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## G = C2×C15⋊Q16order 480 = 25·3·5

### Direct product of C2 and C15⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×C15⋊Q16
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — C15⋊Q16 — C2×C15⋊Q16
 Lower central C15 — C30 — C60 — C2×C15⋊Q16
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C15⋊Q16
G = < a,b,c,d | a2=b15=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b11, dcd-1=c-1 >

Subgroups: 444 in 120 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C10, C10 [×2], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C15, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8 [×3], C30, C30 [×2], C2×Q16, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8 [×3], C2×C3⋊C8, C3⋊Q16 [×4], C2×Dic6, C6×Q8, C5×Dic3 [×2], C3×Dic5 [×2], C60 [×2], C2×C30, C2×C52C8, C5⋊Q16 [×4], C2×Dic10, Q8×C10, C2×C3⋊Q16, C153C8 [×2], C3×Dic10 [×2], C3×Dic10, C6×Dic5, C5×Dic6 [×2], C5×Dic6, C10×Dic3, C2×C60, C2×C5⋊Q16, C15⋊Q16 [×4], C2×C153C8, C6×Dic10, C10×Dic6, C2×C15⋊Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], Q16 [×2], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C2×Q16, C5⋊D4 [×2], C22×D5, C3⋊Q16 [×2], C2×C3⋊D4, S3×D5, C5⋊Q16 [×2], C2×C5⋊D4, C2×C3⋊Q16, C15⋊D4 [×2], C2×S3×D5, C2×C5⋊Q16, C15⋊Q16 [×2], C2×C15⋊D4, C2×C15⋊Q16

Smallest permutation representation of C2×C15⋊Q16
Regular action on 480 points
Generators in S480
(1 71)(2 72)(3 73)(4 74)(5 75)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 88)(17 89)(18 90)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 82)(26 83)(27 84)(28 85)(29 86)(30 87)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 105)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 118)(47 119)(48 120)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(121 192)(122 193)(123 194)(124 195)(125 181)(126 182)(127 183)(128 184)(129 185)(130 186)(131 187)(132 188)(133 189)(134 190)(135 191)(136 196)(137 197)(138 198)(139 199)(140 200)(141 201)(142 202)(143 203)(144 204)(145 205)(146 206)(147 207)(148 208)(149 209)(150 210)(151 211)(152 212)(153 213)(154 214)(155 215)(156 216)(157 217)(158 218)(159 219)(160 220)(161 221)(162 222)(163 223)(164 224)(165 225)(166 226)(167 227)(168 228)(169 229)(170 230)(171 231)(172 232)(173 233)(174 234)(175 235)(176 236)(177 237)(178 238)(179 239)(180 240)(241 312)(242 313)(243 314)(244 315)(245 301)(246 302)(247 303)(248 304)(249 305)(250 306)(251 307)(252 308)(253 309)(254 310)(255 311)(256 319)(257 320)(258 321)(259 322)(260 323)(261 324)(262 325)(263 326)(264 327)(265 328)(266 329)(267 330)(268 316)(269 317)(270 318)(271 338)(272 339)(273 340)(274 341)(275 342)(276 343)(277 344)(278 345)(279 331)(280 332)(281 333)(282 334)(283 335)(284 336)(285 337)(286 352)(287 353)(288 354)(289 355)(290 356)(291 357)(292 358)(293 359)(294 360)(295 346)(296 347)(297 348)(298 349)(299 350)(300 351)(361 428)(362 429)(363 430)(364 431)(365 432)(366 433)(367 434)(368 435)(369 421)(370 422)(371 423)(372 424)(373 425)(374 426)(375 427)(376 443)(377 444)(378 445)(379 446)(380 447)(381 448)(382 449)(383 450)(384 436)(385 437)(386 438)(387 439)(388 440)(389 441)(390 442)(391 451)(392 452)(393 453)(394 454)(395 455)(396 456)(397 457)(398 458)(399 459)(400 460)(401 461)(402 462)(403 463)(404 464)(405 465)(406 474)(407 475)(408 476)(409 477)(410 478)(411 479)(412 480)(413 466)(414 467)(415 468)(416 469)(417 470)(418 471)(419 472)(420 473)
(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)
(1 451 36 440 17 467 50 425)(2 465 37 439 18 466 51 424)(3 464 38 438 19 480 52 423)(4 463 39 437 20 479 53 422)(5 462 40 436 21 478 54 421)(6 461 41 450 22 477 55 435)(7 460 42 449 23 476 56 434)(8 459 43 448 24 475 57 433)(9 458 44 447 25 474 58 432)(10 457 45 446 26 473 59 431)(11 456 31 445 27 472 60 430)(12 455 32 444 28 471 46 429)(13 454 33 443 29 470 47 428)(14 453 34 442 30 469 48 427)(15 452 35 441 16 468 49 426)(61 401 93 383 79 409 112 368)(62 400 94 382 80 408 113 367)(63 399 95 381 81 407 114 366)(64 398 96 380 82 406 115 365)(65 397 97 379 83 420 116 364)(66 396 98 378 84 419 117 363)(67 395 99 377 85 418 118 362)(68 394 100 376 86 417 119 361)(69 393 101 390 87 416 120 375)(70 392 102 389 88 415 106 374)(71 391 103 388 89 414 107 373)(72 405 104 387 90 413 108 372)(73 404 105 386 76 412 109 371)(74 403 91 385 77 411 110 370)(75 402 92 384 78 410 111 369)(121 333 177 302 143 358 164 328)(122 332 178 301 144 357 165 327)(123 331 179 315 145 356 151 326)(124 345 180 314 146 355 152 325)(125 344 166 313 147 354 153 324)(126 343 167 312 148 353 154 323)(127 342 168 311 149 352 155 322)(128 341 169 310 150 351 156 321)(129 340 170 309 136 350 157 320)(130 339 171 308 137 349 158 319)(131 338 172 307 138 348 159 318)(132 337 173 306 139 347 160 317)(133 336 174 305 140 346 161 316)(134 335 175 304 141 360 162 330)(135 334 176 303 142 359 163 329)(181 277 226 242 207 288 213 261)(182 276 227 241 208 287 214 260)(183 275 228 255 209 286 215 259)(184 274 229 254 210 300 216 258)(185 273 230 253 196 299 217 257)(186 272 231 252 197 298 218 256)(187 271 232 251 198 297 219 270)(188 285 233 250 199 296 220 269)(189 284 234 249 200 295 221 268)(190 283 235 248 201 294 222 267)(191 282 236 247 202 293 223 266)(192 281 237 246 203 292 224 265)(193 280 238 245 204 291 225 264)(194 279 239 244 205 290 211 263)(195 278 240 243 206 289 212 262)
(1 143 17 121)(2 139 18 132)(3 150 19 128)(4 146 20 124)(5 142 21 135)(6 138 22 131)(7 149 23 127)(8 145 24 123)(9 141 25 134)(10 137 26 130)(11 148 27 126)(12 144 28 122)(13 140 29 133)(14 136 30 129)(15 147 16 125)(31 167 60 154)(32 178 46 165)(33 174 47 161)(34 170 48 157)(35 166 49 153)(36 177 50 164)(37 173 51 160)(38 169 52 156)(39 180 53 152)(40 176 54 163)(41 172 55 159)(42 168 56 155)(43 179 57 151)(44 175 58 162)(45 171 59 158)(61 198 79 187)(62 209 80 183)(63 205 81 194)(64 201 82 190)(65 197 83 186)(66 208 84 182)(67 204 85 193)(68 200 86 189)(69 196 87 185)(70 207 88 181)(71 203 89 192)(72 199 90 188)(73 210 76 184)(74 206 77 195)(75 202 78 191)(91 240 110 212)(92 236 111 223)(93 232 112 219)(94 228 113 215)(95 239 114 211)(96 235 115 222)(97 231 116 218)(98 227 117 214)(99 238 118 225)(100 234 119 221)(101 230 120 217)(102 226 106 213)(103 237 107 224)(104 233 108 220)(105 229 109 216)(241 419 260 396)(242 415 261 392)(243 411 262 403)(244 407 263 399)(245 418 264 395)(246 414 265 391)(247 410 266 402)(248 406 267 398)(249 417 268 394)(250 413 269 405)(251 409 270 401)(252 420 256 397)(253 416 257 393)(254 412 258 404)(255 408 259 400)(271 368 297 383)(272 364 298 379)(273 375 299 390)(274 371 300 386)(275 367 286 382)(276 363 287 378)(277 374 288 389)(278 370 289 385)(279 366 290 381)(280 362 291 377)(281 373 292 388)(282 369 293 384)(283 365 294 380)(284 361 295 376)(285 372 296 387)(301 471 327 455)(302 467 328 451)(303 478 329 462)(304 474 330 458)(305 470 316 454)(306 466 317 465)(307 477 318 461)(308 473 319 457)(309 469 320 453)(310 480 321 464)(311 476 322 460)(312 472 323 456)(313 468 324 452)(314 479 325 463)(315 475 326 459)(331 433 356 448)(332 429 357 444)(333 425 358 440)(334 421 359 436)(335 432 360 447)(336 428 346 443)(337 424 347 439)(338 435 348 450)(339 431 349 446)(340 427 350 442)(341 423 351 438)(342 434 352 449)(343 430 353 445)(344 426 354 441)(345 422 355 437)

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

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

G=PermutationGroup([(1,71),(2,72),(3,73),(4,74),(5,75),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,88),(17,89),(18,90),(19,76),(20,77),(21,78),(22,79),(23,80),(24,81),(25,82),(26,83),(27,84),(28,85),(29,86),(30,87),(31,98),(32,99),(33,100),(34,101),(35,102),(36,103),(37,104),(38,105),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,97),(46,118),(47,119),(48,120),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,113),(57,114),(58,115),(59,116),(60,117),(121,192),(122,193),(123,194),(124,195),(125,181),(126,182),(127,183),(128,184),(129,185),(130,186),(131,187),(132,188),(133,189),(134,190),(135,191),(136,196),(137,197),(138,198),(139,199),(140,200),(141,201),(142,202),(143,203),(144,204),(145,205),(146,206),(147,207),(148,208),(149,209),(150,210),(151,211),(152,212),(153,213),(154,214),(155,215),(156,216),(157,217),(158,218),(159,219),(160,220),(161,221),(162,222),(163,223),(164,224),(165,225),(166,226),(167,227),(168,228),(169,229),(170,230),(171,231),(172,232),(173,233),(174,234),(175,235),(176,236),(177,237),(178,238),(179,239),(180,240),(241,312),(242,313),(243,314),(244,315),(245,301),(246,302),(247,303),(248,304),(249,305),(250,306),(251,307),(252,308),(253,309),(254,310),(255,311),(256,319),(257,320),(258,321),(259,322),(260,323),(261,324),(262,325),(263,326),(264,327),(265,328),(266,329),(267,330),(268,316),(269,317),(270,318),(271,338),(272,339),(273,340),(274,341),(275,342),(276,343),(277,344),(278,345),(279,331),(280,332),(281,333),(282,334),(283,335),(284,336),(285,337),(286,352),(287,353),(288,354),(289,355),(290,356),(291,357),(292,358),(293,359),(294,360),(295,346),(296,347),(297,348),(298,349),(299,350),(300,351),(361,428),(362,429),(363,430),(364,431),(365,432),(366,433),(367,434),(368,435),(369,421),(370,422),(371,423),(372,424),(373,425),(374,426),(375,427),(376,443),(377,444),(378,445),(379,446),(380,447),(381,448),(382,449),(383,450),(384,436),(385,437),(386,438),(387,439),(388,440),(389,441),(390,442),(391,451),(392,452),(393,453),(394,454),(395,455),(396,456),(397,457),(398,458),(399,459),(400,460),(401,461),(402,462),(403,463),(404,464),(405,465),(406,474),(407,475),(408,476),(409,477),(410,478),(411,479),(412,480),(413,466),(414,467),(415,468),(416,469),(417,470),(418,471),(419,472),(420,473)], [(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)], [(1,451,36,440,17,467,50,425),(2,465,37,439,18,466,51,424),(3,464,38,438,19,480,52,423),(4,463,39,437,20,479,53,422),(5,462,40,436,21,478,54,421),(6,461,41,450,22,477,55,435),(7,460,42,449,23,476,56,434),(8,459,43,448,24,475,57,433),(9,458,44,447,25,474,58,432),(10,457,45,446,26,473,59,431),(11,456,31,445,27,472,60,430),(12,455,32,444,28,471,46,429),(13,454,33,443,29,470,47,428),(14,453,34,442,30,469,48,427),(15,452,35,441,16,468,49,426),(61,401,93,383,79,409,112,368),(62,400,94,382,80,408,113,367),(63,399,95,381,81,407,114,366),(64,398,96,380,82,406,115,365),(65,397,97,379,83,420,116,364),(66,396,98,378,84,419,117,363),(67,395,99,377,85,418,118,362),(68,394,100,376,86,417,119,361),(69,393,101,390,87,416,120,375),(70,392,102,389,88,415,106,374),(71,391,103,388,89,414,107,373),(72,405,104,387,90,413,108,372),(73,404,105,386,76,412,109,371),(74,403,91,385,77,411,110,370),(75,402,92,384,78,410,111,369),(121,333,177,302,143,358,164,328),(122,332,178,301,144,357,165,327),(123,331,179,315,145,356,151,326),(124,345,180,314,146,355,152,325),(125,344,166,313,147,354,153,324),(126,343,167,312,148,353,154,323),(127,342,168,311,149,352,155,322),(128,341,169,310,150,351,156,321),(129,340,170,309,136,350,157,320),(130,339,171,308,137,349,158,319),(131,338,172,307,138,348,159,318),(132,337,173,306,139,347,160,317),(133,336,174,305,140,346,161,316),(134,335,175,304,141,360,162,330),(135,334,176,303,142,359,163,329),(181,277,226,242,207,288,213,261),(182,276,227,241,208,287,214,260),(183,275,228,255,209,286,215,259),(184,274,229,254,210,300,216,258),(185,273,230,253,196,299,217,257),(186,272,231,252,197,298,218,256),(187,271,232,251,198,297,219,270),(188,285,233,250,199,296,220,269),(189,284,234,249,200,295,221,268),(190,283,235,248,201,294,222,267),(191,282,236,247,202,293,223,266),(192,281,237,246,203,292,224,265),(193,280,238,245,204,291,225,264),(194,279,239,244,205,290,211,263),(195,278,240,243,206,289,212,262)], [(1,143,17,121),(2,139,18,132),(3,150,19,128),(4,146,20,124),(5,142,21,135),(6,138,22,131),(7,149,23,127),(8,145,24,123),(9,141,25,134),(10,137,26,130),(11,148,27,126),(12,144,28,122),(13,140,29,133),(14,136,30,129),(15,147,16,125),(31,167,60,154),(32,178,46,165),(33,174,47,161),(34,170,48,157),(35,166,49,153),(36,177,50,164),(37,173,51,160),(38,169,52,156),(39,180,53,152),(40,176,54,163),(41,172,55,159),(42,168,56,155),(43,179,57,151),(44,175,58,162),(45,171,59,158),(61,198,79,187),(62,209,80,183),(63,205,81,194),(64,201,82,190),(65,197,83,186),(66,208,84,182),(67,204,85,193),(68,200,86,189),(69,196,87,185),(70,207,88,181),(71,203,89,192),(72,199,90,188),(73,210,76,184),(74,206,77,195),(75,202,78,191),(91,240,110,212),(92,236,111,223),(93,232,112,219),(94,228,113,215),(95,239,114,211),(96,235,115,222),(97,231,116,218),(98,227,117,214),(99,238,118,225),(100,234,119,221),(101,230,120,217),(102,226,106,213),(103,237,107,224),(104,233,108,220),(105,229,109,216),(241,419,260,396),(242,415,261,392),(243,411,262,403),(244,407,263,399),(245,418,264,395),(246,414,265,391),(247,410,266,402),(248,406,267,398),(249,417,268,394),(250,413,269,405),(251,409,270,401),(252,420,256,397),(253,416,257,393),(254,412,258,404),(255,408,259,400),(271,368,297,383),(272,364,298,379),(273,375,299,390),(274,371,300,386),(275,367,286,382),(276,363,287,378),(277,374,288,389),(278,370,289,385),(279,366,290,381),(280,362,291,377),(281,373,292,388),(282,369,293,384),(283,365,294,380),(284,361,295,376),(285,372,296,387),(301,471,327,455),(302,467,328,451),(303,478,329,462),(304,474,330,458),(305,470,316,454),(306,466,317,465),(307,477,318,461),(308,473,319,457),(309,469,320,453),(310,480,321,464),(311,476,322,460),(312,472,323,456),(313,468,324,452),(314,479,325,463),(315,475,326,459),(331,433,356,448),(332,429,357,444),(333,425,358,440),(334,421,359,436),(335,432,360,447),(336,428,346,443),(337,424,347,439),(338,435,348,450),(339,431,349,446),(340,427,350,442),(341,423,351,438),(342,434,352,449),(343,430,353,445),(344,426,354,441),(345,422,355,437)])

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 4 4 4 4 4 4 4 type + + + + + + + + + + + - + + - + - - + - image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 Q16 D10 D10 C3⋊D4 C3⋊D4 C5⋊D4 C5⋊D4 C3⋊Q16 S3×D5 C5⋊Q16 C15⋊D4 C2×S3×D5 C15⋊D4 C15⋊Q16 kernel C2×C15⋊Q16 C15⋊Q16 C2×C15⋊3C8 C6×Dic10 C10×Dic6 C2×Dic10 C60 C2×C30 C2×Dic6 Dic10 C2×C20 C30 Dic6 C2×C12 C20 C2×C10 C12 C2×C6 C10 C2×C4 C6 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 2 2 1 4 4 2 2 2 4 4 2 2 4 2 2 2 8

Matrix representation of C2×C15⋊Q16 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 240 240 0 0 0 0 1 0 0 0 0 0 0 0 189 190 0 0 0 0 52 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 47 40 0 0 0 0 234 194 0 0 0 0 0 0 95 129 0 0 0 0 115 146 0 0 0 0 0 0 132 127 0 0 0 0 176 131
,
 194 201 0 0 0 0 7 47 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 150 53 0 0 0 0 62 91

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,1,0,0,0,0,240,0,0,0,0,0,0,0,189,52,0,0,0,0,190,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[47,234,0,0,0,0,40,194,0,0,0,0,0,0,95,115,0,0,0,0,129,146,0,0,0,0,0,0,132,176,0,0,0,0,127,131],[194,7,0,0,0,0,201,47,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,150,62,0,0,0,0,53,91] >;

C2×C15⋊Q16 in GAP, Magma, Sage, TeX

C_2\times C_{15}\rtimes Q_{16}
% in TeX

G:=Group("C2xC15:Q16");
// GroupNames label

G:=SmallGroup(480,394);
// by ID

G=gap.SmallGroup(480,394);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,120,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^15=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,d*b*d^-1=b^11,d*c*d^-1=c^-1>;
// generators/relations

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