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## G = C3×C10.Q16order 480 = 25·3·5

### Direct product of C3 and C10.Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×C10.Q16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C6×Dic10 — C3×C10.Q16
 Lower central C5 — C10 — C20 — C3×C10.Q16
 Upper central C1 — C2×C6 — C2×C12 — C3×C4⋊C4

Generators and relations for C3×C10.Q16
G = < a,b,c,d | a3=b10=c8=1, d2=b5c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b5c-1 >

Subgroups: 240 in 84 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C10 [×3], C12 [×2], C12 [×3], C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5 [×2], C20 [×2], C20, C2×C10, C24, C2×C12, C2×C12 [×2], C3×Q8 [×3], C30 [×3], Q8⋊C4, C52C8, Dic10 [×2], Dic10, C2×Dic5, C2×C20, C2×C20, C3×C4⋊C4, C2×C24, C6×Q8, C3×Dic5 [×2], C60 [×2], C60, C2×C30, C2×C52C8, C5×C4⋊C4, C2×Dic10, C3×Q8⋊C4, C3×C52C8, C3×Dic10 [×2], C3×Dic10, C6×Dic5, C2×C60, C2×C60, C10.Q16, C6×C52C8, C15×C4⋊C4, C6×Dic10, C3×C10.Q16
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, SD16, Q16, D10, C2×C12, C3×D4 [×2], C3×D5, Q8⋊C4, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C3×SD16, C3×Q16, C6×D5, D10⋊C4, D4.D5, C5⋊Q16, C3×Q8⋊C4, D5×C12, C3×D20, C3×C5⋊D4, C10.Q16, C3×D10⋊C4, C3×D4.D5, C3×C5⋊Q16, C3×C10.Q16

Smallest permutation representation of C3×C10.Q16
Regular action on 480 points
Generators in S480
(1 117 77)(2 118 78)(3 119 79)(4 120 80)(5 111 71)(6 112 72)(7 113 73)(8 114 74)(9 115 75)(10 116 76)(11 109 69)(12 110 70)(13 101 61)(14 102 62)(15 103 63)(16 104 64)(17 105 65)(18 106 66)(19 107 67)(20 108 68)(21 444 404)(22 445 405)(23 446 406)(24 447 407)(25 448 408)(26 449 409)(27 450 410)(28 441 401)(29 442 402)(30 443 403)(31 93 53)(32 94 54)(33 95 55)(34 96 56)(35 97 57)(36 98 58)(37 99 59)(38 100 60)(39 91 51)(40 92 52)(41 121 81)(42 122 82)(43 123 83)(44 124 84)(45 125 85)(46 126 86)(47 127 87)(48 128 88)(49 129 89)(50 130 90)(131 211 171)(132 212 172)(133 213 173)(134 214 174)(135 215 175)(136 216 176)(137 217 177)(138 218 178)(139 219 179)(140 220 180)(141 221 181)(142 222 182)(143 223 183)(144 224 184)(145 225 185)(146 226 186)(147 227 187)(148 228 188)(149 229 189)(150 230 190)(151 231 191)(152 232 192)(153 233 193)(154 234 194)(155 235 195)(156 236 196)(157 237 197)(158 238 198)(159 239 199)(160 240 200)(161 250 201)(162 241 202)(163 242 203)(164 243 204)(165 244 205)(166 245 206)(167 246 207)(168 247 208)(169 248 209)(170 249 210)(251 331 291)(252 332 292)(253 333 293)(254 334 294)(255 335 295)(256 336 296)(257 337 297)(258 338 298)(259 339 299)(260 340 300)(261 341 301)(262 342 302)(263 343 303)(264 344 304)(265 345 305)(266 346 306)(267 347 307)(268 348 308)(269 349 309)(270 350 310)(271 351 311)(272 352 312)(273 353 313)(274 354 314)(275 355 315)(276 356 316)(277 357 317)(278 358 318)(279 359 319)(280 360 320)(281 361 321)(282 362 322)(283 363 323)(284 364 324)(285 365 325)(286 366 326)(287 367 327)(288 368 328)(289 369 329)(290 370 330)(371 451 411)(372 452 412)(373 453 413)(374 454 414)(375 455 415)(376 456 416)(377 457 417)(378 458 418)(379 459 419)(380 460 420)(381 461 421)(382 462 422)(383 463 423)(384 464 424)(385 465 425)(386 466 426)(387 467 427)(388 468 428)(389 469 429)(390 470 430)(391 471 431)(392 472 432)(393 473 433)(394 474 434)(395 475 435)(396 476 436)(397 477 437)(398 478 438)(399 479 439)(400 480 440)
(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)
(1 410 45 388 14 375 38 393)(2 409 46 387 15 374 39 392)(3 408 47 386 16 373 40 391)(4 407 48 385 17 372 31 400)(5 406 49 384 18 371 32 399)(6 405 50 383 19 380 33 398)(7 404 41 382 20 379 34 397)(8 403 42 381 11 378 35 396)(9 402 43 390 12 377 36 395)(10 401 44 389 13 376 37 394)(21 121 462 108 459 96 477 113)(22 130 463 107 460 95 478 112)(23 129 464 106 451 94 479 111)(24 128 465 105 452 93 480 120)(25 127 466 104 453 92 471 119)(26 126 467 103 454 91 472 118)(27 125 468 102 455 100 473 117)(28 124 469 101 456 99 474 116)(29 123 470 110 457 98 475 115)(30 122 461 109 458 97 476 114)(51 432 78 449 86 427 63 414)(52 431 79 448 87 426 64 413)(53 440 80 447 88 425 65 412)(54 439 71 446 89 424 66 411)(55 438 72 445 90 423 67 420)(56 437 73 444 81 422 68 419)(57 436 74 443 82 421 69 418)(58 435 75 442 83 430 70 417)(59 434 76 441 84 429 61 416)(60 433 77 450 85 428 62 415)(131 277 143 259 166 262 158 284)(132 276 144 258 167 261 159 283)(133 275 145 257 168 270 160 282)(134 274 146 256 169 269 151 281)(135 273 147 255 170 268 152 290)(136 272 148 254 161 267 153 289)(137 271 149 253 162 266 154 288)(138 280 150 252 163 265 155 287)(139 279 141 251 164 264 156 286)(140 278 142 260 165 263 157 285)(171 317 183 299 206 302 198 324)(172 316 184 298 207 301 199 323)(173 315 185 297 208 310 200 322)(174 314 186 296 209 309 191 321)(175 313 187 295 210 308 192 330)(176 312 188 294 201 307 193 329)(177 311 189 293 202 306 194 328)(178 320 190 292 203 305 195 327)(179 319 181 291 204 304 196 326)(180 318 182 300 205 303 197 325)(211 357 223 339 245 342 238 364)(212 356 224 338 246 341 239 363)(213 355 225 337 247 350 240 362)(214 354 226 336 248 349 231 361)(215 353 227 335 249 348 232 370)(216 352 228 334 250 347 233 369)(217 351 229 333 241 346 234 368)(218 360 230 332 242 345 235 367)(219 359 221 331 243 344 236 366)(220 358 222 340 244 343 237 365)
(1 147 19 157)(2 148 20 158)(3 149 11 159)(4 150 12 160)(5 141 13 151)(6 142 14 152)(7 143 15 153)(8 144 16 154)(9 145 17 155)(10 146 18 156)(21 352 454 342)(22 353 455 343)(23 354 456 344)(24 355 457 345)(25 356 458 346)(26 357 459 347)(27 358 460 348)(28 359 451 349)(29 360 452 350)(30 351 453 341)(31 163 43 133)(32 164 44 134)(33 165 45 135)(34 166 46 136)(35 167 47 137)(36 168 48 138)(37 169 49 139)(38 170 50 140)(39 161 41 131)(40 162 42 132)(51 201 81 171)(52 202 82 172)(53 203 83 173)(54 204 84 174)(55 205 85 175)(56 206 86 176)(57 207 87 177)(58 208 88 178)(59 209 89 179)(60 210 90 180)(61 191 71 181)(62 192 72 182)(63 193 73 183)(64 194 74 184)(65 195 75 185)(66 196 76 186)(67 197 77 187)(68 198 78 188)(69 199 79 189)(70 200 80 190)(91 250 121 211)(92 241 122 212)(93 242 123 213)(94 243 124 214)(95 244 125 215)(96 245 126 216)(97 246 127 217)(98 247 128 218)(99 248 129 219)(100 249 130 220)(101 231 111 221)(102 232 112 222)(103 233 113 223)(104 234 114 224)(105 235 115 225)(106 236 116 226)(107 237 117 227)(108 238 118 228)(109 239 119 229)(110 240 120 230)(251 384 281 394)(252 385 282 395)(253 386 283 396)(254 387 284 397)(255 388 285 398)(256 389 286 399)(257 390 287 400)(258 381 288 391)(259 382 289 392)(260 383 290 393)(261 403 271 373)(262 404 272 374)(263 405 273 375)(264 406 274 376)(265 407 275 377)(266 408 276 378)(267 409 277 379)(268 410 278 380)(269 401 279 371)(270 402 280 372)(291 424 321 434)(292 425 322 435)(293 426 323 436)(294 427 324 437)(295 428 325 438)(296 429 326 439)(297 430 327 440)(298 421 328 431)(299 422 329 432)(300 423 330 433)(301 443 311 413)(302 444 312 414)(303 445 313 415)(304 446 314 416)(305 447 315 417)(306 448 316 418)(307 449 317 419)(308 450 318 420)(309 441 319 411)(310 442 320 412)(331 464 361 474)(332 465 362 475)(333 466 363 476)(334 467 364 477)(335 468 365 478)(336 469 366 479)(337 470 367 480)(338 461 368 471)(339 462 369 472)(340 463 370 473)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 6A ··· 6F 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 15A 15B 15C 15D 20A ··· 20L 24A ··· 24H 30A ··· 30L 60A ··· 60X order 1 2 2 2 3 3 4 4 4 4 4 4 5 5 6 ··· 6 8 8 8 8 10 ··· 10 12 12 12 12 12 12 12 12 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 1 1 2 2 4 4 20 20 2 2 1 ··· 1 10 10 10 10 2 ··· 2 2 2 2 2 4 4 4 4 20 20 20 20 2 2 2 2 4 ··· 4 10 ··· 10 2 ··· 2 4 ··· 4

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + - - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D4 D5 SD16 Q16 D10 C3×D4 C3×D4 C3×D5 C4×D5 D20 C5⋊D4 C3×SD16 C3×Q16 C6×D5 D5×C12 C3×D20 C3×C5⋊D4 D4.D5 C5⋊Q16 C3×D4.D5 C3×C5⋊Q16 kernel C3×C10.Q16 C6×C5⋊2C8 C15×C4⋊C4 C6×Dic10 C10.Q16 C3×Dic10 C2×C5⋊2C8 C5×C4⋊C4 C2×Dic10 Dic10 C60 C2×C30 C3×C4⋊C4 C30 C30 C2×C12 C20 C2×C10 C4⋊C4 C12 C12 C2×C6 C10 C10 C2×C4 C4 C4 C22 C6 C6 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 8 2 2 4 4

Matrix representation of C3×C10.Q16 in GL5(𝔽241)

 1 0 0 0 0 0 225 0 0 0 0 0 225 0 0 0 0 0 1 0 0 0 0 0 1
,
 240 0 0 0 0 0 0 189 0 0 0 51 51 0 0 0 0 0 1 0 0 0 0 0 1
,
 64 0 0 0 0 0 128 117 0 0 0 68 113 0 0 0 0 0 11 230 0 0 0 11 11
,
 64 0 0 0 0 0 76 103 0 0 0 89 165 0 0 0 0 0 194 21 0 0 0 21 47

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,225,0,0,0,0,0,225,0,0,0,0,0,1,0,0,0,0,0,1],[240,0,0,0,0,0,0,51,0,0,0,189,51,0,0,0,0,0,1,0,0,0,0,0,1],[64,0,0,0,0,0,128,68,0,0,0,117,113,0,0,0,0,0,11,11,0,0,0,230,11],[64,0,0,0,0,0,76,89,0,0,0,103,165,0,0,0,0,0,194,21,0,0,0,21,47] >;

C3×C10.Q16 in GAP, Magma, Sage, TeX

C_3\times C_{10}.Q_{16}
% in TeX

G:=Group("C3xC10.Q16");
// GroupNames label

G:=SmallGroup(480,88);
// by ID

G=gap.SmallGroup(480,88);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,336,365,92,2524,1271,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^10=c^8=1,d^2=b^5*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=b^5*c^-1>;
// generators/relations

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