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## G = C2×C15⋊7Q16order 480 = 25·3·5

### Direct product of C2 and C15⋊7Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×C15⋊7Q16
 Chief series C1 — C5 — C15 — C30 — C60 — Dic30 — C2×Dic30 — C2×C15⋊7Q16
 Lower central C15 — C30 — C60 — C2×C15⋊7Q16
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for C2×C157Q16
G = < a,b,c,d | a2=b15=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 532 in 120 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], Q8 [×4], C10, C10 [×2], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C15, C2×C8, Q16 [×4], C2×Q8, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C3⋊C8 [×2], Dic6 [×3], C2×Dic3, C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C30, C30 [×2], C2×Q16, C52C8 [×2], Dic10 [×3], C2×Dic5, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C2×C3⋊C8, C3⋊Q16 [×4], C2×Dic6, C6×Q8, Dic15 [×2], C60 [×2], C60 [×2], C2×C30, C2×C52C8, C5⋊Q16 [×4], C2×Dic10, Q8×C10, C2×C3⋊Q16, C153C8 [×2], Dic30 [×2], Dic30, C2×Dic15, C2×C60, C2×C60, Q8×C15 [×2], Q8×C15, C2×C5⋊Q16, C2×C153C8, C157Q16 [×4], C2×Dic30, Q8×C30, C2×C157Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], Q16 [×2], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C2×Q16, C5⋊D4 [×2], C22×D5, C3⋊Q16 [×2], C2×C3⋊D4, D30 [×3], C5⋊Q16 [×2], C2×C5⋊D4, C2×C3⋊Q16, C157D4 [×2], C22×D15, C2×C5⋊Q16, C157Q16 [×2], C2×C157D4, C2×C157Q16

Smallest permutation representation of C2×C157Q16
Regular action on 480 points
Generators in S480
(1 124)(2 125)(3 126)(4 127)(5 128)(6 129)(7 130)(8 131)(9 132)(10 133)(11 134)(12 135)(13 121)(14 122)(15 123)(16 146)(17 147)(18 148)(19 149)(20 150)(21 136)(22 137)(23 138)(24 139)(25 140)(26 141)(27 142)(28 143)(29 144)(30 145)(31 158)(32 159)(33 160)(34 161)(35 162)(36 163)(37 164)(38 165)(39 151)(40 152)(41 153)(42 154)(43 155)(44 156)(45 157)(46 171)(47 172)(48 173)(49 174)(50 175)(51 176)(52 177)(53 178)(54 179)(55 180)(56 166)(57 167)(58 168)(59 169)(60 170)(61 182)(62 183)(63 184)(64 185)(65 186)(66 187)(67 188)(68 189)(69 190)(70 191)(71 192)(72 193)(73 194)(74 195)(75 181)(76 201)(77 202)(78 203)(79 204)(80 205)(81 206)(82 207)(83 208)(84 209)(85 210)(86 196)(87 197)(88 198)(89 199)(90 200)(91 219)(92 220)(93 221)(94 222)(95 223)(96 224)(97 225)(98 211)(99 212)(100 213)(101 214)(102 215)(103 216)(104 217)(105 218)(106 230)(107 231)(108 232)(109 233)(110 234)(111 235)(112 236)(113 237)(114 238)(115 239)(116 240)(117 226)(118 227)(119 228)(120 229)(241 361)(242 362)(243 363)(244 364)(245 365)(246 366)(247 367)(248 368)(249 369)(250 370)(251 371)(252 372)(253 373)(254 374)(255 375)(256 377)(257 378)(258 379)(259 380)(260 381)(261 382)(262 383)(263 384)(264 385)(265 386)(266 387)(267 388)(268 389)(269 390)(270 376)(271 398)(272 399)(273 400)(274 401)(275 402)(276 403)(277 404)(278 405)(279 391)(280 392)(281 393)(282 394)(283 395)(284 396)(285 397)(286 413)(287 414)(288 415)(289 416)(290 417)(291 418)(292 419)(293 420)(294 406)(295 407)(296 408)(297 409)(298 410)(299 411)(300 412)(301 432)(302 433)(303 434)(304 435)(305 421)(306 422)(307 423)(308 424)(309 425)(310 426)(311 427)(312 428)(313 429)(314 430)(315 431)(316 448)(317 449)(318 450)(319 436)(320 437)(321 438)(322 439)(323 440)(324 441)(325 442)(326 443)(327 444)(328 445)(329 446)(330 447)(331 459)(332 460)(333 461)(334 462)(335 463)(336 464)(337 465)(338 451)(339 452)(340 453)(341 454)(342 455)(343 456)(344 457)(345 458)(346 466)(347 467)(348 468)(349 469)(350 470)(351 471)(352 472)(353 473)(354 474)(355 475)(356 476)(357 477)(358 478)(359 479)(360 480)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)(16 17 18 19 20 21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75)(76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135)(136 137 138 139 140 141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165)(166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195)(196 197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225)(226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255)(256 257 258 259 260 261 262 263 264 265 266 267 268 269 270)(271 272 273 274 275 276 277 278 279 280 281 282 283 284 285)(286 287 288 289 290 291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310 311 312 313 314 315)(316 317 318 319 320 321 322 323 324 325 326 327 328 329 330)(331 332 333 334 335 336 337 338 339 340 341 342 343 344 345)(346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375)(376 377 378 379 380 381 382 383 384 385 386 387 388 389 390)(391 392 393 394 395 396 397 398 399 400 401 402 403 404 405)(406 407 408 409 410 411 412 413 414 415 416 417 418 419 420)(421 422 423 424 425 426 427 428 429 430 431 432 433 434 435)(436 437 438 439 440 441 442 443 444 445 446 447 448 449 450)(451 452 453 454 455 456 457 458 459 460 461 462 463 464 465)(466 467 468 469 470 471 472 473 474 475 476 477 478 479 480)
(1 452 42 448 30 478 54 433)(2 451 43 447 16 477 55 432)(3 465 44 446 17 476 56 431)(4 464 45 445 18 475 57 430)(5 463 31 444 19 474 58 429)(6 462 32 443 20 473 59 428)(7 461 33 442 21 472 60 427)(8 460 34 441 22 471 46 426)(9 459 35 440 23 470 47 425)(10 458 36 439 24 469 48 424)(11 457 37 438 25 468 49 423)(12 456 38 437 26 467 50 422)(13 455 39 436 27 466 51 421)(14 454 40 450 28 480 52 435)(15 453 41 449 29 479 53 434)(61 402 109 365 77 406 103 388)(62 401 110 364 78 420 104 387)(63 400 111 363 79 419 105 386)(64 399 112 362 80 418 91 385)(65 398 113 361 81 417 92 384)(66 397 114 375 82 416 93 383)(67 396 115 374 83 415 94 382)(68 395 116 373 84 414 95 381)(69 394 117 372 85 413 96 380)(70 393 118 371 86 412 97 379)(71 392 119 370 87 411 98 378)(72 391 120 369 88 410 99 377)(73 405 106 368 89 409 100 376)(74 404 107 367 90 408 101 390)(75 403 108 366 76 407 102 389)(121 342 151 319 142 346 176 305)(122 341 152 318 143 360 177 304)(123 340 153 317 144 359 178 303)(124 339 154 316 145 358 179 302)(125 338 155 330 146 357 180 301)(126 337 156 329 147 356 166 315)(127 336 157 328 148 355 167 314)(128 335 158 327 149 354 168 313)(129 334 159 326 150 353 169 312)(130 333 160 325 136 352 170 311)(131 332 161 324 137 351 171 310)(132 331 162 323 138 350 172 309)(133 345 163 322 139 349 173 308)(134 344 164 321 140 348 174 307)(135 343 165 320 141 347 175 306)(181 276 232 246 201 295 215 268)(182 275 233 245 202 294 216 267)(183 274 234 244 203 293 217 266)(184 273 235 243 204 292 218 265)(185 272 236 242 205 291 219 264)(186 271 237 241 206 290 220 263)(187 285 238 255 207 289 221 262)(188 284 239 254 208 288 222 261)(189 283 240 253 209 287 223 260)(190 282 226 252 210 286 224 259)(191 281 227 251 196 300 225 258)(192 280 228 250 197 299 211 257)(193 279 229 249 198 298 212 256)(194 278 230 248 199 297 213 270)(195 277 231 247 200 296 214 269)
(1 198 30 193)(2 199 16 194)(3 200 17 195)(4 201 18 181)(5 202 19 182)(6 203 20 183)(7 204 21 184)(8 205 22 185)(9 206 23 186)(10 207 24 187)(11 208 25 188)(12 209 26 189)(13 210 27 190)(14 196 28 191)(15 197 29 192)(31 233 58 216)(32 234 59 217)(33 235 60 218)(34 236 46 219)(35 237 47 220)(36 238 48 221)(37 239 49 222)(38 240 50 223)(39 226 51 224)(40 227 52 225)(41 228 53 211)(42 229 54 212)(43 230 55 213)(44 231 56 214)(45 232 57 215)(61 128 77 149)(62 129 78 150)(63 130 79 136)(64 131 80 137)(65 132 81 138)(66 133 82 139)(67 134 83 140)(68 135 84 141)(69 121 85 142)(70 122 86 143)(71 123 87 144)(72 124 88 145)(73 125 89 146)(74 126 90 147)(75 127 76 148)(91 161 112 171)(92 162 113 172)(93 163 114 173)(94 164 115 174)(95 165 116 175)(96 151 117 176)(97 152 118 177)(98 153 119 178)(99 154 120 179)(100 155 106 180)(101 156 107 166)(102 157 108 167)(103 158 109 168)(104 159 110 169)(105 160 111 170)(241 470 263 459)(242 471 264 460)(243 472 265 461)(244 473 266 462)(245 474 267 463)(246 475 268 464)(247 476 269 465)(248 477 270 451)(249 478 256 452)(250 479 257 453)(251 480 258 454)(252 466 259 455)(253 467 260 456)(254 468 261 457)(255 469 262 458)(271 425 290 440)(272 426 291 441)(273 427 292 442)(274 428 293 443)(275 429 294 444)(276 430 295 445)(277 431 296 446)(278 432 297 447)(279 433 298 448)(280 434 299 449)(281 435 300 450)(282 421 286 436)(283 422 287 437)(284 423 288 438)(285 424 289 439)(301 409 330 405)(302 410 316 391)(303 411 317 392)(304 412 318 393)(305 413 319 394)(306 414 320 395)(307 415 321 396)(308 416 322 397)(309 417 323 398)(310 418 324 399)(311 419 325 400)(312 420 326 401)(313 406 327 402)(314 407 328 403)(315 408 329 404)(331 361 350 384)(332 362 351 385)(333 363 352 386)(334 364 353 387)(335 365 354 388)(336 366 355 389)(337 367 356 390)(338 368 357 376)(339 369 358 377)(340 370 359 378)(341 371 360 379)(342 372 346 380)(343 373 347 381)(344 374 348 382)(345 375 349 383)

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

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

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

84 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 ··· 12F 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X 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 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 4 4 60 60 2 2 2 2 2 30 30 30 30 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - + + + + + - - - image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 Q16 D10 D10 C3⋊D4 C3⋊D4 D15 C5⋊D4 C5⋊D4 D30 D30 C15⋊7D4 C15⋊7D4 C3⋊Q16 C5⋊Q16 C15⋊7Q16 kernel C2×C15⋊7Q16 C2×C15⋊3C8 C15⋊7Q16 C2×Dic30 Q8×C30 Q8×C10 C60 C2×C30 C6×Q8 C2×C20 C5×Q8 C30 C2×C12 C3×Q8 C20 C2×C10 C2×Q8 C12 C2×C6 C2×C4 Q8 C4 C22 C10 C6 C2 # reps 1 1 4 1 1 1 1 1 2 1 2 4 2 4 2 2 4 4 4 4 8 8 8 2 4 8

Matrix representation of C2×C157Q16 in GL5(𝔽241)

 240 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 84 173 0 0 0 215 110
,
 1 0 0 0 0 0 230 11 0 0 0 230 230 0 0 0 0 0 15 39 0 0 0 192 226
,
 240 0 0 0 0 0 209 136 0 0 0 136 32 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,84,215,0,0,0,173,110],[1,0,0,0,0,0,230,230,0,0,0,11,230,0,0,0,0,0,15,192,0,0,0,39,226],[240,0,0,0,0,0,209,136,0,0,0,136,32,0,0,0,0,0,1,0,0,0,0,0,1] >;

C2×C157Q16 in GAP, Magma, Sage, TeX

C_2\times C_{15}\rtimes_7Q_{16}
% in TeX

G:=Group("C2xC15:7Q16");
// GroupNames label

G:=SmallGroup(480,908);
// by ID

G=gap.SmallGroup(480,908);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,100,675,185,80,2693,18822]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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