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G = C3×Dic5.Q8order 480 = 25·3·5

Direct product of C3 and Dic5.Q8

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

Aliases: C3×Dic5.Q8, C6.49(Q8×D5), C4⋊Dic5.6C6, C10.11(C6×Q8), C30.101(C2×Q8), (C3×Dic5).8Q8, Dic5.1(C3×Q8), (C2×C12).277D10, (C4×Dic5).11C6, C10.D4.5C6, C1518(C42.C2), C30.233(C4○D4), C6.118(C4○D20), (C2×C30).347C23, (C2×C60).420C22, (C12×Dic5).25C2, C6.114(D42D5), (C6×Dic5).239C22, C2.5(C3×Q8×D5), (C5×C4⋊C4).6C6, C4⋊C4.5(C3×D5), (C2×C4).8(C6×D5), C52(C3×C42.C2), (C3×C4⋊C4).12D5, (C15×C4⋊C4).13C2, C22.47(D5×C2×C6), (C2×C20).53(C2×C6), C10.10(C3×C4○D4), C2.13(C3×C4○D20), C2.11(C3×D42D5), (C3×C4⋊Dic5).20C2, (C2×C10).30(C22×C6), (C2×Dic5).10(C2×C6), (C2×C6).343(C22×D5), (C3×C10.D4).15C2, SmallGroup(480,682)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×Dic5.Q8
C1C5C10C2×C10C2×C30C6×Dic5C12×Dic5 — C3×Dic5.Q8
C5C2×C10 — C3×Dic5.Q8
C1C2×C6C3×C4⋊C4

Generators and relations for C3×Dic5.Q8
 G = < a,b,c,d,e | a3=b10=d4=1, c2=b5, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b5c, ce=ec, ede-1=b5d-1 >

Subgroups: 288 in 112 conjugacy classes, 62 normal (58 characteristic)
C1, C2 [×3], C3, C4 [×8], C22, C5, C6 [×3], C2×C4 [×3], C2×C4 [×4], C10 [×3], C12 [×8], C2×C6, C15, C42, C4⋊C4, C4⋊C4 [×5], Dic5 [×2], Dic5 [×3], C20 [×3], C2×C10, C2×C12 [×3], C2×C12 [×4], C30 [×3], C42.C2, C2×Dic5 [×4], C2×C20 [×3], C4×C12, C3×C4⋊C4, C3×C4⋊C4 [×5], C3×Dic5 [×2], C3×Dic5 [×3], C60 [×3], C2×C30, C4×Dic5, C10.D4 [×4], C4⋊Dic5, C5×C4⋊C4, C3×C42.C2, C6×Dic5 [×4], C2×C60 [×3], Dic5.Q8, C12×Dic5, C3×C10.D4 [×4], C3×C4⋊Dic5, C15×C4⋊C4, C3×Dic5.Q8
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], Q8 [×2], C23, D5, C2×C6 [×7], C2×Q8, C4○D4 [×2], D10 [×3], C3×Q8 [×2], C22×C6, C3×D5, C42.C2, C22×D5, C6×Q8, C3×C4○D4 [×2], C6×D5 [×3], C4○D20, D42D5, Q8×D5, C3×C42.C2, D5×C2×C6, Dic5.Q8, C3×C4○D20, C3×D42D5, C3×Q8×D5, C3×Dic5.Q8

Smallest permutation representation of C3×Dic5.Q8
Regular action on 480 points
Generators in S480
(1 62 42)(2 63 43)(3 64 44)(4 65 45)(5 66 46)(6 67 47)(7 68 48)(8 69 49)(9 70 50)(10 61 41)(11 468 448)(12 469 449)(13 470 450)(14 461 441)(15 462 442)(16 463 443)(17 464 444)(18 465 445)(19 466 446)(20 467 447)(21 53 33)(22 54 34)(23 55 35)(24 56 36)(25 57 37)(26 58 38)(27 59 39)(28 60 40)(29 51 31)(30 52 32)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)(81 121 101)(82 122 102)(83 123 103)(84 124 104)(85 125 105)(86 126 106)(87 127 107)(88 128 108)(89 129 109)(90 130 110)(131 171 151)(132 172 152)(133 173 153)(134 174 154)(135 175 155)(136 176 156)(137 177 157)(138 178 158)(139 179 159)(140 180 160)(141 181 161)(142 182 162)(143 183 163)(144 184 164)(145 185 165)(146 186 166)(147 187 167)(148 188 168)(149 189 169)(150 190 170)(191 231 211)(192 232 212)(193 233 213)(194 234 214)(195 235 215)(196 236 216)(197 237 217)(198 238 218)(199 239 219)(200 240 220)(201 244 221)(202 245 222)(203 246 223)(204 247 224)(205 248 225)(206 249 226)(207 250 227)(208 241 228)(209 242 229)(210 243 230)(251 291 271)(252 292 272)(253 293 273)(254 294 274)(255 295 275)(256 296 276)(257 297 277)(258 298 278)(259 299 279)(260 300 280)(261 301 281)(262 302 282)(263 303 283)(264 304 284)(265 305 285)(266 306 286)(267 307 287)(268 308 288)(269 309 289)(270 310 290)(311 351 331)(312 352 332)(313 353 333)(314 354 334)(315 355 335)(316 356 336)(317 357 337)(318 358 338)(319 359 339)(320 360 340)(321 361 341)(322 362 342)(323 363 343)(324 364 344)(325 365 345)(326 366 346)(327 367 347)(328 368 348)(329 369 349)(330 370 350)(371 411 391)(372 412 392)(373 413 393)(374 414 394)(375 415 395)(376 416 396)(377 417 397)(378 418 398)(379 419 399)(380 420 400)(381 421 401)(382 422 402)(383 423 403)(384 424 404)(385 425 405)(386 426 406)(387 427 407)(388 428 408)(389 429 409)(390 430 410)(431 471 451)(432 472 452)(433 473 453)(434 474 454)(435 475 455)(436 476 456)(437 477 457)(438 478 458)(439 479 459)(440 480 460)
(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)
(1 379 6 374)(2 378 7 373)(3 377 8 372)(4 376 9 371)(5 375 10 380)(11 111 16 116)(12 120 17 115)(13 119 18 114)(14 118 19 113)(15 117 20 112)(21 381 26 386)(22 390 27 385)(23 389 28 384)(24 388 29 383)(25 387 30 382)(31 403 36 408)(32 402 37 407)(33 401 38 406)(34 410 39 405)(35 409 40 404)(41 400 46 395)(42 399 47 394)(43 398 48 393)(44 397 49 392)(45 396 50 391)(51 423 56 428)(52 422 57 427)(53 421 58 426)(54 430 59 425)(55 429 60 424)(61 420 66 415)(62 419 67 414)(63 418 68 413)(64 417 69 412)(65 416 70 411)(71 443 76 448)(72 442 77 447)(73 441 78 446)(74 450 79 445)(75 449 80 444)(81 440 86 435)(82 439 87 434)(83 438 88 433)(84 437 89 432)(85 436 90 431)(91 463 96 468)(92 462 97 467)(93 461 98 466)(94 470 99 465)(95 469 100 464)(101 460 106 455)(102 459 107 454)(103 458 108 453)(104 457 109 452)(105 456 110 451)(121 480 126 475)(122 479 127 474)(123 478 128 473)(124 477 129 472)(125 476 130 471)(131 259 136 254)(132 258 137 253)(133 257 138 252)(134 256 139 251)(135 255 140 260)(141 264 146 269)(142 263 147 268)(143 262 148 267)(144 261 149 266)(145 270 150 265)(151 279 156 274)(152 278 157 273)(153 277 158 272)(154 276 159 271)(155 275 160 280)(161 284 166 289)(162 283 167 288)(163 282 168 287)(164 281 169 286)(165 290 170 285)(171 299 176 294)(172 298 177 293)(173 297 178 292)(174 296 179 291)(175 295 180 300)(181 304 186 309)(182 303 187 308)(183 302 188 307)(184 301 189 306)(185 310 190 305)(191 319 196 314)(192 318 197 313)(193 317 198 312)(194 316 199 311)(195 315 200 320)(201 324 206 329)(202 323 207 328)(203 322 208 327)(204 321 209 326)(205 330 210 325)(211 339 216 334)(212 338 217 333)(213 337 218 332)(214 336 219 331)(215 335 220 340)(221 344 226 349)(222 343 227 348)(223 342 228 347)(224 341 229 346)(225 350 230 345)(231 359 236 354)(232 358 237 353)(233 357 238 352)(234 356 239 351)(235 355 240 360)(241 367 246 362)(242 366 247 361)(243 365 248 370)(244 364 249 369)(245 363 250 368)
(1 76 24 82)(2 77 25 83)(3 78 26 84)(4 79 27 85)(5 80 28 86)(6 71 29 87)(7 72 30 88)(8 73 21 89)(9 74 22 90)(10 75 23 81)(11 423 479 414)(12 424 480 415)(13 425 471 416)(14 426 472 417)(15 427 473 418)(16 428 474 419)(17 429 475 420)(18 430 476 411)(19 421 477 412)(20 422 478 413)(31 107 47 91)(32 108 48 92)(33 109 49 93)(34 110 50 94)(35 101 41 95)(36 102 42 96)(37 103 43 97)(38 104 44 98)(39 105 45 99)(40 106 46 100)(51 127 67 111)(52 128 68 112)(53 129 69 113)(54 130 70 114)(55 121 61 115)(56 122 62 116)(57 123 63 117)(58 124 64 118)(59 125 65 119)(60 126 66 120)(131 191 147 207)(132 192 148 208)(133 193 149 209)(134 194 150 210)(135 195 141 201)(136 196 142 202)(137 197 143 203)(138 198 144 204)(139 199 145 205)(140 200 146 206)(151 211 167 227)(152 212 168 228)(153 213 169 229)(154 214 170 230)(155 215 161 221)(156 216 162 222)(157 217 163 223)(158 218 164 224)(159 219 165 225)(160 220 166 226)(171 231 187 250)(172 232 188 241)(173 233 189 242)(174 234 190 243)(175 235 181 244)(176 236 182 245)(177 237 183 246)(178 238 184 247)(179 239 185 248)(180 240 186 249)(251 316 270 325)(252 317 261 326)(253 318 262 327)(254 319 263 328)(255 320 264 329)(256 311 265 330)(257 312 266 321)(258 313 267 322)(259 314 268 323)(260 315 269 324)(271 336 290 345)(272 337 281 346)(273 338 282 347)(274 339 283 348)(275 340 284 349)(276 331 285 350)(277 332 286 341)(278 333 287 342)(279 334 288 343)(280 335 289 344)(291 356 310 365)(292 357 301 366)(293 358 302 367)(294 359 303 368)(295 360 304 369)(296 351 305 370)(297 352 306 361)(298 353 307 362)(299 354 308 363)(300 355 309 364)(371 445 390 436)(372 446 381 437)(373 447 382 438)(374 448 383 439)(375 449 384 440)(376 450 385 431)(377 441 386 432)(378 442 387 433)(379 443 388 434)(380 444 389 435)(391 465 410 456)(392 466 401 457)(393 467 402 458)(394 468 403 459)(395 469 404 460)(396 470 405 451)(397 461 406 452)(398 462 407 453)(399 463 408 454)(400 464 409 455)
(1 254 24 263)(2 253 25 262)(3 252 26 261)(4 251 27 270)(5 260 28 269)(6 259 29 268)(7 258 30 267)(8 257 21 266)(9 256 22 265)(10 255 23 264)(11 250 479 231)(12 249 480 240)(13 248 471 239)(14 247 472 238)(15 246 473 237)(16 245 474 236)(17 244 475 235)(18 243 476 234)(19 242 477 233)(20 241 478 232)(31 288 47 279)(32 287 48 278)(33 286 49 277)(34 285 50 276)(35 284 41 275)(36 283 42 274)(37 282 43 273)(38 281 44 272)(39 290 45 271)(40 289 46 280)(51 308 67 299)(52 307 68 298)(53 306 69 297)(54 305 70 296)(55 304 61 295)(56 303 62 294)(57 302 63 293)(58 301 64 292)(59 310 65 291)(60 309 66 300)(71 328 87 319)(72 327 88 318)(73 326 89 317)(74 325 90 316)(75 324 81 315)(76 323 82 314)(77 322 83 313)(78 321 84 312)(79 330 85 311)(80 329 86 320)(91 348 107 339)(92 347 108 338)(93 346 109 337)(94 345 110 336)(95 344 101 335)(96 343 102 334)(97 342 103 333)(98 341 104 332)(99 350 105 331)(100 349 106 340)(111 368 127 359)(112 367 128 358)(113 366 129 357)(114 365 130 356)(115 364 121 355)(116 363 122 354)(117 362 123 353)(118 361 124 352)(119 370 125 351)(120 369 126 360)(131 388 147 379)(132 387 148 378)(133 386 149 377)(134 385 150 376)(135 384 141 375)(136 383 142 374)(137 382 143 373)(138 381 144 372)(139 390 145 371)(140 389 146 380)(151 408 167 399)(152 407 168 398)(153 406 169 397)(154 405 170 396)(155 404 161 395)(156 403 162 394)(157 402 163 393)(158 401 164 392)(159 410 165 391)(160 409 166 400)(171 428 187 419)(172 427 188 418)(173 426 189 417)(174 425 190 416)(175 424 181 415)(176 423 182 414)(177 422 183 413)(178 421 184 412)(179 430 185 411)(180 429 186 420)(191 448 207 439)(192 447 208 438)(193 446 209 437)(194 445 210 436)(195 444 201 435)(196 443 202 434)(197 442 203 433)(198 441 204 432)(199 450 205 431)(200 449 206 440)(211 468 227 459)(212 467 228 458)(213 466 229 457)(214 465 230 456)(215 464 221 455)(216 463 222 454)(217 462 223 453)(218 461 224 452)(219 470 225 451)(220 469 226 460)

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

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

102 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H4I4J5A5B6A···6F10A···10F12A12B12C12D12E12F12G12H12I···12P12Q12R12S12T15A15B15C15D20A···20L30A···30L60A···60X
order1222334444444444556···610···10121212121212121212···12121212121515151520···2030···3060···60
size1111112244101010102020221···12···22222444410···102020202022224···42···24···4

102 irreducible representations

dim111111111122222222224444
type+++++-++--
imageC1C2C2C2C2C3C6C6C6C6Q8D5C4○D4D10C3×Q8C3×D5C3×C4○D4C6×D5C4○D20C3×C4○D20D42D5Q8×D5C3×D42D5C3×Q8×D5
kernelC3×Dic5.Q8C12×Dic5C3×C10.D4C3×C4⋊Dic5C15×C4⋊C4Dic5.Q8C4×Dic5C10.D4C4⋊Dic5C5×C4⋊C4C3×Dic5C3×C4⋊C4C30C2×C12Dic5C4⋊C4C10C2×C4C6C2C6C6C2C2
# reps11411228222246448128162244

Matrix representation of C3×Dic5.Q8 in GL4(𝔽61) generated by

13000
01300
00470
00047
,
18100
60000
0010
0001
,
25800
225900
0010
0001
,
304400
173100
001559
005246
,
393300
22200
00533
00198
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,47,0,0,0,0,47],[18,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[2,22,0,0,58,59,0,0,0,0,1,0,0,0,0,1],[30,17,0,0,44,31,0,0,0,0,15,52,0,0,59,46],[39,2,0,0,33,22,0,0,0,0,53,19,0,0,3,8] >;

C3×Dic5.Q8 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_5.Q_8
% in TeX

G:=Group("C3xDic5.Q8");
// GroupNames label

G:=SmallGroup(480,682);
// by ID

G=gap.SmallGroup(480,682);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,176,590,555,268,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^10=d^4=1,c^2=b^5,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e^-1=b^5*d^-1>;
// generators/relations

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