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G = C15×Q32order 480 = 25·3·5

Direct product of C15 and Q32

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C15×Q32, C16.C30, C80.2C6, Q16.C30, C240.5C2, C48.2C10, C30.62D8, C60.195D4, C120.105C22, C8.4(C2×C30), C4.3(D4×C15), C6.17(C5×D8), C2.5(C15×D8), C40.26(C2×C6), C10.17(C3×D8), C20.38(C3×D4), C12.38(C5×D4), (C5×Q16).2C6, C24.21(C2×C10), (C3×Q16).2C10, (C15×Q16).4C2, SmallGroup(480,216)

Series: Derived Chief Lower central Upper central

C1C8 — C15×Q32
C1C2C4C8C40C120C15×Q16 — C15×Q32
C1C2C4C8 — C15×Q32
C1C30C60C120 — C15×Q32

Generators and relations for C15×Q32
 G = < a,b,c | a15=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >

4C4
4C4
2Q8
2Q8
4C12
4C12
4C20
4C20
2C3×Q8
2C3×Q8
2C5×Q8
2C5×Q8
4C60
4C60
2Q8×C15
2Q8×C15

Smallest permutation representation of C15×Q32
Regular action on 480 points
Generators in S480
(1 386 99 65 472 454 301 412 175 228 147 113 445 243 272)(2 387 100 66 473 455 302 413 176 229 148 114 446 244 257)(3 388 101 67 474 456 303 414 161 230 149 115 447 245 258)(4 389 102 68 475 457 304 415 162 231 150 116 448 246 259)(5 390 103 69 476 458 289 416 163 232 151 117 433 247 260)(6 391 104 70 477 459 290 401 164 233 152 118 434 248 261)(7 392 105 71 478 460 291 402 165 234 153 119 435 249 262)(8 393 106 72 479 461 292 403 166 235 154 120 436 250 263)(9 394 107 73 480 462 293 404 167 236 155 121 437 251 264)(10 395 108 74 465 463 294 405 168 237 156 122 438 252 265)(11 396 109 75 466 464 295 406 169 238 157 123 439 253 266)(12 397 110 76 467 449 296 407 170 239 158 124 440 254 267)(13 398 111 77 468 450 297 408 171 240 159 125 441 255 268)(14 399 112 78 469 451 298 409 172 225 160 126 442 256 269)(15 400 97 79 470 452 299 410 173 226 145 127 443 241 270)(16 385 98 80 471 453 300 411 174 227 146 128 444 242 271)(17 197 323 36 134 352 378 223 84 59 283 314 353 190 429)(18 198 324 37 135 337 379 224 85 60 284 315 354 191 430)(19 199 325 38 136 338 380 209 86 61 285 316 355 192 431)(20 200 326 39 137 339 381 210 87 62 286 317 356 177 432)(21 201 327 40 138 340 382 211 88 63 287 318 357 178 417)(22 202 328 41 139 341 383 212 89 64 288 319 358 179 418)(23 203 329 42 140 342 384 213 90 49 273 320 359 180 419)(24 204 330 43 141 343 369 214 91 50 274 305 360 181 420)(25 205 331 44 142 344 370 215 92 51 275 306 361 182 421)(26 206 332 45 143 345 371 216 93 52 276 307 362 183 422)(27 207 333 46 144 346 372 217 94 53 277 308 363 184 423)(28 208 334 47 129 347 373 218 95 54 278 309 364 185 424)(29 193 335 48 130 348 374 219 96 55 279 310 365 186 425)(30 194 336 33 131 349 375 220 81 56 280 311 366 187 426)(31 195 321 34 132 350 376 221 82 57 281 312 367 188 427)(32 196 322 35 133 351 377 222 83 58 282 313 368 189 428)
(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)
(1 360 9 368)(2 359 10 367)(3 358 11 366)(4 357 12 365)(5 356 13 364)(6 355 14 363)(7 354 15 362)(8 353 16 361)(17 80 25 72)(18 79 26 71)(19 78 27 70)(20 77 28 69)(21 76 29 68)(22 75 30 67)(23 74 31 66)(24 73 32 65)(33 303 41 295)(34 302 42 294)(35 301 43 293)(36 300 44 292)(37 299 45 291)(38 298 46 290)(39 297 47 289)(40 296 48 304)(49 438 57 446)(50 437 58 445)(51 436 59 444)(52 435 60 443)(53 434 61 442)(54 433 62 441)(55 448 63 440)(56 447 64 439)(81 115 89 123)(82 114 90 122)(83 113 91 121)(84 128 92 120)(85 127 93 119)(86 126 94 118)(87 125 95 117)(88 124 96 116)(97 422 105 430)(98 421 106 429)(99 420 107 428)(100 419 108 427)(101 418 109 426)(102 417 110 425)(103 432 111 424)(104 431 112 423)(129 416 137 408)(130 415 138 407)(131 414 139 406)(132 413 140 405)(133 412 141 404)(134 411 142 403)(135 410 143 402)(136 409 144 401)(145 216 153 224)(146 215 154 223)(147 214 155 222)(148 213 156 221)(149 212 157 220)(150 211 158 219)(151 210 159 218)(152 209 160 217)(161 341 169 349)(162 340 170 348)(163 339 171 347)(164 338 172 346)(165 337 173 345)(166 352 174 344)(167 351 175 343)(168 350 176 342)(177 398 185 390)(178 397 186 389)(179 396 187 388)(180 395 188 387)(181 394 189 386)(182 393 190 385)(183 392 191 400)(184 391 192 399)(193 475 201 467)(194 474 202 466)(195 473 203 465)(196 472 204 480)(197 471 205 479)(198 470 206 478)(199 469 207 477)(200 468 208 476)(225 372 233 380)(226 371 234 379)(227 370 235 378)(228 369 236 377)(229 384 237 376)(230 383 238 375)(231 382 239 374)(232 381 240 373)(241 276 249 284)(242 275 250 283)(243 274 251 282)(244 273 252 281)(245 288 253 280)(246 287 254 279)(247 286 255 278)(248 285 256 277)(257 320 265 312)(258 319 266 311)(259 318 267 310)(260 317 268 309)(261 316 269 308)(262 315 270 307)(263 314 271 306)(264 313 272 305)(321 455 329 463)(322 454 330 462)(323 453 331 461)(324 452 332 460)(325 451 333 459)(326 450 334 458)(327 449 335 457)(328 464 336 456)

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

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

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

165 conjugacy classes

class 1  2 3A3B4A4B4C5A5B5C5D6A6B8A8B10A10B10C10D12A12B12C12D12E12F15A···15H16A16B16C16D20A20B20C20D20E···20L24A24B24C24D30A···30H40A···40H48A···48H60A···60H60I···60X80A···80P120A···120P240A···240AF
order1233444555566881010101012121212121215···15161616162020202020···202424242430···3040···4048···4860···6060···6080···80120···120240···240
size11112881111112211112288881···1222222228···822221···12···22···22···28···82···22···22···2

165 irreducible representations

dim111111111111222222222222
type+++++-
imageC1C2C2C3C5C6C6C10C10C15C30C30D4D8C3×D4Q32C5×D4C3×D8C5×D8C3×Q32D4×C15C5×Q32C15×D8C15×Q32
kernelC15×Q32C240C15×Q16C5×Q32C3×Q32C80C5×Q16C48C3×Q16Q32C16Q16C60C30C20C15C12C10C6C5C4C3C2C1
# reps1122424488816122444888161632

Matrix representation of C15×Q32 in GL2(𝔽31) generated by

190
019
,
2010
416
,
07
220
G:=sub<GL(2,GF(31))| [19,0,0,19],[20,4,10,16],[0,22,7,0] >;

C15×Q32 in GAP, Magma, Sage, TeX

C_{15}\times Q_{32}
% in TeX

G:=Group("C15xQ32");
// GroupNames label

G:=SmallGroup(480,216);
// by ID

G=gap.SmallGroup(480,216);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,1680,869,1688,6304,3161,242,15125,7572,124]);
// Polycyclic

G:=Group<a,b,c|a^15=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C15×Q32 in TeX

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