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G = C2×C30.4Q8order 480 = 25·3·5

Direct product of C2 and C30.4Q8

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

Aliases: C2×C30.4Q8, C23.35D30, C22.4Dic30, C307(C4⋊C4), (C2×C4).66D30, (C2×C30).14Q8, C30.70(C2×Q8), (C2×C20).382D6, (C2×C30).142D4, C30.374(C2×D4), (C22×C60).8C2, (C22×C20).9S3, C2.2(C2×Dic30), (C22×C12).5D5, (C22×C4).5D15, C104(Dic3⋊C4), Dic1520(C2×C4), (C2×Dic15)⋊10C4, (C2×C12).381D10, C63(C10.D4), C6.38(C2×Dic10), C10.38(C2×Dic6), (C2×C10).13Dic6, (C2×C6).13Dic10, C22.16(C4×D15), (C2×C60).463C22, C30.169(C22×C4), (C2×C30).297C23, (C22×C6).114D10, (C22×C10).132D6, C22.19(C157D4), (C22×Dic15).4C2, C22.20(C22×D15), (C22×C30).137C22, (C2×Dic15).167C22, C1517(C2×C4⋊C4), C6.74(C2×C4×D5), C55(C2×Dic3⋊C4), C2.18(C2×C4×D15), C10.106(S3×C2×C4), (C2×C6).35(C4×D5), C2.1(C2×C157D4), C34(C2×C10.D4), C6.97(C2×C5⋊D4), (C2×C10).60(C4×S3), C10.97(C2×C3⋊D4), (C2×C30).141(C2×C4), (C2×C6).74(C5⋊D4), (C2×C10).74(C3⋊D4), (C2×C6).293(C22×D5), (C2×C10).292(C22×S3), SmallGroup(480,888)

Series: Derived Chief Lower central Upper central

C1C30 — C2×C30.4Q8
C1C5C15C30C2×C30C2×Dic15C22×Dic15 — C2×C30.4Q8
C15C30 — C2×C30.4Q8
C1C23C22×C4

Generators and relations for C2×C30.4Q8
 G = < a,b,c,d | a2=b30=c4=1, d2=b15c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b15c-1 >

Subgroups: 756 in 184 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×8], C22, C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×12], C23, C10 [×3], C10 [×4], Dic3 [×6], C12 [×2], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4, C22×C4 [×2], Dic5 [×6], C20 [×2], C2×C10, C2×C10 [×6], C2×Dic3 [×10], C2×C12 [×2], C2×C12 [×2], C22×C6, C30 [×3], C30 [×4], C2×C4⋊C4, C2×Dic5 [×10], C2×C20 [×2], C2×C20 [×2], C22×C10, Dic3⋊C4 [×4], C22×Dic3 [×2], C22×C12, Dic15 [×4], Dic15 [×2], C60 [×2], C2×C30, C2×C30 [×6], C10.D4 [×4], C22×Dic5 [×2], C22×C20, C2×Dic3⋊C4, C2×Dic15 [×8], C2×Dic15 [×2], C2×C60 [×2], C2×C60 [×2], C22×C30, C2×C10.D4, C30.4Q8 [×4], C22×Dic15 [×2], C22×C60, C2×C30.4Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, D15, C2×C4⋊C4, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, D30 [×3], C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×Dic3⋊C4, Dic30 [×2], C4×D15 [×2], C157D4 [×2], C22×D15, C2×C10.D4, C30.4Q8 [×4], C2×Dic30, C2×C4×D15, C2×C157D4, C2×C30.4Q8

Smallest permutation representation of C2×C30.4Q8
Regular action on 480 points
Generators in S480
(1 432)(2 433)(3 434)(4 435)(5 436)(6 437)(7 438)(8 439)(9 440)(10 441)(11 442)(12 443)(13 444)(14 445)(15 446)(16 447)(17 448)(18 449)(19 450)(20 421)(21 422)(22 423)(23 424)(24 425)(25 426)(26 427)(27 428)(28 429)(29 430)(30 431)(31 298)(32 299)(33 300)(34 271)(35 272)(36 273)(37 274)(38 275)(39 276)(40 277)(41 278)(42 279)(43 280)(44 281)(45 282)(46 283)(47 284)(48 285)(49 286)(50 287)(51 288)(52 289)(53 290)(54 291)(55 292)(56 293)(57 294)(58 295)(59 296)(60 297)(61 467)(62 468)(63 469)(64 470)(65 471)(66 472)(67 473)(68 474)(69 475)(70 476)(71 477)(72 478)(73 479)(74 480)(75 451)(76 452)(77 453)(78 454)(79 455)(80 456)(81 457)(82 458)(83 459)(84 460)(85 461)(86 462)(87 463)(88 464)(89 465)(90 466)(91 412)(92 413)(93 414)(94 415)(95 416)(96 417)(97 418)(98 419)(99 420)(100 391)(101 392)(102 393)(103 394)(104 395)(105 396)(106 397)(107 398)(108 399)(109 400)(110 401)(111 402)(112 403)(113 404)(114 405)(115 406)(116 407)(117 408)(118 409)(119 410)(120 411)(121 244)(122 245)(123 246)(124 247)(125 248)(126 249)(127 250)(128 251)(129 252)(130 253)(131 254)(132 255)(133 256)(134 257)(135 258)(136 259)(137 260)(138 261)(139 262)(140 263)(141 264)(142 265)(143 266)(144 267)(145 268)(146 269)(147 270)(148 241)(149 242)(150 243)(151 323)(152 324)(153 325)(154 326)(155 327)(156 328)(157 329)(158 330)(159 301)(160 302)(161 303)(162 304)(163 305)(164 306)(165 307)(166 308)(167 309)(168 310)(169 311)(170 312)(171 313)(172 314)(173 315)(174 316)(175 317)(176 318)(177 319)(178 320)(179 321)(180 322)(181 389)(182 390)(183 361)(184 362)(185 363)(186 364)(187 365)(188 366)(189 367)(190 368)(191 369)(192 370)(193 371)(194 372)(195 373)(196 374)(197 375)(198 376)(199 377)(200 378)(201 379)(202 380)(203 381)(204 382)(205 383)(206 384)(207 385)(208 386)(209 387)(210 388)(211 339)(212 340)(213 341)(214 342)(215 343)(216 344)(217 345)(218 346)(219 347)(220 348)(221 349)(222 350)(223 351)(224 352)(225 353)(226 354)(227 355)(228 356)(229 357)(230 358)(231 359)(232 360)(233 331)(234 332)(235 333)(236 334)(237 335)(238 336)(239 337)(240 338)
(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)
(1 335 125 58)(2 336 126 59)(3 337 127 60)(4 338 128 31)(5 339 129 32)(6 340 130 33)(7 341 131 34)(8 342 132 35)(9 343 133 36)(10 344 134 37)(11 345 135 38)(12 346 136 39)(13 347 137 40)(14 348 138 41)(15 349 139 42)(16 350 140 43)(17 351 141 44)(18 352 142 45)(19 353 143 46)(20 354 144 47)(21 355 145 48)(22 356 146 49)(23 357 147 50)(24 358 148 51)(25 359 149 52)(26 360 150 53)(27 331 121 54)(28 332 122 55)(29 333 123 56)(30 334 124 57)(61 193 154 392)(62 194 155 393)(63 195 156 394)(64 196 157 395)(65 197 158 396)(66 198 159 397)(67 199 160 398)(68 200 161 399)(69 201 162 400)(70 202 163 401)(71 203 164 402)(72 204 165 403)(73 205 166 404)(74 206 167 405)(75 207 168 406)(76 208 169 407)(77 209 170 408)(78 210 171 409)(79 181 172 410)(80 182 173 411)(81 183 174 412)(82 184 175 413)(83 185 176 414)(84 186 177 415)(85 187 178 416)(86 188 179 417)(87 189 180 418)(88 190 151 419)(89 191 152 420)(90 192 153 391)(91 457 361 316)(92 458 362 317)(93 459 363 318)(94 460 364 319)(95 461 365 320)(96 462 366 321)(97 463 367 322)(98 464 368 323)(99 465 369 324)(100 466 370 325)(101 467 371 326)(102 468 372 327)(103 469 373 328)(104 470 374 329)(105 471 375 330)(106 472 376 301)(107 473 377 302)(108 474 378 303)(109 475 379 304)(110 476 380 305)(111 477 381 306)(112 478 382 307)(113 479 383 308)(114 480 384 309)(115 451 385 310)(116 452 386 311)(117 453 387 312)(118 454 388 313)(119 455 389 314)(120 456 390 315)(211 252 299 436)(212 253 300 437)(213 254 271 438)(214 255 272 439)(215 256 273 440)(216 257 274 441)(217 258 275 442)(218 259 276 443)(219 260 277 444)(220 261 278 445)(221 262 279 446)(222 263 280 447)(223 264 281 448)(224 265 282 449)(225 266 283 450)(226 267 284 421)(227 268 285 422)(228 269 286 423)(229 270 287 424)(230 241 288 425)(231 242 289 426)(232 243 290 427)(233 244 291 428)(234 245 292 429)(235 246 293 430)(236 247 294 431)(237 248 295 432)(238 249 296 433)(239 250 297 434)(240 251 298 435)
(1 86 140 164)(2 85 141 163)(3 84 142 162)(4 83 143 161)(5 82 144 160)(6 81 145 159)(7 80 146 158)(8 79 147 157)(9 78 148 156)(10 77 149 155)(11 76 150 154)(12 75 121 153)(13 74 122 152)(14 73 123 151)(15 72 124 180)(16 71 125 179)(17 70 126 178)(18 69 127 177)(19 68 128 176)(20 67 129 175)(21 66 130 174)(22 65 131 173)(23 64 132 172)(24 63 133 171)(25 62 134 170)(26 61 135 169)(27 90 136 168)(28 89 137 167)(29 88 138 166)(30 87 139 165)(31 200 353 414)(32 199 354 413)(33 198 355 412)(34 197 356 411)(35 196 357 410)(36 195 358 409)(37 194 359 408)(38 193 360 407)(39 192 331 406)(40 191 332 405)(41 190 333 404)(42 189 334 403)(43 188 335 402)(44 187 336 401)(45 186 337 400)(46 185 338 399)(47 184 339 398)(48 183 340 397)(49 182 341 396)(50 181 342 395)(51 210 343 394)(52 209 344 393)(53 208 345 392)(54 207 346 391)(55 206 347 420)(56 205 348 419)(57 204 349 418)(58 203 350 417)(59 202 351 416)(60 201 352 415)(91 300 376 227)(92 299 377 226)(93 298 378 225)(94 297 379 224)(95 296 380 223)(96 295 381 222)(97 294 382 221)(98 293 383 220)(99 292 384 219)(100 291 385 218)(101 290 386 217)(102 289 387 216)(103 288 388 215)(104 287 389 214)(105 286 390 213)(106 285 361 212)(107 284 362 211)(108 283 363 240)(109 282 364 239)(110 281 365 238)(111 280 366 237)(112 279 367 236)(113 278 368 235)(114 277 369 234)(115 276 370 233)(116 275 371 232)(117 274 372 231)(118 273 373 230)(119 272 374 229)(120 271 375 228)(241 328 440 454)(242 327 441 453)(243 326 442 452)(244 325 443 451)(245 324 444 480)(246 323 445 479)(247 322 446 478)(248 321 447 477)(249 320 448 476)(250 319 449 475)(251 318 450 474)(252 317 421 473)(253 316 422 472)(254 315 423 471)(255 314 424 470)(256 313 425 469)(257 312 426 468)(258 311 427 467)(259 310 428 466)(260 309 429 465)(261 308 430 464)(262 307 431 463)(263 306 432 462)(264 305 433 461)(265 304 434 460)(266 303 435 459)(267 302 436 458)(268 301 437 457)(269 330 438 456)(270 329 439 455)

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

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

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

132 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E···4L5A5B6A···6G10A···10N12A···12H15A15B15C15D20A···20P30A···30AB60A···60AF
order12···2344444···4556···610···1012···121515151520···2030···3060···60
size11···12222230···30222···22···22···222222···22···22···2

132 irreducible representations

dim1111122222222222222222222
type++++++-+++++-+-++-
imageC1C2C2C2C4S3D4Q8D5D6D6D10D10Dic6C4×S3C3⋊D4D15Dic10C4×D5C5⋊D4D30D30Dic30C4×D15C157D4
kernelC2×C30.4Q8C30.4Q8C22×Dic15C22×C60C2×Dic15C22×C20C2×C30C2×C30C22×C12C2×C20C22×C10C2×C12C22×C6C2×C10C2×C10C2×C10C22×C4C2×C6C2×C6C2×C6C2×C4C23C22C22C22
# reps1421812222142444488884161616

Matrix representation of C2×C30.4Q8 in GL5(𝔽61)

600000
060000
006000
000600
000060
,
10000
060000
006000
000308
000624
,
10000
00100
01000
000110
000011
,
600000
0483200
0291300
000335
000228

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,30,6,0,0,0,8,24],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,11],[60,0,0,0,0,0,48,29,0,0,0,32,13,0,0,0,0,0,33,2,0,0,0,5,28] >;

C2×C30.4Q8 in GAP, Magma, Sage, TeX

C_2\times C_{30}._4Q_8
% in TeX

G:=Group("C2xC30.4Q8");
// GroupNames label

G:=SmallGroup(480,888);
// by ID

G=gap.SmallGroup(480,888);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,422,58,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^30=c^4=1,d^2=b^15*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^15*c^-1>;
// generators/relations

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