direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×C5⋊C16, C10⋊C48, C30⋊2C16, C60.6C8, C20.3C24, C5⋊2(C2×C48), C15⋊9(C2×C16), C12.6(C5⋊C8), (C2×C30).3C8, C4.17(C6×F5), C5⋊2C8.5C12, C60.70(C2×C4), C30.39(C2×C8), (C2×C20).9C12, (C2×C10).1C24, (C2×C60).20C4, C10.5(C2×C24), C12.70(C2×F5), (C2×C12).20F5, C20.17(C2×C12), C4.3(C3×C5⋊C8), C2.1(C6×C5⋊C8), C6.11(C2×C5⋊C8), (C2×C6).4(C5⋊C8), C22.2(C3×C5⋊C8), (C2×C4).9(C3×F5), (C6×C5⋊2C8).26C2, (C3×C5⋊2C8).12C4, (C2×C5⋊2C8).12C6, C5⋊2C8.15(C2×C6), (C3×C5⋊2C8).59C22, SmallGroup(480,277)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C6×C5⋊C16 |
Generators and relations for C6×C5⋊C16
G = < a,b,c | a6=b5=c16=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C6×C5⋊C16
►Regular action on 480 points
Generators in S480
(1 111 446 249 420 142)(2 112 447 250 421 143)(3 97 448 251 422 144)(4 98 433 252 423 129)(5 99 434 253 424 130)(6 100 435 254 425 131)(7 101 436 255 426 132)(8 102 437 256 427 133)(9 103 438 241 428 134)(10 104 439 242 429 135)(11 105 440 243 430 136)(12 106 441 244 431 137)(13 107 442 245 432 138)(14 108 443 246 417 139)(15 109 444 247 418 140)(16 110 445 248 419 141)(17 261 306 173 195 214)(18 262 307 174 196 215)(19 263 308 175 197 216)(20 264 309 176 198 217)(21 265 310 161 199 218)(22 266 311 162 200 219)(23 267 312 163 201 220)(24 268 313 164 202 221)(25 269 314 165 203 222)(26 270 315 166 204 223)(27 271 316 167 205 224)(28 272 317 168 206 209)(29 257 318 169 207 210)(30 258 319 170 208 211)(31 259 320 171 193 212)(32 260 305 172 194 213)(33 70 301 453 328 181)(34 71 302 454 329 182)(35 72 303 455 330 183)(36 73 304 456 331 184)(37 74 289 457 332 185)(38 75 290 458 333 186)(39 76 291 459 334 187)(40 77 292 460 335 188)(41 78 293 461 336 189)(42 79 294 462 321 190)(43 80 295 463 322 191)(44 65 296 464 323 192)(45 66 297 449 324 177)(46 67 298 450 325 178)(47 68 299 451 326 179)(48 69 300 452 327 180)(49 408 384 94 470 121)(50 409 369 95 471 122)(51 410 370 96 472 123)(52 411 371 81 473 124)(53 412 372 82 474 125)(54 413 373 83 475 126)(55 414 374 84 476 127)(56 415 375 85 477 128)(57 416 376 86 478 113)(58 401 377 87 479 114)(59 402 378 88 480 115)(60 403 379 89 465 116)(61 404 380 90 466 117)(62 405 381 91 467 118)(63 406 382 92 468 119)(64 407 383 93 469 120)(145 288 239 354 385 344)(146 273 240 355 386 345)(147 274 225 356 387 346)(148 275 226 357 388 347)(149 276 227 358 389 348)(150 277 228 359 390 349)(151 278 229 360 391 350)(152 279 230 361 392 351)(153 280 231 362 393 352)(154 281 232 363 394 337)(155 282 233 364 395 338)(156 283 234 365 396 339)(157 284 235 366 397 340)(158 285 236 367 398 341)(159 286 237 368 399 342)(160 287 238 353 400 343)
(1 461 386 318 87)(2 319 462 88 387)(3 89 320 388 463)(4 389 90 464 305)(5 449 390 306 91)(6 307 450 92 391)(7 93 308 392 451)(8 393 94 452 309)(9 453 394 310 95)(10 311 454 96 395)(11 81 312 396 455)(12 397 82 456 313)(13 457 398 314 83)(14 315 458 84 399)(15 85 316 400 459)(16 385 86 460 317)(17 405 424 66 228)(18 67 406 229 425)(19 230 68 426 407)(20 427 231 408 69)(21 409 428 70 232)(22 71 410 233 429)(23 234 72 430 411)(24 431 235 412 73)(25 413 432 74 236)(26 75 414 237 417)(27 238 76 418 415)(28 419 239 416 77)(29 401 420 78 240)(30 79 402 225 421)(31 226 80 422 403)(32 423 227 404 65)(33 281 218 50 241)(34 51 282 242 219)(35 243 52 220 283)(36 221 244 284 53)(37 285 222 54 245)(38 55 286 246 223)(39 247 56 224 287)(40 209 248 288 57)(41 273 210 58 249)(42 59 274 250 211)(43 251 60 212 275)(44 213 252 276 61)(45 277 214 62 253)(46 63 278 254 215)(47 255 64 216 279)(48 217 256 280 49)(97 465 171 347 322)(98 348 466 323 172)(99 324 349 173 467)(100 174 325 468 350)(101 469 175 351 326)(102 352 470 327 176)(103 328 337 161 471)(104 162 329 472 338)(105 473 163 339 330)(106 340 474 331 164)(107 332 341 165 475)(108 166 333 476 342)(109 477 167 343 334)(110 344 478 335 168)(111 336 345 169 479)(112 170 321 480 346)(113 188 206 445 145)(114 446 189 146 207)(115 147 447 208 190)(116 193 148 191 448)(117 192 194 433 149)(118 434 177 150 195)(119 151 435 196 178)(120 197 152 179 436)(121 180 198 437 153)(122 438 181 154 199)(123 155 439 200 182)(124 201 156 183 440)(125 184 202 441 157)(126 442 185 158 203)(127 159 443 204 186)(128 205 160 187 444)(129 358 380 296 260)(130 297 359 261 381)(131 262 298 382 360)(132 383 263 361 299)(133 362 384 300 264)(134 301 363 265 369)(135 266 302 370 364)(136 371 267 365 303)(137 366 372 304 268)(138 289 367 269 373)(139 270 290 374 368)(140 375 271 353 291)(141 354 376 292 272)(142 293 355 257 377)(143 258 294 378 356)(144 379 259 357 295)
(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)
G:=sub<Sym(480)| (1,111,446,249,420,142)(2,112,447,250,421,143)(3,97,448,251,422,144)(4,98,433,252,423,129)(5,99,434,253,424,130)(6,100,435,254,425,131)(7,101,436,255,426,132)(8,102,437,256,427,133)(9,103,438,241,428,134)(10,104,439,242,429,135)(11,105,440,243,430,136)(12,106,441,244,431,137)(13,107,442,245,432,138)(14,108,443,246,417,139)(15,109,444,247,418,140)(16,110,445,248,419,141)(17,261,306,173,195,214)(18,262,307,174,196,215)(19,263,308,175,197,216)(20,264,309,176,198,217)(21,265,310,161,199,218)(22,266,311,162,200,219)(23,267,312,163,201,220)(24,268,313,164,202,221)(25,269,314,165,203,222)(26,270,315,166,204,223)(27,271,316,167,205,224)(28,272,317,168,206,209)(29,257,318,169,207,210)(30,258,319,170,208,211)(31,259,320,171,193,212)(32,260,305,172,194,213)(33,70,301,453,328,181)(34,71,302,454,329,182)(35,72,303,455,330,183)(36,73,304,456,331,184)(37,74,289,457,332,185)(38,75,290,458,333,186)(39,76,291,459,334,187)(40,77,292,460,335,188)(41,78,293,461,336,189)(42,79,294,462,321,190)(43,80,295,463,322,191)(44,65,296,464,323,192)(45,66,297,449,324,177)(46,67,298,450,325,178)(47,68,299,451,326,179)(48,69,300,452,327,180)(49,408,384,94,470,121)(50,409,369,95,471,122)(51,410,370,96,472,123)(52,411,371,81,473,124)(53,412,372,82,474,125)(54,413,373,83,475,126)(55,414,374,84,476,127)(56,415,375,85,477,128)(57,416,376,86,478,113)(58,401,377,87,479,114)(59,402,378,88,480,115)(60,403,379,89,465,116)(61,404,380,90,466,117)(62,405,381,91,467,118)(63,406,382,92,468,119)(64,407,383,93,469,120)(145,288,239,354,385,344)(146,273,240,355,386,345)(147,274,225,356,387,346)(148,275,226,357,388,347)(149,276,227,358,389,348)(150,277,228,359,390,349)(151,278,229,360,391,350)(152,279,230,361,392,351)(153,280,231,362,393,352)(154,281,232,363,394,337)(155,282,233,364,395,338)(156,283,234,365,396,339)(157,284,235,366,397,340)(158,285,236,367,398,341)(159,286,237,368,399,342)(160,287,238,353,400,343), (1,461,386,318,87)(2,319,462,88,387)(3,89,320,388,463)(4,389,90,464,305)(5,449,390,306,91)(6,307,450,92,391)(7,93,308,392,451)(8,393,94,452,309)(9,453,394,310,95)(10,311,454,96,395)(11,81,312,396,455)(12,397,82,456,313)(13,457,398,314,83)(14,315,458,84,399)(15,85,316,400,459)(16,385,86,460,317)(17,405,424,66,228)(18,67,406,229,425)(19,230,68,426,407)(20,427,231,408,69)(21,409,428,70,232)(22,71,410,233,429)(23,234,72,430,411)(24,431,235,412,73)(25,413,432,74,236)(26,75,414,237,417)(27,238,76,418,415)(28,419,239,416,77)(29,401,420,78,240)(30,79,402,225,421)(31,226,80,422,403)(32,423,227,404,65)(33,281,218,50,241)(34,51,282,242,219)(35,243,52,220,283)(36,221,244,284,53)(37,285,222,54,245)(38,55,286,246,223)(39,247,56,224,287)(40,209,248,288,57)(41,273,210,58,249)(42,59,274,250,211)(43,251,60,212,275)(44,213,252,276,61)(45,277,214,62,253)(46,63,278,254,215)(47,255,64,216,279)(48,217,256,280,49)(97,465,171,347,322)(98,348,466,323,172)(99,324,349,173,467)(100,174,325,468,350)(101,469,175,351,326)(102,352,470,327,176)(103,328,337,161,471)(104,162,329,472,338)(105,473,163,339,330)(106,340,474,331,164)(107,332,341,165,475)(108,166,333,476,342)(109,477,167,343,334)(110,344,478,335,168)(111,336,345,169,479)(112,170,321,480,346)(113,188,206,445,145)(114,446,189,146,207)(115,147,447,208,190)(116,193,148,191,448)(117,192,194,433,149)(118,434,177,150,195)(119,151,435,196,178)(120,197,152,179,436)(121,180,198,437,153)(122,438,181,154,199)(123,155,439,200,182)(124,201,156,183,440)(125,184,202,441,157)(126,442,185,158,203)(127,159,443,204,186)(128,205,160,187,444)(129,358,380,296,260)(130,297,359,261,381)(131,262,298,382,360)(132,383,263,361,299)(133,362,384,300,264)(134,301,363,265,369)(135,266,302,370,364)(136,371,267,365,303)(137,366,372,304,268)(138,289,367,269,373)(139,270,290,374,368)(140,375,271,353,291)(141,354,376,292,272)(142,293,355,257,377)(143,258,294,378,356)(144,379,259,357,295), (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)>;
G:=Group( (1,111,446,249,420,142)(2,112,447,250,421,143)(3,97,448,251,422,144)(4,98,433,252,423,129)(5,99,434,253,424,130)(6,100,435,254,425,131)(7,101,436,255,426,132)(8,102,437,256,427,133)(9,103,438,241,428,134)(10,104,439,242,429,135)(11,105,440,243,430,136)(12,106,441,244,431,137)(13,107,442,245,432,138)(14,108,443,246,417,139)(15,109,444,247,418,140)(16,110,445,248,419,141)(17,261,306,173,195,214)(18,262,307,174,196,215)(19,263,308,175,197,216)(20,264,309,176,198,217)(21,265,310,161,199,218)(22,266,311,162,200,219)(23,267,312,163,201,220)(24,268,313,164,202,221)(25,269,314,165,203,222)(26,270,315,166,204,223)(27,271,316,167,205,224)(28,272,317,168,206,209)(29,257,318,169,207,210)(30,258,319,170,208,211)(31,259,320,171,193,212)(32,260,305,172,194,213)(33,70,301,453,328,181)(34,71,302,454,329,182)(35,72,303,455,330,183)(36,73,304,456,331,184)(37,74,289,457,332,185)(38,75,290,458,333,186)(39,76,291,459,334,187)(40,77,292,460,335,188)(41,78,293,461,336,189)(42,79,294,462,321,190)(43,80,295,463,322,191)(44,65,296,464,323,192)(45,66,297,449,324,177)(46,67,298,450,325,178)(47,68,299,451,326,179)(48,69,300,452,327,180)(49,408,384,94,470,121)(50,409,369,95,471,122)(51,410,370,96,472,123)(52,411,371,81,473,124)(53,412,372,82,474,125)(54,413,373,83,475,126)(55,414,374,84,476,127)(56,415,375,85,477,128)(57,416,376,86,478,113)(58,401,377,87,479,114)(59,402,378,88,480,115)(60,403,379,89,465,116)(61,404,380,90,466,117)(62,405,381,91,467,118)(63,406,382,92,468,119)(64,407,383,93,469,120)(145,288,239,354,385,344)(146,273,240,355,386,345)(147,274,225,356,387,346)(148,275,226,357,388,347)(149,276,227,358,389,348)(150,277,228,359,390,349)(151,278,229,360,391,350)(152,279,230,361,392,351)(153,280,231,362,393,352)(154,281,232,363,394,337)(155,282,233,364,395,338)(156,283,234,365,396,339)(157,284,235,366,397,340)(158,285,236,367,398,341)(159,286,237,368,399,342)(160,287,238,353,400,343), (1,461,386,318,87)(2,319,462,88,387)(3,89,320,388,463)(4,389,90,464,305)(5,449,390,306,91)(6,307,450,92,391)(7,93,308,392,451)(8,393,94,452,309)(9,453,394,310,95)(10,311,454,96,395)(11,81,312,396,455)(12,397,82,456,313)(13,457,398,314,83)(14,315,458,84,399)(15,85,316,400,459)(16,385,86,460,317)(17,405,424,66,228)(18,67,406,229,425)(19,230,68,426,407)(20,427,231,408,69)(21,409,428,70,232)(22,71,410,233,429)(23,234,72,430,411)(24,431,235,412,73)(25,413,432,74,236)(26,75,414,237,417)(27,238,76,418,415)(28,419,239,416,77)(29,401,420,78,240)(30,79,402,225,421)(31,226,80,422,403)(32,423,227,404,65)(33,281,218,50,241)(34,51,282,242,219)(35,243,52,220,283)(36,221,244,284,53)(37,285,222,54,245)(38,55,286,246,223)(39,247,56,224,287)(40,209,248,288,57)(41,273,210,58,249)(42,59,274,250,211)(43,251,60,212,275)(44,213,252,276,61)(45,277,214,62,253)(46,63,278,254,215)(47,255,64,216,279)(48,217,256,280,49)(97,465,171,347,322)(98,348,466,323,172)(99,324,349,173,467)(100,174,325,468,350)(101,469,175,351,326)(102,352,470,327,176)(103,328,337,161,471)(104,162,329,472,338)(105,473,163,339,330)(106,340,474,331,164)(107,332,341,165,475)(108,166,333,476,342)(109,477,167,343,334)(110,344,478,335,168)(111,336,345,169,479)(112,170,321,480,346)(113,188,206,445,145)(114,446,189,146,207)(115,147,447,208,190)(116,193,148,191,448)(117,192,194,433,149)(118,434,177,150,195)(119,151,435,196,178)(120,197,152,179,436)(121,180,198,437,153)(122,438,181,154,199)(123,155,439,200,182)(124,201,156,183,440)(125,184,202,441,157)(126,442,185,158,203)(127,159,443,204,186)(128,205,160,187,444)(129,358,380,296,260)(130,297,359,261,381)(131,262,298,382,360)(132,383,263,361,299)(133,362,384,300,264)(134,301,363,265,369)(135,266,302,370,364)(136,371,267,365,303)(137,366,372,304,268)(138,289,367,269,373)(139,270,290,374,368)(140,375,271,353,291)(141,354,376,292,272)(142,293,355,257,377)(143,258,294,378,356)(144,379,259,357,295), (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) );
G=PermutationGroup([[(1,111,446,249,420,142),(2,112,447,250,421,143),(3,97,448,251,422,144),(4,98,433,252,423,129),(5,99,434,253,424,130),(6,100,435,254,425,131),(7,101,436,255,426,132),(8,102,437,256,427,133),(9,103,438,241,428,134),(10,104,439,242,429,135),(11,105,440,243,430,136),(12,106,441,244,431,137),(13,107,442,245,432,138),(14,108,443,246,417,139),(15,109,444,247,418,140),(16,110,445,248,419,141),(17,261,306,173,195,214),(18,262,307,174,196,215),(19,263,308,175,197,216),(20,264,309,176,198,217),(21,265,310,161,199,218),(22,266,311,162,200,219),(23,267,312,163,201,220),(24,268,313,164,202,221),(25,269,314,165,203,222),(26,270,315,166,204,223),(27,271,316,167,205,224),(28,272,317,168,206,209),(29,257,318,169,207,210),(30,258,319,170,208,211),(31,259,320,171,193,212),(32,260,305,172,194,213),(33,70,301,453,328,181),(34,71,302,454,329,182),(35,72,303,455,330,183),(36,73,304,456,331,184),(37,74,289,457,332,185),(38,75,290,458,333,186),(39,76,291,459,334,187),(40,77,292,460,335,188),(41,78,293,461,336,189),(42,79,294,462,321,190),(43,80,295,463,322,191),(44,65,296,464,323,192),(45,66,297,449,324,177),(46,67,298,450,325,178),(47,68,299,451,326,179),(48,69,300,452,327,180),(49,408,384,94,470,121),(50,409,369,95,471,122),(51,410,370,96,472,123),(52,411,371,81,473,124),(53,412,372,82,474,125),(54,413,373,83,475,126),(55,414,374,84,476,127),(56,415,375,85,477,128),(57,416,376,86,478,113),(58,401,377,87,479,114),(59,402,378,88,480,115),(60,403,379,89,465,116),(61,404,380,90,466,117),(62,405,381,91,467,118),(63,406,382,92,468,119),(64,407,383,93,469,120),(145,288,239,354,385,344),(146,273,240,355,386,345),(147,274,225,356,387,346),(148,275,226,357,388,347),(149,276,227,358,389,348),(150,277,228,359,390,349),(151,278,229,360,391,350),(152,279,230,361,392,351),(153,280,231,362,393,352),(154,281,232,363,394,337),(155,282,233,364,395,338),(156,283,234,365,396,339),(157,284,235,366,397,340),(158,285,236,367,398,341),(159,286,237,368,399,342),(160,287,238,353,400,343)], [(1,461,386,318,87),(2,319,462,88,387),(3,89,320,388,463),(4,389,90,464,305),(5,449,390,306,91),(6,307,450,92,391),(7,93,308,392,451),(8,393,94,452,309),(9,453,394,310,95),(10,311,454,96,395),(11,81,312,396,455),(12,397,82,456,313),(13,457,398,314,83),(14,315,458,84,399),(15,85,316,400,459),(16,385,86,460,317),(17,405,424,66,228),(18,67,406,229,425),(19,230,68,426,407),(20,427,231,408,69),(21,409,428,70,232),(22,71,410,233,429),(23,234,72,430,411),(24,431,235,412,73),(25,413,432,74,236),(26,75,414,237,417),(27,238,76,418,415),(28,419,239,416,77),(29,401,420,78,240),(30,79,402,225,421),(31,226,80,422,403),(32,423,227,404,65),(33,281,218,50,241),(34,51,282,242,219),(35,243,52,220,283),(36,221,244,284,53),(37,285,222,54,245),(38,55,286,246,223),(39,247,56,224,287),(40,209,248,288,57),(41,273,210,58,249),(42,59,274,250,211),(43,251,60,212,275),(44,213,252,276,61),(45,277,214,62,253),(46,63,278,254,215),(47,255,64,216,279),(48,217,256,280,49),(97,465,171,347,322),(98,348,466,323,172),(99,324,349,173,467),(100,174,325,468,350),(101,469,175,351,326),(102,352,470,327,176),(103,328,337,161,471),(104,162,329,472,338),(105,473,163,339,330),(106,340,474,331,164),(107,332,341,165,475),(108,166,333,476,342),(109,477,167,343,334),(110,344,478,335,168),(111,336,345,169,479),(112,170,321,480,346),(113,188,206,445,145),(114,446,189,146,207),(115,147,447,208,190),(116,193,148,191,448),(117,192,194,433,149),(118,434,177,150,195),(119,151,435,196,178),(120,197,152,179,436),(121,180,198,437,153),(122,438,181,154,199),(123,155,439,200,182),(124,201,156,183,440),(125,184,202,441,157),(126,442,185,158,203),(127,159,443,204,186),(128,205,160,187,444),(129,358,380,296,260),(130,297,359,261,381),(131,262,298,382,360),(132,383,263,361,299),(133,362,384,300,264),(134,301,363,265,369),(135,266,302,370,364),(136,371,267,365,303),(137,366,372,304,268),(138,289,367,269,373),(139,270,290,374,368),(140,375,271,353,291),(141,354,376,292,272),(142,293,355,257,377),(143,258,294,378,356),(144,379,259,357,295)], [(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)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5 | 6A | ··· | 6F | 8A | ··· | 8H | 10A | 10B | 10C | 12A | ··· | 12H | 15A | 15B | 16A | ··· | 16P | 20A | 20B | 20C | 20D | 24A | ··· | 24P | 30A | ··· | 30F | 48A | ··· | 48AF | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 6 | ··· | 6 | 8 | ··· | 8 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 15 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 48 | ··· | 48 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 5 | ··· | 5 | 4 | ··· | 4 | 5 | ··· | 5 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | |||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C8 | C12 | C12 | C16 | C24 | C24 | C48 | F5 | C5⋊C8 | C2×F5 | C5⋊C8 | C3×F5 | C5⋊C16 | C3×C5⋊C8 | C6×F5 | C3×C5⋊C8 | C3×C5⋊C16 |
| kernel | C6×C5⋊C16 | C3×C5⋊C16 | C6×C5⋊2C8 | C2×C5⋊C16 | C3×C5⋊2C8 | C2×C60 | C5⋊C16 | C2×C5⋊2C8 | C60 | C2×C30 | C5⋊2C8 | C2×C20 | C30 | C20 | C2×C10 | C10 | C2×C12 | C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 16 | 8 | 8 | 32 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of C6×C5⋊C16 ►in GL5(𝔽241)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 1 | 240 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 82 | 173 | 80 | 33 |
| 0 | 162 | 206 | 194 | 115 |
| 0 | 35 | 47 | 126 | 195 |
| 0 | 208 | 127 | 159 | 68 |
G:=sub<GL(5,GF(241))| [16,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,240,240,240,240],[1,0,0,0,0,0,82,162,35,208,0,173,206,47,127,0,80,194,126,159,0,33,115,195,68] >;
C6×C5⋊C16 in GAP, Magma, Sage, TeX
C_6\times C_5\rtimes C_{16} % in TeX
G:=Group("C6xC5:C16"); // GroupNames label
G:=SmallGroup(480,277);
// by ID
G=gap.SmallGroup(480,277);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,80,102,9414,1595]);
// Polycyclic
G:=Group<a,b,c|a^6=b^5=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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