direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×C3⋊Q16, C30⋊9Q16, C60.153D4, C60.232C23, C6⋊2(C5×Q16), C3⋊3(C10×Q16), C15⋊18(C2×Q16), C12.20(C5×D4), C6.55(D4×C10), (C6×Q8).3C10, (C5×Q8).56D6, (C2×C20).365D6, (C2×C30).186D4, C30.438(C2×D4), Q8.12(S3×C10), (Q8×C30).13C2, (Q8×C10).12S3, C20.74(C3⋊D4), (C2×Dic6).8C10, C20.205(C22×S3), C12.16(C22×C10), (C2×C60).366C22, Dic6.10(C2×C10), (C10×Dic6).18C2, (Q8×C15).50C22, (C5×Dic6).52C22, (C2×C3⋊C8).6C10, C3⋊C8.9(C2×C10), C4.16(S3×C2×C10), C4.9(C5×C3⋊D4), (C10×C3⋊C8).18C2, (C2×C6).43(C5×D4), (C2×Q8).5(C5×S3), (C2×C4).54(S3×C10), C2.19(C10×C3⋊D4), (C5×C3⋊C8).45C22, (C3×Q8).7(C2×C10), (C2×C12).39(C2×C10), C10.140(C2×C3⋊D4), C22.24(C5×C3⋊D4), (C2×C10).96(C3⋊D4), SmallGroup(480,822)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C3⋊Q16
G = < a,b,c,d | a10=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 228 in 120 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, C10, C10, Dic3, C12, C12, C2×C6, C15, C2×C8, Q16, C2×Q8, C2×Q8, C20, C20, C2×C10, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C30, C30, C2×Q16, C40, C2×C20, C2×C20, C5×Q8, C5×Q8, C2×C3⋊C8, C3⋊Q16, C2×Dic6, C6×Q8, C5×Dic3, C60, C60, C2×C30, C2×C40, C5×Q16, Q8×C10, Q8×C10, C2×C3⋊Q16, C5×C3⋊C8, C5×Dic6, C5×Dic6, C10×Dic3, C2×C60, C2×C60, Q8×C15, Q8×C15, C10×Q16, C10×C3⋊C8, C5×C3⋊Q16, C10×Dic6, Q8×C30, C10×C3⋊Q16
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, Q16, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C2×Q16, C5×D4, C22×C10, C3⋊Q16, C2×C3⋊D4, S3×C10, C5×Q16, D4×C10, C2×C3⋊Q16, C5×C3⋊D4, S3×C2×C10, C10×Q16, C5×C3⋊Q16, C10×C3⋊D4, C10×C3⋊Q16
►Regular action on 480 points
Generators in S480
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170)(171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190)(191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230)(231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250)(251 252 253 254 255 256 257 258 259 260)(261 262 263 264 265 266 267 268 269 270)(271 272 273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288 289 290)(291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310)(311 312 313 314 315 316 317 318 319 320)(321 322 323 324 325 326 327 328 329 330)(331 332 333 334 335 336 337 338 339 340)(341 342 343 344 345 346 347 348 349 350)(351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370)(371 372 373 374 375 376 377 378 379 380)(381 382 383 384 385 386 387 388 389 390)(391 392 393 394 395 396 397 398 399 400)(401 402 403 404 405 406 407 408 409 410)(411 412 413 414 415 416 417 418 419 420)(421 422 423 424 425 426 427 428 429 430)(431 432 433 434 435 436 437 438 439 440)(441 442 443 444 445 446 447 448 449 450)(451 452 453 454 455 456 457 458 459 460)(461 462 463 464 465 466 467 468 469 470)(471 472 473 474 475 476 477 478 479 480)
(1 365 75)(2 366 76)(3 367 77)(4 368 78)(5 369 79)(6 370 80)(7 361 71)(8 362 72)(9 363 73)(10 364 74)(11 468 440)(12 469 431)(13 470 432)(14 461 433)(15 462 434)(16 463 435)(17 464 436)(18 465 437)(19 466 438)(20 467 439)(21 149 450)(22 150 441)(23 141 442)(24 142 443)(25 143 444)(26 144 445)(27 145 446)(28 146 447)(29 147 448)(30 148 449)(31 416 332)(32 417 333)(33 418 334)(34 419 335)(35 420 336)(36 411 337)(37 412 338)(38 413 339)(39 414 340)(40 415 331)(41 342 163)(42 343 164)(43 344 165)(44 345 166)(45 346 167)(46 347 168)(47 348 169)(48 349 170)(49 350 161)(50 341 162)(51 219 409)(52 220 410)(53 211 401)(54 212 402)(55 213 403)(56 214 404)(57 215 405)(58 216 406)(59 217 407)(60 218 408)(61 183 83)(62 184 84)(63 185 85)(64 186 86)(65 187 87)(66 188 88)(67 189 89)(68 190 90)(69 181 81)(70 182 82)(91 175 293)(92 176 294)(93 177 295)(94 178 296)(95 179 297)(96 180 298)(97 171 299)(98 172 300)(99 173 291)(100 174 292)(101 385 375)(102 386 376)(103 387 377)(104 388 378)(105 389 379)(106 390 380)(107 381 371)(108 382 372)(109 383 373)(110 384 374)(111 239 243)(112 240 244)(113 231 245)(114 232 246)(115 233 247)(116 234 248)(117 235 249)(118 236 250)(119 237 241)(120 238 242)(121 478 360)(122 479 351)(123 480 352)(124 471 353)(125 472 354)(126 473 355)(127 474 356)(128 475 357)(129 476 358)(130 477 359)(131 264 320)(132 265 311)(133 266 312)(134 267 313)(135 268 314)(136 269 315)(137 270 316)(138 261 317)(139 262 318)(140 263 319)(151 452 275)(152 453 276)(153 454 277)(154 455 278)(155 456 279)(156 457 280)(157 458 271)(158 459 272)(159 460 273)(160 451 274)(191 229 395)(192 230 396)(193 221 397)(194 222 398)(195 223 399)(196 224 400)(197 225 391)(198 226 392)(199 227 393)(200 228 394)(201 283 259)(202 284 260)(203 285 251)(204 286 252)(205 287 253)(206 288 254)(207 289 255)(208 290 256)(209 281 257)(210 282 258)(301 323 423)(302 324 424)(303 325 425)(304 326 426)(305 327 427)(306 328 428)(307 329 429)(308 330 430)(309 321 421)(310 322 422)
(1 467 181 170 243 30 91 451)(2 468 182 161 244 21 92 452)(3 469 183 162 245 22 93 453)(4 470 184 163 246 23 94 454)(5 461 185 164 247 24 95 455)(6 462 186 165 248 25 96 456)(7 463 187 166 249 26 97 457)(8 464 188 167 250 27 98 458)(9 465 189 168 241 28 99 459)(10 466 190 169 242 29 100 460)(11 82 350 112 450 176 151 366)(12 83 341 113 441 177 152 367)(13 84 342 114 442 178 153 368)(14 85 343 115 443 179 154 369)(15 86 344 116 444 180 155 370)(16 87 345 117 445 171 156 361)(17 88 346 118 446 172 157 362)(18 89 347 119 447 173 158 363)(19 90 348 120 448 174 159 364)(20 81 349 111 449 175 160 365)(31 200 480 54 310 282 316 384)(32 191 471 55 301 283 317 385)(33 192 472 56 302 284 318 386)(34 193 473 57 303 285 319 387)(35 194 474 58 304 286 320 388)(36 195 475 59 305 287 311 389)(37 196 476 60 306 288 312 390)(38 197 477 51 307 289 313 381)(39 198 478 52 308 290 314 382)(40 199 479 53 309 281 315 383)(41 232 141 296 277 78 432 62)(42 233 142 297 278 79 433 63)(43 234 143 298 279 80 434 64)(44 235 144 299 280 71 435 65)(45 236 145 300 271 72 436 66)(46 237 146 291 272 73 437 67)(47 238 147 292 273 74 438 68)(48 239 148 293 274 75 439 69)(49 240 149 294 275 76 440 70)(50 231 150 295 276 77 431 61)(101 417 395 353 403 323 201 138)(102 418 396 354 404 324 202 139)(103 419 397 355 405 325 203 140)(104 420 398 356 406 326 204 131)(105 411 399 357 407 327 205 132)(106 412 400 358 408 328 206 133)(107 413 391 359 409 329 207 134)(108 414 392 360 410 330 208 135)(109 415 393 351 401 321 209 136)(110 416 394 352 402 322 210 137)(121 220 430 256 268 372 340 226)(122 211 421 257 269 373 331 227)(123 212 422 258 270 374 332 228)(124 213 423 259 261 375 333 229)(125 214 424 260 262 376 334 230)(126 215 425 251 263 377 335 221)(127 216 426 252 264 378 336 222)(128 217 427 253 265 379 337 223)(129 218 428 254 266 380 338 224)(130 219 429 255 267 371 339 225)
(1 378 243 216)(2 379 244 217)(3 380 245 218)(4 371 246 219)(5 372 247 220)(6 373 248 211)(7 374 249 212)(8 375 250 213)(9 376 241 214)(10 377 242 215)(11 132 450 357)(12 133 441 358)(13 134 442 359)(14 135 443 360)(15 136 444 351)(16 137 445 352)(17 138 446 353)(18 139 447 354)(19 140 448 355)(20 131 449 356)(21 128 468 265)(22 129 469 266)(23 130 470 267)(24 121 461 268)(25 122 462 269)(26 123 463 270)(27 124 464 261)(28 125 465 262)(29 126 466 263)(30 127 467 264)(31 44 310 280)(32 45 301 271)(33 46 302 272)(34 47 303 273)(35 48 304 274)(36 49 305 275)(37 50 306 276)(38 41 307 277)(39 42 308 278)(40 43 309 279)(51 78 381 232)(52 79 382 233)(53 80 383 234)(54 71 384 235)(55 72 385 236)(56 73 386 237)(57 74 387 238)(58 75 388 239)(59 76 389 240)(60 77 390 231)(61 288 295 196)(62 289 296 197)(63 290 297 198)(64 281 298 199)(65 282 299 200)(66 283 300 191)(67 284 291 192)(68 285 292 193)(69 286 293 194)(70 287 294 195)(81 204 175 398)(82 205 176 399)(83 206 177 400)(84 207 178 391)(85 208 179 392)(86 209 180 393)(87 210 171 394)(88 201 172 395)(89 202 173 396)(90 203 174 397)(91 222 181 252)(92 223 182 253)(93 224 183 254)(94 225 184 255)(95 226 185 256)(96 227 186 257)(97 228 187 258)(98 229 188 259)(99 230 189 260)(100 221 190 251)(101 118 403 362)(102 119 404 363)(103 120 405 364)(104 111 406 365)(105 112 407 366)(106 113 408 367)(107 114 409 368)(108 115 410 369)(109 116 401 370)(110 117 402 361)(141 477 432 313)(142 478 433 314)(143 479 434 315)(144 480 435 316)(145 471 436 317)(146 472 437 318)(147 473 438 319)(148 474 439 320)(149 475 440 311)(150 476 431 312)(151 411 350 327)(152 412 341 328)(153 413 342 329)(154 414 343 330)(155 415 344 321)(156 416 345 322)(157 417 346 323)(158 418 347 324)(159 419 348 325)(160 420 349 326)(161 427 452 337)(162 428 453 338)(163 429 454 339)(164 430 455 340)(165 421 456 331)(166 422 457 332)(167 423 458 333)(168 424 459 334)(169 425 460 335)(170 426 451 336)
G:=sub<Sym(480)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190)(191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230)(231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250)(251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290)(291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310)(311,312,313,314,315,316,317,318,319,320)(321,322,323,324,325,326,327,328,329,330)(331,332,333,334,335,336,337,338,339,340)(341,342,343,344,345,346,347,348,349,350)(351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370)(371,372,373,374,375,376,377,378,379,380)(381,382,383,384,385,386,387,388,389,390)(391,392,393,394,395,396,397,398,399,400)(401,402,403,404,405,406,407,408,409,410)(411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430)(431,432,433,434,435,436,437,438,439,440)(441,442,443,444,445,446,447,448,449,450)(451,452,453,454,455,456,457,458,459,460)(461,462,463,464,465,466,467,468,469,470)(471,472,473,474,475,476,477,478,479,480), (1,365,75)(2,366,76)(3,367,77)(4,368,78)(5,369,79)(6,370,80)(7,361,71)(8,362,72)(9,363,73)(10,364,74)(11,468,440)(12,469,431)(13,470,432)(14,461,433)(15,462,434)(16,463,435)(17,464,436)(18,465,437)(19,466,438)(20,467,439)(21,149,450)(22,150,441)(23,141,442)(24,142,443)(25,143,444)(26,144,445)(27,145,446)(28,146,447)(29,147,448)(30,148,449)(31,416,332)(32,417,333)(33,418,334)(34,419,335)(35,420,336)(36,411,337)(37,412,338)(38,413,339)(39,414,340)(40,415,331)(41,342,163)(42,343,164)(43,344,165)(44,345,166)(45,346,167)(46,347,168)(47,348,169)(48,349,170)(49,350,161)(50,341,162)(51,219,409)(52,220,410)(53,211,401)(54,212,402)(55,213,403)(56,214,404)(57,215,405)(58,216,406)(59,217,407)(60,218,408)(61,183,83)(62,184,84)(63,185,85)(64,186,86)(65,187,87)(66,188,88)(67,189,89)(68,190,90)(69,181,81)(70,182,82)(91,175,293)(92,176,294)(93,177,295)(94,178,296)(95,179,297)(96,180,298)(97,171,299)(98,172,300)(99,173,291)(100,174,292)(101,385,375)(102,386,376)(103,387,377)(104,388,378)(105,389,379)(106,390,380)(107,381,371)(108,382,372)(109,383,373)(110,384,374)(111,239,243)(112,240,244)(113,231,245)(114,232,246)(115,233,247)(116,234,248)(117,235,249)(118,236,250)(119,237,241)(120,238,242)(121,478,360)(122,479,351)(123,480,352)(124,471,353)(125,472,354)(126,473,355)(127,474,356)(128,475,357)(129,476,358)(130,477,359)(131,264,320)(132,265,311)(133,266,312)(134,267,313)(135,268,314)(136,269,315)(137,270,316)(138,261,317)(139,262,318)(140,263,319)(151,452,275)(152,453,276)(153,454,277)(154,455,278)(155,456,279)(156,457,280)(157,458,271)(158,459,272)(159,460,273)(160,451,274)(191,229,395)(192,230,396)(193,221,397)(194,222,398)(195,223,399)(196,224,400)(197,225,391)(198,226,392)(199,227,393)(200,228,394)(201,283,259)(202,284,260)(203,285,251)(204,286,252)(205,287,253)(206,288,254)(207,289,255)(208,290,256)(209,281,257)(210,282,258)(301,323,423)(302,324,424)(303,325,425)(304,326,426)(305,327,427)(306,328,428)(307,329,429)(308,330,430)(309,321,421)(310,322,422), (1,467,181,170,243,30,91,451)(2,468,182,161,244,21,92,452)(3,469,183,162,245,22,93,453)(4,470,184,163,246,23,94,454)(5,461,185,164,247,24,95,455)(6,462,186,165,248,25,96,456)(7,463,187,166,249,26,97,457)(8,464,188,167,250,27,98,458)(9,465,189,168,241,28,99,459)(10,466,190,169,242,29,100,460)(11,82,350,112,450,176,151,366)(12,83,341,113,441,177,152,367)(13,84,342,114,442,178,153,368)(14,85,343,115,443,179,154,369)(15,86,344,116,444,180,155,370)(16,87,345,117,445,171,156,361)(17,88,346,118,446,172,157,362)(18,89,347,119,447,173,158,363)(19,90,348,120,448,174,159,364)(20,81,349,111,449,175,160,365)(31,200,480,54,310,282,316,384)(32,191,471,55,301,283,317,385)(33,192,472,56,302,284,318,386)(34,193,473,57,303,285,319,387)(35,194,474,58,304,286,320,388)(36,195,475,59,305,287,311,389)(37,196,476,60,306,288,312,390)(38,197,477,51,307,289,313,381)(39,198,478,52,308,290,314,382)(40,199,479,53,309,281,315,383)(41,232,141,296,277,78,432,62)(42,233,142,297,278,79,433,63)(43,234,143,298,279,80,434,64)(44,235,144,299,280,71,435,65)(45,236,145,300,271,72,436,66)(46,237,146,291,272,73,437,67)(47,238,147,292,273,74,438,68)(48,239,148,293,274,75,439,69)(49,240,149,294,275,76,440,70)(50,231,150,295,276,77,431,61)(101,417,395,353,403,323,201,138)(102,418,396,354,404,324,202,139)(103,419,397,355,405,325,203,140)(104,420,398,356,406,326,204,131)(105,411,399,357,407,327,205,132)(106,412,400,358,408,328,206,133)(107,413,391,359,409,329,207,134)(108,414,392,360,410,330,208,135)(109,415,393,351,401,321,209,136)(110,416,394,352,402,322,210,137)(121,220,430,256,268,372,340,226)(122,211,421,257,269,373,331,227)(123,212,422,258,270,374,332,228)(124,213,423,259,261,375,333,229)(125,214,424,260,262,376,334,230)(126,215,425,251,263,377,335,221)(127,216,426,252,264,378,336,222)(128,217,427,253,265,379,337,223)(129,218,428,254,266,380,338,224)(130,219,429,255,267,371,339,225), (1,378,243,216)(2,379,244,217)(3,380,245,218)(4,371,246,219)(5,372,247,220)(6,373,248,211)(7,374,249,212)(8,375,250,213)(9,376,241,214)(10,377,242,215)(11,132,450,357)(12,133,441,358)(13,134,442,359)(14,135,443,360)(15,136,444,351)(16,137,445,352)(17,138,446,353)(18,139,447,354)(19,140,448,355)(20,131,449,356)(21,128,468,265)(22,129,469,266)(23,130,470,267)(24,121,461,268)(25,122,462,269)(26,123,463,270)(27,124,464,261)(28,125,465,262)(29,126,466,263)(30,127,467,264)(31,44,310,280)(32,45,301,271)(33,46,302,272)(34,47,303,273)(35,48,304,274)(36,49,305,275)(37,50,306,276)(38,41,307,277)(39,42,308,278)(40,43,309,279)(51,78,381,232)(52,79,382,233)(53,80,383,234)(54,71,384,235)(55,72,385,236)(56,73,386,237)(57,74,387,238)(58,75,388,239)(59,76,389,240)(60,77,390,231)(61,288,295,196)(62,289,296,197)(63,290,297,198)(64,281,298,199)(65,282,299,200)(66,283,300,191)(67,284,291,192)(68,285,292,193)(69,286,293,194)(70,287,294,195)(81,204,175,398)(82,205,176,399)(83,206,177,400)(84,207,178,391)(85,208,179,392)(86,209,180,393)(87,210,171,394)(88,201,172,395)(89,202,173,396)(90,203,174,397)(91,222,181,252)(92,223,182,253)(93,224,183,254)(94,225,184,255)(95,226,185,256)(96,227,186,257)(97,228,187,258)(98,229,188,259)(99,230,189,260)(100,221,190,251)(101,118,403,362)(102,119,404,363)(103,120,405,364)(104,111,406,365)(105,112,407,366)(106,113,408,367)(107,114,409,368)(108,115,410,369)(109,116,401,370)(110,117,402,361)(141,477,432,313)(142,478,433,314)(143,479,434,315)(144,480,435,316)(145,471,436,317)(146,472,437,318)(147,473,438,319)(148,474,439,320)(149,475,440,311)(150,476,431,312)(151,411,350,327)(152,412,341,328)(153,413,342,329)(154,414,343,330)(155,415,344,321)(156,416,345,322)(157,417,346,323)(158,418,347,324)(159,419,348,325)(160,420,349,326)(161,427,452,337)(162,428,453,338)(163,429,454,339)(164,430,455,340)(165,421,456,331)(166,422,457,332)(167,423,458,333)(168,424,459,334)(169,425,460,335)(170,426,451,336)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190)(191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230)(231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250)(251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290)(291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310)(311,312,313,314,315,316,317,318,319,320)(321,322,323,324,325,326,327,328,329,330)(331,332,333,334,335,336,337,338,339,340)(341,342,343,344,345,346,347,348,349,350)(351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370)(371,372,373,374,375,376,377,378,379,380)(381,382,383,384,385,386,387,388,389,390)(391,392,393,394,395,396,397,398,399,400)(401,402,403,404,405,406,407,408,409,410)(411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430)(431,432,433,434,435,436,437,438,439,440)(441,442,443,444,445,446,447,448,449,450)(451,452,453,454,455,456,457,458,459,460)(461,462,463,464,465,466,467,468,469,470)(471,472,473,474,475,476,477,478,479,480), (1,365,75)(2,366,76)(3,367,77)(4,368,78)(5,369,79)(6,370,80)(7,361,71)(8,362,72)(9,363,73)(10,364,74)(11,468,440)(12,469,431)(13,470,432)(14,461,433)(15,462,434)(16,463,435)(17,464,436)(18,465,437)(19,466,438)(20,467,439)(21,149,450)(22,150,441)(23,141,442)(24,142,443)(25,143,444)(26,144,445)(27,145,446)(28,146,447)(29,147,448)(30,148,449)(31,416,332)(32,417,333)(33,418,334)(34,419,335)(35,420,336)(36,411,337)(37,412,338)(38,413,339)(39,414,340)(40,415,331)(41,342,163)(42,343,164)(43,344,165)(44,345,166)(45,346,167)(46,347,168)(47,348,169)(48,349,170)(49,350,161)(50,341,162)(51,219,409)(52,220,410)(53,211,401)(54,212,402)(55,213,403)(56,214,404)(57,215,405)(58,216,406)(59,217,407)(60,218,408)(61,183,83)(62,184,84)(63,185,85)(64,186,86)(65,187,87)(66,188,88)(67,189,89)(68,190,90)(69,181,81)(70,182,82)(91,175,293)(92,176,294)(93,177,295)(94,178,296)(95,179,297)(96,180,298)(97,171,299)(98,172,300)(99,173,291)(100,174,292)(101,385,375)(102,386,376)(103,387,377)(104,388,378)(105,389,379)(106,390,380)(107,381,371)(108,382,372)(109,383,373)(110,384,374)(111,239,243)(112,240,244)(113,231,245)(114,232,246)(115,233,247)(116,234,248)(117,235,249)(118,236,250)(119,237,241)(120,238,242)(121,478,360)(122,479,351)(123,480,352)(124,471,353)(125,472,354)(126,473,355)(127,474,356)(128,475,357)(129,476,358)(130,477,359)(131,264,320)(132,265,311)(133,266,312)(134,267,313)(135,268,314)(136,269,315)(137,270,316)(138,261,317)(139,262,318)(140,263,319)(151,452,275)(152,453,276)(153,454,277)(154,455,278)(155,456,279)(156,457,280)(157,458,271)(158,459,272)(159,460,273)(160,451,274)(191,229,395)(192,230,396)(193,221,397)(194,222,398)(195,223,399)(196,224,400)(197,225,391)(198,226,392)(199,227,393)(200,228,394)(201,283,259)(202,284,260)(203,285,251)(204,286,252)(205,287,253)(206,288,254)(207,289,255)(208,290,256)(209,281,257)(210,282,258)(301,323,423)(302,324,424)(303,325,425)(304,326,426)(305,327,427)(306,328,428)(307,329,429)(308,330,430)(309,321,421)(310,322,422), (1,467,181,170,243,30,91,451)(2,468,182,161,244,21,92,452)(3,469,183,162,245,22,93,453)(4,470,184,163,246,23,94,454)(5,461,185,164,247,24,95,455)(6,462,186,165,248,25,96,456)(7,463,187,166,249,26,97,457)(8,464,188,167,250,27,98,458)(9,465,189,168,241,28,99,459)(10,466,190,169,242,29,100,460)(11,82,350,112,450,176,151,366)(12,83,341,113,441,177,152,367)(13,84,342,114,442,178,153,368)(14,85,343,115,443,179,154,369)(15,86,344,116,444,180,155,370)(16,87,345,117,445,171,156,361)(17,88,346,118,446,172,157,362)(18,89,347,119,447,173,158,363)(19,90,348,120,448,174,159,364)(20,81,349,111,449,175,160,365)(31,200,480,54,310,282,316,384)(32,191,471,55,301,283,317,385)(33,192,472,56,302,284,318,386)(34,193,473,57,303,285,319,387)(35,194,474,58,304,286,320,388)(36,195,475,59,305,287,311,389)(37,196,476,60,306,288,312,390)(38,197,477,51,307,289,313,381)(39,198,478,52,308,290,314,382)(40,199,479,53,309,281,315,383)(41,232,141,296,277,78,432,62)(42,233,142,297,278,79,433,63)(43,234,143,298,279,80,434,64)(44,235,144,299,280,71,435,65)(45,236,145,300,271,72,436,66)(46,237,146,291,272,73,437,67)(47,238,147,292,273,74,438,68)(48,239,148,293,274,75,439,69)(49,240,149,294,275,76,440,70)(50,231,150,295,276,77,431,61)(101,417,395,353,403,323,201,138)(102,418,396,354,404,324,202,139)(103,419,397,355,405,325,203,140)(104,420,398,356,406,326,204,131)(105,411,399,357,407,327,205,132)(106,412,400,358,408,328,206,133)(107,413,391,359,409,329,207,134)(108,414,392,360,410,330,208,135)(109,415,393,351,401,321,209,136)(110,416,394,352,402,322,210,137)(121,220,430,256,268,372,340,226)(122,211,421,257,269,373,331,227)(123,212,422,258,270,374,332,228)(124,213,423,259,261,375,333,229)(125,214,424,260,262,376,334,230)(126,215,425,251,263,377,335,221)(127,216,426,252,264,378,336,222)(128,217,427,253,265,379,337,223)(129,218,428,254,266,380,338,224)(130,219,429,255,267,371,339,225), (1,378,243,216)(2,379,244,217)(3,380,245,218)(4,371,246,219)(5,372,247,220)(6,373,248,211)(7,374,249,212)(8,375,250,213)(9,376,241,214)(10,377,242,215)(11,132,450,357)(12,133,441,358)(13,134,442,359)(14,135,443,360)(15,136,444,351)(16,137,445,352)(17,138,446,353)(18,139,447,354)(19,140,448,355)(20,131,449,356)(21,128,468,265)(22,129,469,266)(23,130,470,267)(24,121,461,268)(25,122,462,269)(26,123,463,270)(27,124,464,261)(28,125,465,262)(29,126,466,263)(30,127,467,264)(31,44,310,280)(32,45,301,271)(33,46,302,272)(34,47,303,273)(35,48,304,274)(36,49,305,275)(37,50,306,276)(38,41,307,277)(39,42,308,278)(40,43,309,279)(51,78,381,232)(52,79,382,233)(53,80,383,234)(54,71,384,235)(55,72,385,236)(56,73,386,237)(57,74,387,238)(58,75,388,239)(59,76,389,240)(60,77,390,231)(61,288,295,196)(62,289,296,197)(63,290,297,198)(64,281,298,199)(65,282,299,200)(66,283,300,191)(67,284,291,192)(68,285,292,193)(69,286,293,194)(70,287,294,195)(81,204,175,398)(82,205,176,399)(83,206,177,400)(84,207,178,391)(85,208,179,392)(86,209,180,393)(87,210,171,394)(88,201,172,395)(89,202,173,396)(90,203,174,397)(91,222,181,252)(92,223,182,253)(93,224,183,254)(94,225,184,255)(95,226,185,256)(96,227,186,257)(97,228,187,258)(98,229,188,259)(99,230,189,260)(100,221,190,251)(101,118,403,362)(102,119,404,363)(103,120,405,364)(104,111,406,365)(105,112,407,366)(106,113,408,367)(107,114,409,368)(108,115,410,369)(109,116,401,370)(110,117,402,361)(141,477,432,313)(142,478,433,314)(143,479,434,315)(144,480,435,316)(145,471,436,317)(146,472,437,318)(147,473,438,319)(148,474,439,320)(149,475,440,311)(150,476,431,312)(151,411,350,327)(152,412,341,328)(153,413,342,329)(154,414,343,330)(155,415,344,321)(156,416,345,322)(157,417,346,323)(158,418,347,324)(159,419,348,325)(160,420,349,326)(161,427,452,337)(162,428,453,338)(163,429,454,339)(164,430,455,340)(165,421,456,331)(166,422,457,332)(167,423,458,333)(168,424,459,334)(169,425,460,335)(170,426,451,336) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170),(171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190),(191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220),(221,222,223,224,225,226,227,228,229,230),(231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250),(251,252,253,254,255,256,257,258,259,260),(261,262,263,264,265,266,267,268,269,270),(271,272,273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288,289,290),(291,292,293,294,295,296,297,298,299,300),(301,302,303,304,305,306,307,308,309,310),(311,312,313,314,315,316,317,318,319,320),(321,322,323,324,325,326,327,328,329,330),(331,332,333,334,335,336,337,338,339,340),(341,342,343,344,345,346,347,348,349,350),(351,352,353,354,355,356,357,358,359,360),(361,362,363,364,365,366,367,368,369,370),(371,372,373,374,375,376,377,378,379,380),(381,382,383,384,385,386,387,388,389,390),(391,392,393,394,395,396,397,398,399,400),(401,402,403,404,405,406,407,408,409,410),(411,412,413,414,415,416,417,418,419,420),(421,422,423,424,425,426,427,428,429,430),(431,432,433,434,435,436,437,438,439,440),(441,442,443,444,445,446,447,448,449,450),(451,452,453,454,455,456,457,458,459,460),(461,462,463,464,465,466,467,468,469,470),(471,472,473,474,475,476,477,478,479,480)], [(1,365,75),(2,366,76),(3,367,77),(4,368,78),(5,369,79),(6,370,80),(7,361,71),(8,362,72),(9,363,73),(10,364,74),(11,468,440),(12,469,431),(13,470,432),(14,461,433),(15,462,434),(16,463,435),(17,464,436),(18,465,437),(19,466,438),(20,467,439),(21,149,450),(22,150,441),(23,141,442),(24,142,443),(25,143,444),(26,144,445),(27,145,446),(28,146,447),(29,147,448),(30,148,449),(31,416,332),(32,417,333),(33,418,334),(34,419,335),(35,420,336),(36,411,337),(37,412,338),(38,413,339),(39,414,340),(40,415,331),(41,342,163),(42,343,164),(43,344,165),(44,345,166),(45,346,167),(46,347,168),(47,348,169),(48,349,170),(49,350,161),(50,341,162),(51,219,409),(52,220,410),(53,211,401),(54,212,402),(55,213,403),(56,214,404),(57,215,405),(58,216,406),(59,217,407),(60,218,408),(61,183,83),(62,184,84),(63,185,85),(64,186,86),(65,187,87),(66,188,88),(67,189,89),(68,190,90),(69,181,81),(70,182,82),(91,175,293),(92,176,294),(93,177,295),(94,178,296),(95,179,297),(96,180,298),(97,171,299),(98,172,300),(99,173,291),(100,174,292),(101,385,375),(102,386,376),(103,387,377),(104,388,378),(105,389,379),(106,390,380),(107,381,371),(108,382,372),(109,383,373),(110,384,374),(111,239,243),(112,240,244),(113,231,245),(114,232,246),(115,233,247),(116,234,248),(117,235,249),(118,236,250),(119,237,241),(120,238,242),(121,478,360),(122,479,351),(123,480,352),(124,471,353),(125,472,354),(126,473,355),(127,474,356),(128,475,357),(129,476,358),(130,477,359),(131,264,320),(132,265,311),(133,266,312),(134,267,313),(135,268,314),(136,269,315),(137,270,316),(138,261,317),(139,262,318),(140,263,319),(151,452,275),(152,453,276),(153,454,277),(154,455,278),(155,456,279),(156,457,280),(157,458,271),(158,459,272),(159,460,273),(160,451,274),(191,229,395),(192,230,396),(193,221,397),(194,222,398),(195,223,399),(196,224,400),(197,225,391),(198,226,392),(199,227,393),(200,228,394),(201,283,259),(202,284,260),(203,285,251),(204,286,252),(205,287,253),(206,288,254),(207,289,255),(208,290,256),(209,281,257),(210,282,258),(301,323,423),(302,324,424),(303,325,425),(304,326,426),(305,327,427),(306,328,428),(307,329,429),(308,330,430),(309,321,421),(310,322,422)], [(1,467,181,170,243,30,91,451),(2,468,182,161,244,21,92,452),(3,469,183,162,245,22,93,453),(4,470,184,163,246,23,94,454),(5,461,185,164,247,24,95,455),(6,462,186,165,248,25,96,456),(7,463,187,166,249,26,97,457),(8,464,188,167,250,27,98,458),(9,465,189,168,241,28,99,459),(10,466,190,169,242,29,100,460),(11,82,350,112,450,176,151,366),(12,83,341,113,441,177,152,367),(13,84,342,114,442,178,153,368),(14,85,343,115,443,179,154,369),(15,86,344,116,444,180,155,370),(16,87,345,117,445,171,156,361),(17,88,346,118,446,172,157,362),(18,89,347,119,447,173,158,363),(19,90,348,120,448,174,159,364),(20,81,349,111,449,175,160,365),(31,200,480,54,310,282,316,384),(32,191,471,55,301,283,317,385),(33,192,472,56,302,284,318,386),(34,193,473,57,303,285,319,387),(35,194,474,58,304,286,320,388),(36,195,475,59,305,287,311,389),(37,196,476,60,306,288,312,390),(38,197,477,51,307,289,313,381),(39,198,478,52,308,290,314,382),(40,199,479,53,309,281,315,383),(41,232,141,296,277,78,432,62),(42,233,142,297,278,79,433,63),(43,234,143,298,279,80,434,64),(44,235,144,299,280,71,435,65),(45,236,145,300,271,72,436,66),(46,237,146,291,272,73,437,67),(47,238,147,292,273,74,438,68),(48,239,148,293,274,75,439,69),(49,240,149,294,275,76,440,70),(50,231,150,295,276,77,431,61),(101,417,395,353,403,323,201,138),(102,418,396,354,404,324,202,139),(103,419,397,355,405,325,203,140),(104,420,398,356,406,326,204,131),(105,411,399,357,407,327,205,132),(106,412,400,358,408,328,206,133),(107,413,391,359,409,329,207,134),(108,414,392,360,410,330,208,135),(109,415,393,351,401,321,209,136),(110,416,394,352,402,322,210,137),(121,220,430,256,268,372,340,226),(122,211,421,257,269,373,331,227),(123,212,422,258,270,374,332,228),(124,213,423,259,261,375,333,229),(125,214,424,260,262,376,334,230),(126,215,425,251,263,377,335,221),(127,216,426,252,264,378,336,222),(128,217,427,253,265,379,337,223),(129,218,428,254,266,380,338,224),(130,219,429,255,267,371,339,225)], [(1,378,243,216),(2,379,244,217),(3,380,245,218),(4,371,246,219),(5,372,247,220),(6,373,248,211),(7,374,249,212),(8,375,250,213),(9,376,241,214),(10,377,242,215),(11,132,450,357),(12,133,441,358),(13,134,442,359),(14,135,443,360),(15,136,444,351),(16,137,445,352),(17,138,446,353),(18,139,447,354),(19,140,448,355),(20,131,449,356),(21,128,468,265),(22,129,469,266),(23,130,470,267),(24,121,461,268),(25,122,462,269),(26,123,463,270),(27,124,464,261),(28,125,465,262),(29,126,466,263),(30,127,467,264),(31,44,310,280),(32,45,301,271),(33,46,302,272),(34,47,303,273),(35,48,304,274),(36,49,305,275),(37,50,306,276),(38,41,307,277),(39,42,308,278),(40,43,309,279),(51,78,381,232),(52,79,382,233),(53,80,383,234),(54,71,384,235),(55,72,385,236),(56,73,386,237),(57,74,387,238),(58,75,388,239),(59,76,389,240),(60,77,390,231),(61,288,295,196),(62,289,296,197),(63,290,297,198),(64,281,298,199),(65,282,299,200),(66,283,300,191),(67,284,291,192),(68,285,292,193),(69,286,293,194),(70,287,294,195),(81,204,175,398),(82,205,176,399),(83,206,177,400),(84,207,178,391),(85,208,179,392),(86,209,180,393),(87,210,171,394),(88,201,172,395),(89,202,173,396),(90,203,174,397),(91,222,181,252),(92,223,182,253),(93,224,183,254),(94,225,184,255),(95,226,185,256),(96,227,186,257),(97,228,187,258),(98,229,188,259),(99,230,189,260),(100,221,190,251),(101,118,403,362),(102,119,404,363),(103,120,405,364),(104,111,406,365),(105,112,407,366),(106,113,408,367),(107,114,409,368),(108,115,410,369),(109,116,401,370),(110,117,402,361),(141,477,432,313),(142,478,433,314),(143,479,434,315),(144,480,435,316),(145,471,436,317),(146,472,437,318),(147,473,438,319),(148,474,439,320),(149,475,440,311),(150,476,431,312),(151,411,350,327),(152,412,341,328),(153,413,342,329),(154,414,343,330),(155,415,344,321),(156,416,345,322),(157,417,346,323),(158,418,347,324),(159,419,348,325),(160,420,349,326),(161,427,452,337),(162,428,453,338),(163,429,454,339),(164,430,455,340),(165,421,456,331),(166,422,457,332),(167,423,458,333),(168,424,459,334),(169,425,460,335),(170,426,451,336)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | ··· | 20X | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
120 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | Q16 | C3⋊D4 | C3⋊D4 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C5×Q16 | C5×C3⋊D4 | C5×C3⋊D4 | C3⋊Q16 | C5×C3⋊Q16 |
| kernel | C10×C3⋊Q16 | C10×C3⋊C8 | C5×C3⋊Q16 | C10×Dic6 | Q8×C30 | C2×C3⋊Q16 | C2×C3⋊C8 | C3⋊Q16 | C2×Dic6 | C6×Q8 | Q8×C10 | C60 | C2×C30 | C2×C20 | C5×Q8 | C30 | C20 | C2×C10 | C2×Q8 | C12 | C2×C6 | C2×C4 | Q8 | C6 | C4 | C22 | C10 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 16 | 8 | 8 | 2 | 8 |
Matrix representation of C10×C3⋊Q16 ►in GL4(𝔽241) generated by
| 150 | 0 | 0 | 0 |
| 0 | 150 | 0 | 0 |
| 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 87 |
| 240 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 219 | 18 | 0 | 0 |
| 40 | 22 | 0 | 0 |
| 0 | 0 | 0 | 168 |
| 0 | 0 | 208 | 22 |
| 70 | 140 | 0 | 0 |
| 101 | 171 | 0 | 0 |
| 0 | 0 | 76 | 46 |
| 0 | 0 | 194 | 165 |
G:=sub<GL(4,GF(241))| [150,0,0,0,0,150,0,0,0,0,87,0,0,0,0,87],[240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[219,40,0,0,18,22,0,0,0,0,0,208,0,0,168,22],[70,101,0,0,140,171,0,0,0,0,76,194,0,0,46,165] >;
C10×C3⋊Q16 in GAP, Magma, Sage, TeX
C_{10}\times C_3\rtimes Q_{16} % in TeX
G:=Group("C10xC3:Q16"); // GroupNames label
G:=SmallGroup(480,822);
// by ID
G=gap.SmallGroup(480,822);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,926,436,4204,1068,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^3=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations