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## G = C15×C4.Q8order 480 = 25·3·5

### Direct product of C15 and C4.Q8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C15×C4.Q8
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — C2×C60 — C15×C4⋊C4 — C15×C4.Q8
 Lower central C1 — C2 — C4 — C15×C4.Q8
 Upper central C1 — C2×C30 — C2×C60 — C15×C4.Q8

Generators and relations for C15×C4.Q8
G = < a,b,c,d | a15=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 104 in 72 conjugacy classes, 56 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C20, C20, C2×C10, C24, C2×C12, C2×C12, C30, C30, C4.Q8, C40, C2×C20, C2×C20, C3×C4⋊C4, C2×C24, C60, C60, C2×C30, C5×C4⋊C4, C2×C40, C3×C4.Q8, C120, C2×C60, C2×C60, C5×C4.Q8, C15×C4⋊C4, C2×C120, C15×C4.Q8
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, Q8, C10, C12, C2×C6, C15, C4⋊C4, SD16, C20, C2×C10, C2×C12, C3×D4, C3×Q8, C30, C4.Q8, C2×C20, C5×D4, C5×Q8, C3×C4⋊C4, C3×SD16, C60, C2×C30, C5×C4⋊C4, C5×SD16, C3×C4.Q8, C2×C60, D4×C15, Q8×C15, C5×C4.Q8, C15×C4⋊C4, C15×SD16, C15×C4.Q8

Smallest permutation representation of C15×C4.Q8
Regular action on 480 points
Generators in S480
(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 72 391 341)(2 73 392 342)(3 74 393 343)(4 75 394 344)(5 61 395 345)(6 62 396 331)(7 63 397 332)(8 64 398 333)(9 65 399 334)(10 66 400 335)(11 67 401 336)(12 68 402 337)(13 69 403 338)(14 70 404 339)(15 71 405 340)(16 281 60 248)(17 282 46 249)(18 283 47 250)(19 284 48 251)(20 285 49 252)(21 271 50 253)(22 272 51 254)(23 273 52 255)(24 274 53 241)(25 275 54 242)(26 276 55 243)(27 277 56 244)(28 278 57 245)(29 279 58 246)(30 280 59 247)(31 124 364 177)(32 125 365 178)(33 126 366 179)(34 127 367 180)(35 128 368 166)(36 129 369 167)(37 130 370 168)(38 131 371 169)(39 132 372 170)(40 133 373 171)(41 134 374 172)(42 135 375 173)(43 121 361 174)(44 122 362 175)(45 123 363 176)(76 159 294 359)(77 160 295 360)(78 161 296 346)(79 162 297 347)(80 163 298 348)(81 164 299 349)(82 165 300 350)(83 151 286 351)(84 152 287 352)(85 153 288 353)(86 154 289 354)(87 155 290 355)(88 156 291 356)(89 157 292 357)(90 158 293 358)(91 438 197 149)(92 439 198 150)(93 440 199 136)(94 441 200 137)(95 442 201 138)(96 443 202 139)(97 444 203 140)(98 445 204 141)(99 446 205 142)(100 447 206 143)(101 448 207 144)(102 449 208 145)(103 450 209 146)(104 436 210 147)(105 437 196 148)(106 387 463 309)(107 388 464 310)(108 389 465 311)(109 390 451 312)(110 376 452 313)(111 377 453 314)(112 378 454 315)(113 379 455 301)(114 380 456 302)(115 381 457 303)(116 382 458 304)(117 383 459 305)(118 384 460 306)(119 385 461 307)(120 386 462 308)(181 227 479 268)(182 228 480 269)(183 229 466 270)(184 230 467 256)(185 231 468 257)(186 232 469 258)(187 233 470 259)(188 234 471 260)(189 235 472 261)(190 236 473 262)(191 237 474 263)(192 238 475 264)(193 239 476 265)(194 240 477 266)(195 226 478 267)(211 435 411 328)(212 421 412 329)(213 422 413 330)(214 423 414 316)(215 424 415 317)(216 425 416 318)(217 426 417 319)(218 427 418 320)(219 428 419 321)(220 429 420 322)(221 430 406 323)(222 431 407 324)(223 432 408 325)(224 433 409 326)(225 434 410 327)
(1 371 76 380 391 38 294 302)(2 372 77 381 392 39 295 303)(3 373 78 382 393 40 296 304)(4 374 79 383 394 41 297 305)(5 375 80 384 395 42 298 306)(6 361 81 385 396 43 299 307)(7 362 82 386 397 44 300 308)(8 363 83 387 398 45 286 309)(9 364 84 388 399 31 287 310)(10 365 85 389 400 32 288 311)(11 366 86 390 401 33 289 312)(12 367 87 376 402 34 290 313)(13 368 88 377 403 35 291 314)(14 369 89 378 404 36 292 315)(15 370 90 379 405 37 293 301)(16 265 102 411 60 239 208 211)(17 266 103 412 46 240 209 212)(18 267 104 413 47 226 210 213)(19 268 105 414 48 227 196 214)(20 269 91 415 49 228 197 215)(21 270 92 416 50 229 198 216)(22 256 93 417 51 230 199 217)(23 257 94 418 52 231 200 218)(24 258 95 419 53 232 201 219)(25 259 96 420 54 233 202 220)(26 260 97 406 55 234 203 221)(27 261 98 407 56 235 204 222)(28 262 99 408 57 236 205 223)(29 263 100 409 58 237 206 224)(30 264 101 410 59 238 207 225)(61 173 163 460 345 135 348 118)(62 174 164 461 331 121 349 119)(63 175 165 462 332 122 350 120)(64 176 151 463 333 123 351 106)(65 177 152 464 334 124 352 107)(66 178 153 465 335 125 353 108)(67 179 154 451 336 126 354 109)(68 180 155 452 337 127 355 110)(69 166 156 453 338 128 356 111)(70 167 157 454 339 129 357 112)(71 168 158 455 340 130 358 113)(72 169 159 456 341 131 359 114)(73 170 160 457 342 132 360 115)(74 171 161 458 343 133 346 116)(75 172 162 459 344 134 347 117)(136 426 272 184 440 319 254 467)(137 427 273 185 441 320 255 468)(138 428 274 186 442 321 241 469)(139 429 275 187 443 322 242 470)(140 430 276 188 444 323 243 471)(141 431 277 189 445 324 244 472)(142 432 278 190 446 325 245 473)(143 433 279 191 447 326 246 474)(144 434 280 192 448 327 247 475)(145 435 281 193 449 328 248 476)(146 421 282 194 450 329 249 477)(147 422 283 195 436 330 250 478)(148 423 284 181 437 316 251 479)(149 424 285 182 438 317 252 480)(150 425 271 183 439 318 253 466)
(1 442 359 53)(2 443 360 54)(3 444 346 55)(4 445 347 56)(5 446 348 57)(6 447 349 58)(7 448 350 59)(8 449 351 60)(9 450 352 46)(10 436 353 47)(11 437 354 48)(12 438 355 49)(13 439 356 50)(14 440 357 51)(15 441 358 52)(16 398 145 151)(17 399 146 152)(18 400 147 153)(19 401 148 154)(20 402 149 155)(21 403 150 156)(22 404 136 157)(23 405 137 158)(24 391 138 159)(25 392 139 160)(26 393 140 161)(27 394 141 162)(28 395 142 163)(29 396 143 164)(30 397 144 165)(31 194 464 412)(32 195 465 413)(33 181 451 414)(34 182 452 415)(35 183 453 416)(36 184 454 417)(37 185 455 418)(38 186 456 419)(39 187 457 420)(40 188 458 406)(41 189 459 407)(42 190 460 408)(43 191 461 409)(44 192 462 410)(45 193 463 411)(61 99 80 278)(62 100 81 279)(63 101 82 280)(64 102 83 281)(65 103 84 282)(66 104 85 283)(67 105 86 284)(68 91 87 285)(69 92 88 271)(70 93 89 272)(71 94 90 273)(72 95 76 274)(73 96 77 275)(74 97 78 276)(75 98 79 277)(106 211 363 476)(107 212 364 477)(108 213 365 478)(109 214 366 479)(110 215 367 480)(111 216 368 466)(112 217 369 467)(113 218 370 468)(114 219 371 469)(115 220 372 470)(116 221 373 471)(117 222 374 472)(118 223 375 473)(119 224 361 474)(120 225 362 475)(121 263 307 433)(122 264 308 434)(123 265 309 435)(124 266 310 421)(125 267 311 422)(126 268 312 423)(127 269 313 424)(128 270 314 425)(129 256 315 426)(130 257 301 427)(131 258 302 428)(132 259 303 429)(133 260 304 430)(134 261 305 431)(135 262 306 432)(166 229 377 318)(167 230 378 319)(168 231 379 320)(169 232 380 321)(170 233 381 322)(171 234 382 323)(172 235 383 324)(173 236 384 325)(174 237 385 326)(175 238 386 327)(176 239 387 328)(177 240 388 329)(178 226 389 330)(179 227 390 316)(180 228 376 317)(196 289 251 336)(197 290 252 337)(198 291 253 338)(199 292 254 339)(200 293 255 340)(201 294 241 341)(202 295 242 342)(203 296 243 343)(204 297 244 344)(205 298 245 345)(206 299 246 331)(207 300 247 332)(208 286 248 333)(209 287 249 334)(210 288 250 335)

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

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

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

210 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A ··· 6F 8A 8B 8C 8D 10A ··· 10L 12A 12B 12C 12D 12E ··· 12L 15A ··· 15H 20A ··· 20H 20I ··· 20X 24A ··· 24H 30A ··· 30X 40A ··· 40P 60A ··· 60P 60Q ··· 60AV 120A ··· 120AF order 1 2 2 2 3 3 4 4 4 4 4 4 5 5 5 5 6 ··· 6 8 8 8 8 10 ··· 10 12 12 12 12 12 ··· 12 15 ··· 15 20 ··· 20 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 120 ··· 120 size 1 1 1 1 1 1 2 2 4 4 4 4 1 1 1 1 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2

210 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + - + image C1 C2 C2 C3 C4 C5 C6 C6 C10 C10 C12 C15 C20 C30 C30 C60 Q8 D4 SD16 C3×Q8 C3×D4 C5×Q8 C5×D4 C3×SD16 C5×SD16 Q8×C15 D4×C15 C15×SD16 kernel C15×C4.Q8 C15×C4⋊C4 C2×C120 C5×C4.Q8 C120 C3×C4.Q8 C5×C4⋊C4 C2×C40 C3×C4⋊C4 C2×C24 C40 C4.Q8 C24 C4⋊C4 C2×C8 C8 C60 C2×C30 C30 C20 C2×C10 C12 C2×C6 C10 C6 C4 C22 C2 # reps 1 2 1 2 4 4 4 2 8 4 8 8 16 16 8 32 1 1 4 2 2 4 4 8 16 8 8 32

Matrix representation of C15×C4.Q8 in GL5(𝔽241)

 87 0 0 0 0 0 1 0 0 0 0 0 225 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 240 0 0 0 0 0 1 0 0 0 0 0 1 239 0 0 0 1 240
,
 240 0 0 0 0 0 1 0 0 0 0 0 240 0 0 0 0 0 0 38 0 0 0 222 38
,
 1 0 0 0 0 0 177 0 0 0 0 0 240 0 0 0 0 0 73 6 0 0 0 76 168

G:=sub<GL(5,GF(241))| [87,0,0,0,0,0,1,0,0,0,0,0,225,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,240,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,239,240],[240,0,0,0,0,0,1,0,0,0,0,0,240,0,0,0,0,0,0,222,0,0,0,38,38],[1,0,0,0,0,0,177,0,0,0,0,0,240,0,0,0,0,0,73,76,0,0,0,6,168] >;

C15×C4.Q8 in GAP, Magma, Sage, TeX

C_{15}\times C_4.Q_8
% in TeX

G:=Group("C15xC4.Q8");
// GroupNames label

G:=SmallGroup(480,209);
// by ID

G=gap.SmallGroup(480,209);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,428,10504,172]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

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