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G = C2×C15⋊7Q16  order 480 = 25·3·5

Direct product of C2 and C15⋊7Q16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C15⋊7Q16, C30⋊7Q16, C60.20D4, Q8.12D30, C60.79C23, Dic30.41C22, C15⋊16(C2×Q16), (C6×Q8).3D5, (C2×C4).54D30, C6⋊3(C5⋊Q16), (C2×Q8).5D15, (Q8×C10).7S3, (C5×Q8).50D6, (Q8×C30).3C2, (C2×C20).153D6, C30.387(C2×D4), C10⋊3(C3⋊Q16), (C2×C30).150D4, C4.9(C15⋊7D4), (C3×Q8).33D10, (C2×C12).152D10, C20.45(C3⋊D4), C12.47(C5⋊D4), (C2×C60).79C22, (C2×Dic30).9C2, C4.16(C22×D15), C20.117(C22×S3), C15⋊3C8.35C22, C12.117(C22×D5), (Q8×C15).38C22, C22.24(C15⋊7D4), C3⋊4(C2×C5⋊Q16), C5⋊4(C2×C3⋊Q16), (C2×C15⋊3C8).6C2, C6.114(C2×C5⋊D4), C2.19(C2×C15⋊7D4), C10.114(C2×C3⋊D4), (C2×C6).82(C5⋊D4), (C2×C10).82(C3⋊D4), SmallGroup(480,908)

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×C15⋊7Q16
 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, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, C10, C10, Dic3, C12, C12, C2×C6, C15, C2×C8, Q16, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C30, C30, C2×Q16, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C2×C3⋊C8, C3⋊Q16, C2×Dic6, C6×Q8, Dic15, C60, C60, C2×C30, C2×C5⋊2C8, C5⋊Q16, C2×Dic10, Q8×C10, C2×C3⋊Q16, C15⋊3C8, Dic30, Dic30, C2×Dic15, C2×C60, C2×C60, Q8×C15, Q8×C15, C2×C5⋊Q16, C2×C15⋊3C8, C15⋊7Q16, C2×Dic30, Q8×C30, C2×C15⋊7Q16
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, Q16, C2×D4, D10, C3⋊D4, C22×S3, D15, C2×Q16, C5⋊D4, C22×D5, C3⋊Q16, C2×C3⋊D4, D30, C5⋊Q16, C2×C5⋊D4, C2×C3⋊Q16, C15⋊7D4, C22×D15, C2×C5⋊Q16, C15⋊7Q16, C2×C15⋊7D4, C2×C15⋊7Q16

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order1222344444455666888810···1012···121515151520···2030···3060···60
size111122244606022222303030302···24···422224···42···24···4

84 irreducible representations

dim11111222222222222222222444
type+++++++++++-+++++---
imageC1C2C2C2C2S3D4D4D5D6D6Q16D10D10C3⋊D4C3⋊D4D15C5⋊D4C5⋊D4D30D30C15⋊7D4C15⋊7D4C3⋊Q16C5⋊Q16C15⋊7Q16
kernelC2×C15⋊7Q16C2×C15⋊3C8C15⋊7Q16C2×Dic30Q8×C30Q8×C10C60C2×C30C6×Q8C2×C20C5×Q8C30C2×C12C3×Q8C20C2×C10C2×Q8C12C2×C6C2×C4Q8C4C22C10C6C2
# reps11411111212424224444888248

Matrix representation of C2×C15⋊7Q16 ►in GL5(𝔽241)

2400000
0240000
0024000
00010
00001
,
10000
01000
00100
00084173
000215110
,
10000
02301100
023023000
0001539
000192226
,
2400000
020913600
01363200
00010
00001

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×C15⋊7Q16 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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