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G = C2×C30.Q8  order 480 = 25·3·5

Direct product of C2 and C30.Q8

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

Aliases: C2×C30.Q8, C30⋊4(C4⋊C4), (C2×C30).6Q8, (C6×Dic5)⋊8C4, (C2×C30).72D4, C30.59(C2×Q8), C10⋊2(C4⋊Dic3), C10.66(C2×D12), (C2×C10).46D12, C30.221(C2×D4), (C2×C10).7Dic6, (C2×C6).7Dic10, C23.64(S3×D5), C22.5(C15⋊Q8), Dic5⋊5(C2×Dic3), (C2×Dic5)⋊5Dic3, C6⋊1(C10.D4), C6.26(C2×Dic10), C10.26(C2×Dic6), (C22×C6).86D10, C30.140(C22×C4), (C2×C30).183C23, (C2×Dic5).189D6, (C22×C10).103D6, (C22×Dic5).5S3, (C22×Dic3).3D5, C22.17(D5×Dic3), (C2×Dic3).162D10, C22.25(C5⋊D12), (C22×C30).45C22, C10.30(C22×Dic3), (C6×Dic5).218C22, (C22×Dic15).10C2, (C10×Dic3).198C22, (C2×Dic15).223C22, C15⋊11(C2×C4⋊C4), C5⋊3(C2×C4⋊Dic3), C6.93(C2×C4×D5), C2.2(C2×C15⋊Q8), (C2×C6).56(C4×D5), C3⋊2(C2×C10.D4), C6.20(C2×C5⋊D4), C2.4(C2×C5⋊D12), (C2×C6×Dic5).4C2, C2.17(C2×D5×Dic3), C22.80(C2×S3×D5), (Dic3×C2×C10).4C2, (C2×C30).115(C2×C4), (C3×Dic5)⋊22(C2×C4), (C2×C6).37(C5⋊D4), (C2×C10).39(C2×Dic3), (C2×C6).195(C22×D5), (C2×C10).195(C22×S3), SmallGroup(480,617)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — C2×C30.Q8
Chief series
C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C30.Q8 — C2×C30.Q8
Lower central
C15 — C30 — C2×C30.Q8
Upper central
C1 — C23

Generators and relations for C2×C30.Q8
 G = < a,b,c,d | a2=b30=c4=1, d2=b15c2, ab=ba, ac=ca, ad=da, cbc-1=b11, dbd-1=b19, dcd-1=b15c-1 >

Subgroups: 668 in 184 conjugacy classes, 100 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C30, C30, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C4⋊Dic3, C22×Dic3, C22×Dic3, C22×C12, C5×Dic3, C3×Dic5, Dic15, C2×C30, C2×C30, C10.D4, C22×Dic5, C22×Dic5, C22×C20, C2×C4⋊Dic3, C6×Dic5, C10×Dic3, C10×Dic3, C2×Dic15, C2×Dic15, C22×C30, C2×C10.D4, C30.Q8, C2×C6×Dic5, Dic3×C2×C10, C22×Dic15, C2×C30.Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D5, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, Dic6, D12, C2×Dic3, C22×S3, C2×C4⋊C4, Dic10, C4×D5, C5⋊D4, C22×D5, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, S3×D5, C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×C4⋊Dic3, D5×Dic3, C5⋊D12, C15⋊Q8, C2×S3×D5, C2×C10.D4, C30.Q8, C2×D5×Dic3, C2×C5⋊D12, C2×C15⋊Q8, C2×C30.Q8

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

(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)

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

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

84 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A···6G10A···10N12A···12H15A15B20A···20P30A···30N
order12···23444444444444556···610···1012···12151520···2030···30
size11···1266661010101030303030222···22···210···10446···64···4

84 irreducible representations

dim1111112222222222222244444
type+++++++-+-++++-+-+-+-+
imageC1C2C2C2C2C4S3D4Q8D5Dic3D6D6D10D10Dic6D12Dic10C4×D5C5⋊D4S3×D5D5×Dic3C5⋊D12C15⋊Q8C2×S3×D5
kernelC2×C30.Q8C30.Q8C2×C6×Dic5Dic3×C2×C10C22×Dic15C6×Dic5C22×Dic5C2×C30C2×C30C22×Dic3C2×Dic5C2×Dic5C22×C10C2×Dic3C22×C6C2×C10C2×C10C2×C6C2×C6C2×C6C23C22C22C22C22
# reps1411181222421424488824442

Matrix representation of C2×C30.Q8 ►in GL6(𝔽61)

6000000
0600000
001000
000100
0000600
0000060
,
0600000
110000
0006000
001100
00004460
00004560
,
38590000
21230000
00255100
00263600
00002957
00005832
,
6000000
0600000
0060000
0006000
0000558
00002956

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,60,1,0,0,0,0,0,0,44,45,0,0,0,0,60,60],[38,21,0,0,0,0,59,23,0,0,0,0,0,0,25,26,0,0,0,0,51,36,0,0,0,0,0,0,29,58,0,0,0,0,57,32],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,5,29,0,0,0,0,58,56] >;
 

C2×C30.Q8 in GAP, Magma, Sage, TeX 

C_2\times C_{30}.Q_8
 
% in TeX
 
G:=Group("C2xC30.Q8");
 
// GroupNames label
 
G:=SmallGroup(480,617);
 
// by ID
 
G=gap.SmallGroup(480,617);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,64,1356,18822]);
 
// 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,c*b*c^-1=b^11,d*b*d^-1=b^19,d*c*d^-1=b^15*c^-1>;
 
// generators/relations
 

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