,230,53,226)(50,229,54,225)(51,228,55,232)(52,227,56,231)(57,412,61,416)(58,411,62,415)(59,410,63,414)(60,409,64,413)(65,155,69,159)(66,154,70,158)(67,153,71,157)(68,160,72,156)(73,147,77,151)(74,146,78,150)(75,145,79,149)(76,152,80,148)(81,162,85,166)(82,161,86,165)(83,168,87,164)(84,167,88,163)(89,174,93,170)(90,173,94,169)(91,172,95,176)(92,171,96,175)(97,477,101,473)(98,476,102,480)(99,475,103,479)(100,474,104,478)(105,315,109,319)(106,314,110,318)(107,313,111,317)(108,320,112,316)(121,367,125,363)(122,366,126,362)(123,365,127,361)(124,364,128,368)(129,186,133,190)(130,185,134,189)(131,192,135,188)(132,191,136,187)(137,245,141,241)(138,244,142,248)(139,243,143,247)(140,242,144,246)(177,280,181,276)(178,279,182,275)(179,278,183,274)(180,277,184,273)(193,429,197,425)(194,428,198,432)(195,427,199,431)(196,426,200,430)(201,322,205,326)(202,321,206,325)(203,328,207,324)(204,327,208,323)(217,343,221,339)(218,342,222,338)(219,341,223,337)(220,340,224,344)(233,345,237,349)(234,352,238,348)(235,351,239,347)(236,350,240,346)(249,456,253,452)(250,455,254,451)(251,454,255,450)(252,453,256,449)(257,460,261,464)(258,459,262,463)(259,458,263,462)(260,457,264,461)(265,471,269,467)(266,470,270,466)(267,469,271,465)(268,468,272,472)(281,299,285,303)(282,298,286,302)(283,297,287,301)(284,304,288,300)(289,311,293,307)(290,310,294,306)(291,309,295,305)(292,308,296,312)(329,400,333,396)(330,399,334,395)(331,398,335,394)(332,397,336,393)(353,422,357,418)(354,421,358,417)(355,420,359,424)(356,419,360,423)(369,440,373,436)(370,439,374,435)(371,438,375,434)(372,437,376,433)>;
G:=Group( (1,304,87,323,128,47)(2,297,88,324,121,48)(3,298,81,325,122,41)(4,299,82,326,123,42)(5,300,83,327,124,43)(6,301,84,328,125,44)(7,302,85,321,126,45)(8,303,86,322,127,46)(9,234,28,431,471,73)(10,235,29,432,472,74)(11,236,30,425,465,75)(12,237,31,426,466,76)(13,238,32,427,467,77)(14,239,25,428,468,78)(15,240,26,429,469,79)(16,233,27,430,470,80)(17,419,338,248,477,258)(18,420,339,241,478,259)(19,421,340,242,479,260)(20,422,341,243,480,261)(21,423,342,244,473,262)(22,424,343,245,474,263)(23,417,344,246,475,264)(24,418,337,247,476,257)(33,133,173,68,314,229)(34,134,174,69,315,230)(35,135,175,70,316,231)(36,136,176,71,317,232)(37,129,169,72,318,225)(38,130,170,65,319,226)(39,131,171,66,320,227)(40,132,172,67,313,228)(49,214,185,89,155,105)(50,215,186,90,156,106)(51,216,187,91,157,107)(52,209,188,92,158,108)(53,210,189,93,159,109)(54,211,190,94,160,110)(55,212,191,95,153,111)(56,213,192,96,154,112)(57,394,373,277,450,293)(58,395,374,278,451,294)(59,396,375,279,452,295)(60,397,376,280,453,296)(61,398,369,273,454,289)(62,399,370,274,455,290)(63,400,371,275,456,291)(64,393,372,276,449,292)(97,463,383,356,222,142)(98,464,384,357,223,143)(99,457,377,358,224,144)(100,458,378,359,217,137)(101,459,379,360,218,138)(102,460,380,353,219,139)(103,461,381,354,220,140)(104,462,382,355,221,141)(113,350,392,193,267,145)(114,351,385,194,268,146)(115,352,386,195,269,147)(116,345,387,196,270,148)(117,346,388,197,271,149)(118,347,389,198,272,150)(119,348,390,199,265,151)(120,349,391,200,266,152)(161,201,365,402,445,285)(162,202,366,403,446,286)(163,203,367,404,447,287)(164,204,368,405,448,288)(165,205,361,406,441,281)(166,206,362,407,442,282)(167,207,363,408,443,283)(168,208,364,401,444,284)(177,252,308,413,332,437)(178,253,309,414,333,438)(179,254,310,415,334,439)(180,255,311,416,335,440)(181,256,312,409,336,433)(182,249,305,410,329,434)(183,250,306,411,330,435)(184,251,307,412,331,436), (1,230,248,237,411)(2,412,238,241,231)(3,232,242,239,413)(4,414,240,243,225)(5,226,244,233,415)(6,416,234,245,227)(7,228,246,235,409)(8,410,236,247,229)(9,343,320,44,311)(10,312,45,313,344)(11,337,314,46,305)(12,306,47,315,338)(13,339,316,48,307)(14,308,41,317,340)(15,341,318,42,309)(16,310,43,319,342)(17,466,183,323,174)(18,175,324,184,467)(19,468,177,325,176)(20,169,326,178,469)(21,470,179,327,170)(22,171,328,180,471)(23,472,181,321,172)(24,173,322,182,465)(25,332,298,36,479)(26,480,37,299,333)(27,334,300,38,473)(28,474,39,301,335)(29,336,302,40,475)(30,476,33,303,329)(31,330,304,34,477)(32,478,35,297,331)(49,142,345,58,444)(50,445,59,346,143)(51,144,347,60,446)(52,447,61,348,137)(53,138,349,62,448)(54,441,63,350,139)(55,140,351,64,442)(56,443,57,352,141)(65,423,80,254,124)(66,125,255,73,424)(67,417,74,256,126)(68,127,249,75,418)(69,419,76,250,128)(70,121,251,77,420)(71,421,78,252,122)(72,123,253,79,422)(81,136,260,428,437)(82,438,429,261,129)(83,130,262,430,439)(84,440,431,263,131)(85,132,264,432,433)(86,434,425,257,133)(87,134,258,426,435)(88,436,427,259,135)(89,383,270,278,208)(90,201,279,271,384)(91,377,272,280,202)(92,203,273,265,378)(93,379,266,274,204)(94,205,275,267,380)(95,381,268,276,206)(96,207,277,269,382)(97,387,395,284,214)(98,215,285,396,388)(99,389,397,286,216)(100,209,287,398,390)(101,391,399,288,210)(102,211,281,400,392)(103,385,393,282,212)(104,213,283,394,386)(105,222,116,294,401)(106,402,295,117,223)(107,224,118,296,403)(108,404,289,119,217)(109,218,120,290,405)(110,406,291,113,219)(111,220,114,292,407)(112,408,293,115,221)(145,353,160,361,456)(146,449,362,153,354)(147,355,154,363,450)(148,451,364,155,356)(149,357,156,365,452)(150,453,366,157,358)(151,359,158,367,454)(152,455,368,159,360)(161,375,197,464,186)(162,187,457,198,376)(163,369,199,458,188)(164,189,459,200,370)(165,371,193,460,190)(166,191,461,194,372)(167,373,195,462,192)(168,185,463,196,374), (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), (1,448,5,444)(2,447,6,443)(3,446,7,442)(4,445,8,441)(9,115,13,119)(10,114,14,118)(11,113,15,117)(12,120,16,116)(17,379,21,383)(18,378,22,382)(19,377,23,381)(20,384,24,380)(25,389,29,385)(26,388,30,392)(27,387,31,391)(28,386,32,390)(33,211,37,215)(34,210,38,214)(35,209,39,213)(36,216,40,212)(41,403,45,407)(42,402,46,406)(43,401,47,405)(44,408,48,404)(49,230,53,226)(50,229,54,225)(51,228,55,232)(52,227,56,231)(57,412,61,416)(58,411,62,415)(59,410,63,414)(60,409,64,413)(65,155,69,159)(66,154,70,158)(67,153,71,157)(68,160,72,156)(73,147,77,151)(74,146,78,150)(75,145,79,149)(76,152,80,148)(81,162,85,166)(82,161,86,165)(83,168,87,164)(84,167,88,163)(89,174,93,170)(90,173,94,169)(91,172,95,176)(92,171,96,175)(97,477,101,473)(98,476,102,480)(99,475,103,479)(100,474,104,478)(105,315,109,319)(106,314,110,318)(107,313,111,317)(108,320,112,316)(121,367,125,363)(122,366,126,362)(123,365,127,361)(124,364,128,368)(129,186,133,190)(130,185,134,189)(131,192,135,188)(132,191,136,187)(137,245,141,241)(138,244,142,248)(139,243,143,247)(140,242,144,246)(177,280,181,276)(178,279,182,275)(179,278,183,274)(180,277,184,273)(193,429,197,425)(194,428,198,432)(195,427,199,431)(196,426,200,430)(201,322,205,326)(202,321,206,325)(203,328,207,324)(204,327,208,323)(217,343,221,339)(218,342,222,338)(219,341,223,337)(220,340,224,344)(233,345,237,349)(234,352,238,348)(235,351,239,347)(236,350,240,346)(249,456,253,452)(250,455,254,451)(251,454,255,450)(252,453,256,449)(257,460,261,464)(258,459,262,463)(259,458,263,462)(260,457,264,461)(265,471,269,467)(266,470,270,466)(267,469,271,465)(268,468,272,472)(281,299,285,303)(282,298,286,302)(283,297,287,301)(284,304,288,300)(289,311,293,307)(290,310,294,306)(291,309,295,305)(292,308,296,312)(329,400,333,396)(330,399,334,395)(331,398,335,394)(332,397,336,393)(353,422,357,418)(354,421,358,417)(355,420,359,424)(356,419,360,423)(369,440,373,436)(370,439,374,435)(371,438,375,434)(372,437,376,433) );
G=PermutationGroup([[(1,304,87,323,128,47),(2,297,88,324,121,48),(3,298,81,325,122,41),(4,299,82,326,123,42),(5,300,83,327,124,43),(6,301,84,328,125,44),(7,302,85,321,126,45),(8,303,86,322,127,46),(9,234,28,431,471,73),(10,235,29,432,472,74),(11,236,30,425,465,75),(12,237,31,426,466,76),(13,238,32,427,467,77),(14,239,25,428,468,78),(15,240,26,429,469,79),(16,233,27,430,470,80),(17,419,338,248,477,258),(18,420,339,241,478,259),(19,421,340,242,479,260),(20,422,341,243,480,261),(21,423,342,244,473,262),(22,424,343,245,474,263),(23,417,344,246,475,264),(24,418,337,247,476,257),(33,133,173,68,314,229),(34,134,174,69,315,230),(35,135,175,70,316,231),(36,136,176,71,317,232),(37,129,169,72,318,225),(38,130,170,65,319,226),(39,131,171,66,320,227),(40,132,172,67,313,228),(49,214,185,89,155,105),(50,215,186,90,156,106),(51,216,187,91,157,107),(52,209,188,92,158,108),(53,210,189,93,159,109),(54,211,190,94,160,110),(55,212,191,95,153,111),(56,213,192,96,154,112),(57,394,373,277,450,293),(58,395,374,278,451,294),(59,396,375,279,452,295),(60,397,376,280,453,296),(61,398,369,273,454,289),(62,399,370,274,455,290),(63,400,371,275,456,291),(64,393,372,276,449,292),(97,463,383,356,222,142),(98,464,384,357,223,143),(99,457,377,358,224,144),(100,458,378,359,217,137),(101,459,379,360,218,138),(102,460,380,353,219,139),(103,461,381,354,220,140),(104,462,382,355,221,141),(113,350,392,193,267,145),(114,351,385,194,268,146),(115,352,386,195,269,147),(116,345,387,196,270,148),(117,346,388,197,271,149),(118,347,389,198,272,150),(119,348,390,199,265,151),(120,349,391,200,266,152),(161,201,365,402,445,285),(162,202,366,403,446,286),(163,203,367,404,447,287),(164,204,368,405,448,288),(165,205,361,406,441,281),(166,206,362,407,442,282),(167,207,363,408,443,283),(168,208,364,401,444,284),(177,252,308,413,332,437),(178,253,309,414,333,438),(179,254,310,415,334,439),(180,255,311,416,335,440),(181,256,312,409,336,433),(182,249,305,410,329,434),(183,250,306,411,330,435),(184,251,307,412,331,436)], [(1,230,248,237,411),(2,412,238,241,231),(3,232,242,239,413),(4,414,240,243,225),(5,226,244,233,415),(6,416,234,245,227),(7,228,246,235,409),(8,410,236,247,229),(9,343,320,44,311),(10,312,45,313,344),(11,337,314,46,305),(12,306,47,315,338),(13,339,316,48,307),(14,308,41,317,340),(15,341,318,42,309),(16,310,43,319,342),(17,466,183,323,174),(18,175,324,184,467),(19,468,177,325,176),(20,169,326,178,469),(21,470,179,327,170),(22,171,328,180,471),(23,472,181,321,172),(24,173,322,182,465),(25,332,298,36,479),(26,480,37,299,333),(27,334,300,38,473),(28,474,39,301,335),(29,336,302,40,475),(30,476,33,303,329),(31,330,304,34,477),(32,478,35,297,331),(49,142,345,58,444),(50,445,59,346,143),(51,144,347,60,446),(52,447,61,348,137),(53,138,349,62,448),(54,441,63,350,139),(55,140,351,64,442),(56,443,57,352,141),(65,423,80,254,124),(66,125,255,73,424),(67,417,74,256,126),(68,127,249,75,418),(69,419,76,250,128),(70,121,251,77,420),(71,421,78,252,122),(72,123,253,79,422),(81,136,260,428,437),(82,438,429,261,129),(83,130,262,430,439),(84,440,431,263,131),(85,132,264,432,433),(86,434,425,257,133),(87,134,258,426,435),(88,436,427,259,135),(89,383,270,278,208),(90,201,279,271,384),(91,377,272,280,202),(92,203,273,265,378),(93,379,266,274,204),(94,205,275,267,380),(95,381,268,276,206),(96,207,277,269,382),(97,387,395,284,214),(98,215,285,396,388),(99,389,397,286,216),(100,209,287,398,390),(101,391,399,288,210),(102,211,281,400,392),(103,385,393,282,212),(104,213,283,394,386),(105,222,116,294,401),(106,402,295,117,223),(107,224,118,296,403),(108,404,289,119,217),(109,218,120,290,405),(110,406,291,113,219),(111,220,114,292,407),(112,408,293,115,221),(145,353,160,361,456),(146,449,362,153,354),(147,355,154,363,450),(148,451,364,155,356),(149,357,156,365,452),(150,453,366,157,358),(151,359,158,367,454),(152,455,368,159,360),(161,375,197,464,186),(162,187,457,198,376),(163,369,199,458,188),(164,189,459,200,370),(165,371,193,460,190),(166,191,461,194,372),(167,373,195,462,192),(168,185,463,196,374)], [(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)], [(1,448,5,444),(2,447,6,443),(3,446,7,442),(4,445,8,441),(9,115,13,119),(10,114,14,118),(11,113,15,117),(12,120,16,116),(17,379,21,383),(18,378,22,382),(19,377,23,381),(20,384,24,380),(25,389,29,385),(26,388,30,392),(27,387,31,391),(28,386,32,390),(33,211,37,215),(34,210,38,214),(35,209,39,213),(36,216,40,212),(41,403,45,407),(42,402,46,406),(43,401,47,405),(44,408,48,404),(49,230,53,226),(50,229,54,225),(51,228,55,232),(52,227,56,231),(57,412,61,416),(58,411,62,415),(59,410,63,414),(60,409,64,413),(65,155,69,159),(66,154,70,158),(67,153,71,157),(68,160,72,156),(73,147,77,151),(74,146,78,150),(75,145,79,149),(76,152,80,148),(81,162,85,166),(82,161,86,165),(83,168,87,164),(84,167,88,163),(89,174,93,170),(90,173,94,169),(91,172,95,176),(92,171,96,175),(97,477,101,473),(98,476,102,480),(99,475,103,479),(100,474,104,478),(105,315,109,319),(106,314,110,318),(107,313,111,317),(108,320,112,316),(121,367,125,363),(122,366,126,362),(123,365,127,361),(124,364,128,368),(129,186,133,190),(130,185,134,189),(131,192,135,188),(132,191,136,187),(137,245,141,241),(138,244,142,248),(139,243,143,247),(140,242,144,246),(177,280,181,276),(178,279,182,275),(179,278,183,274),(180,277,184,273),(193,429,197,425),(194,428,198,432),(195,427,199,431),(196,426,200,430),(201,322,205,326),(202,321,206,325),(203,328,207,324),(204,327,208,323),(217,343,221,339),(218,342,222,338),(219,341,223,337),(220,340,224,344),(233,345,237,349),(234,352,238,348),(235,351,239,347),(236,350,240,346),(249,456,253,452),(250,455,254,451),(251,454,255,450),(252,453,256,449),(257,460,261,464),(258,459,262,463),(259,458,263,462),(260,457,264,461),(265,471,269,467),(266,470,270,466),(267,469,271,465),(268,468,272,472),(281,299,285,303),(282,298,286,302),(283,297,287,301),(284,304,288,300),(289,311,293,307),(290,310,294,306),(291,309,295,305),(292,308,296,312),(329,400,333,396),(330,399,334,395),(331,398,335,394),(332,397,336,393),(353,422,357,418),(354,421,358,417),(355,420,359,424),(356,419,360,423),(369,440,373,436),(370,439,374,435),(371,438,375,434),(372,437,376,433)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 24A | ··· | 24H | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | Q16 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C5⋊D4 | C5⋊D4 | C3×Q16 | C6×D5 | C6×D5 | C3×C5⋊D4 | C3×C5⋊D4 | C5⋊Q16 | C3×C5⋊Q16 |
| kernel | C6×C5⋊Q16 | C6×C5⋊2C8 | C3×C5⋊Q16 | C6×Dic10 | Q8×C30 | C2×C5⋊Q16 | C2×C5⋊2C8 | C5⋊Q16 | C2×Dic10 | Q8×C10 | C60 | C2×C30 | C6×Q8 | C30 | C2×C12 | C3×Q8 | C20 | C2×C10 | C2×Q8 | C12 | C2×C6 | C10 | C2×C4 | Q8 | C4 | C22 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 8 | 4 | 8 |
Matrix representation of C6×C5⋊Q16 ►in GL4(𝔽241) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 225 |
| 0 | 240 | 0 | 0 |
| 1 | 51 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 73 | 16 | 0 | 0 |
| 149 | 168 | 0 | 0 |
| 0 | 0 | 22 | 129 |
| 0 | 0 | 170 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 159 |
| 0 | 0 | 0 | 177 |
G:=sub<GL(4,GF(241))| [16,0,0,0,0,16,0,0,0,0,225,0,0,0,0,225],[0,1,0,0,240,51,0,0,0,0,1,0,0,0,0,1],[73,149,0,0,16,168,0,0,0,0,22,170,0,0,129,0],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,159,177] >;
C6×C5⋊Q16 in GAP, Magma, Sage, TeX
C_6\times C_5\rtimes Q_{16} % in TeX
G:=Group("C6xC5:Q16"); // GroupNames label
G:=SmallGroup(480,736);
// by ID
G=gap.SmallGroup(480,736);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,590,268,2524,648,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^5=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations