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G = C2×C30.Q8order 480 = 25·3·5

Direct product of C2 and C30.Q8

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 [×3], C2 [×4], C3, C4 [×8], C22, C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×14], C23, C10 [×3], C10 [×4], Dic3 [×4], C12 [×4], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4 [×3], Dic5 [×4], Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×6], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×6], C22×C6, C30 [×3], C30 [×4], C2×C4⋊C4, C2×Dic5 [×6], C2×Dic5 [×4], C2×C20 [×4], C22×C10, C4⋊Dic3 [×4], C22×Dic3, C22×Dic3, C22×C12, C5×Dic3 [×2], C3×Dic5 [×4], Dic15 [×2], C2×C30, C2×C30 [×6], C10.D4 [×4], C22×Dic5, C22×Dic5, C22×C20, C2×C4⋊Dic3, C6×Dic5 [×6], C10×Dic3 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×Dic15 [×2], C22×C30, C2×C10.D4, C30.Q8 [×4], C2×C6×Dic5, Dic3×C2×C10, C22×Dic15, C2×C30.Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, Dic3 [×4], D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], Dic6 [×2], D12 [×2], C2×Dic3 [×6], C22×S3, C2×C4⋊C4, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, C4⋊Dic3 [×4], C2×Dic6, C2×D12, C22×Dic3, S3×D5, C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×C4⋊Dic3, D5×Dic3 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C2×S3×D5, C2×C10.D4, C30.Q8 [×4], 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 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(61 250)(62 251)(63 252)(64 253)(65 254)(66 255)(67 256)(68 257)(69 258)(70 259)(71 260)(72 261)(73 262)(74 263)(75 264)(76 265)(77 266)(78 267)(79 268)(80 269)(81 270)(82 241)(83 242)(84 243)(85 244)(86 245)(87 246)(88 247)(89 248)(90 249)(91 300)(92 271)(93 272)(94 273)(95 274)(96 275)(97 276)(98 277)(99 278)(100 279)(101 280)(102 281)(103 282)(104 283)(105 284)(106 285)(107 286)(108 287)(109 288)(110 289)(111 290)(112 291)(113 292)(114 293)(115 294)(116 295)(117 296)(118 297)(119 298)(120 299)(121 462)(122 463)(123 464)(124 465)(125 466)(126 467)(127 468)(128 469)(129 470)(130 471)(131 472)(132 473)(133 474)(134 475)(135 476)(136 477)(137 478)(138 479)(139 480)(140 451)(141 452)(142 453)(143 454)(144 455)(145 456)(146 457)(147 458)(148 459)(149 460)(150 461)(151 377)(152 378)(153 379)(154 380)(155 381)(156 382)(157 383)(158 384)(159 385)(160 386)(161 387)(162 388)(163 389)(164 390)(165 361)(166 362)(167 363)(168 364)(169 365)(170 366)(171 367)(172 368)(173 369)(174 370)(175 371)(176 372)(177 373)(178 374)(179 375)(180 376)(181 347)(182 348)(183 349)(184 350)(185 351)(186 352)(187 353)(188 354)(189 355)(190 356)(191 357)(192 358)(193 359)(194 360)(195 331)(196 332)(197 333)(198 334)(199 335)(200 336)(201 337)(202 338)(203 339)(204 340)(205 341)(206 342)(207 343)(208 344)(209 345)(210 346)(211 397)(212 398)(213 399)(214 400)(215 401)(216 402)(217 403)(218 404)(219 405)(220 406)(221 407)(222 408)(223 409)(224 410)(225 411)(226 412)(227 413)(228 414)(229 415)(230 416)(231 417)(232 418)(233 419)(234 420)(235 391)(236 392)(237 393)(238 394)(239 395)(240 396)(301 445)(302 446)(303 447)(304 448)(305 449)(306 450)(307 421)(308 422)(309 423)(310 424)(311 425)(312 426)(313 427)(314 428)(315 429)(316 430)(317 431)(318 432)(319 433)(320 434)(321 435)(322 436)(323 437)(324 438)(325 439)(326 440)(327 441)(328 442)(329 443)(330 444)
(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)
(1 183 100 73)(2 194 101 84)(3 205 102 65)(4 186 103 76)(5 197 104 87)(6 208 105 68)(7 189 106 79)(8 200 107 90)(9 181 108 71)(10 192 109 82)(11 203 110 63)(12 184 111 74)(13 195 112 85)(14 206 113 66)(15 187 114 77)(16 198 115 88)(17 209 116 69)(18 190 117 80)(19 201 118 61)(20 182 119 72)(21 193 120 83)(22 204 91 64)(23 185 92 75)(24 196 93 86)(25 207 94 67)(26 188 95 78)(27 199 96 89)(28 210 97 70)(29 191 98 81)(30 202 99 62)(31 350 290 263)(32 331 291 244)(33 342 292 255)(34 353 293 266)(35 334 294 247)(36 345 295 258)(37 356 296 269)(38 337 297 250)(39 348 298 261)(40 359 299 242)(41 340 300 253)(42 351 271 264)(43 332 272 245)(44 343 273 256)(45 354 274 267)(46 335 275 248)(47 346 276 259)(48 357 277 270)(49 338 278 251)(50 349 279 262)(51 360 280 243)(52 341 281 254)(53 352 282 265)(54 333 283 246)(55 344 284 257)(56 355 285 268)(57 336 286 249)(58 347 287 260)(59 358 288 241)(60 339 289 252)(121 179 216 315)(122 160 217 326)(123 171 218 307)(124 152 219 318)(125 163 220 329)(126 174 221 310)(127 155 222 321)(128 166 223 302)(129 177 224 313)(130 158 225 324)(131 169 226 305)(132 180 227 316)(133 161 228 327)(134 172 229 308)(135 153 230 319)(136 164 231 330)(137 175 232 311)(138 156 233 322)(139 167 234 303)(140 178 235 314)(141 159 236 325)(142 170 237 306)(143 151 238 317)(144 162 239 328)(145 173 240 309)(146 154 211 320)(147 165 212 301)(148 176 213 312)(149 157 214 323)(150 168 215 304)(361 398 445 458)(362 409 446 469)(363 420 447 480)(364 401 448 461)(365 412 449 472)(366 393 450 453)(367 404 421 464)(368 415 422 475)(369 396 423 456)(370 407 424 467)(371 418 425 478)(372 399 426 459)(373 410 427 470)(374 391 428 451)(375 402 429 462)(376 413 430 473)(377 394 431 454)(378 405 432 465)(379 416 433 476)(380 397 434 457)(381 408 435 468)(382 419 436 479)(383 400 437 460)(384 411 438 471)(385 392 439 452)(386 403 440 463)(387 414 441 474)(388 395 442 455)(389 406 443 466)(390 417 444 477)
(1 231 115 121)(2 220 116 140)(3 239 117 129)(4 228 118 148)(5 217 119 137)(6 236 120 126)(7 225 91 145)(8 214 92 134)(9 233 93 123)(10 222 94 142)(11 211 95 131)(12 230 96 150)(13 219 97 139)(14 238 98 128)(15 227 99 147)(16 216 100 136)(17 235 101 125)(18 224 102 144)(19 213 103 133)(20 232 104 122)(21 221 105 141)(22 240 106 130)(23 229 107 149)(24 218 108 138)(25 237 109 127)(26 226 110 146)(27 215 111 135)(28 234 112 124)(29 223 113 143)(30 212 114 132)(31 416 275 461)(32 405 276 480)(33 394 277 469)(34 413 278 458)(35 402 279 477)(36 391 280 466)(37 410 281 455)(38 399 282 474)(39 418 283 463)(40 407 284 452)(41 396 285 471)(42 415 286 460)(43 404 287 479)(44 393 288 468)(45 412 289 457)(46 401 290 476)(47 420 291 465)(48 409 292 454)(49 398 293 473)(50 417 294 462)(51 406 295 451)(52 395 296 470)(53 414 297 459)(54 403 298 478)(55 392 299 467)(56 411 300 456)(57 400 271 475)(58 419 272 464)(59 408 273 453)(60 397 274 472)(61 327 186 176)(62 316 187 165)(63 305 188 154)(64 324 189 173)(65 313 190 162)(66 302 191 151)(67 321 192 170)(68 310 193 159)(69 329 194 178)(70 318 195 167)(71 307 196 156)(72 326 197 175)(73 315 198 164)(74 304 199 153)(75 323 200 172)(76 312 201 161)(77 301 202 180)(78 320 203 169)(79 309 204 158)(80 328 205 177)(81 317 206 166)(82 306 207 155)(83 325 208 174)(84 314 209 163)(85 303 210 152)(86 322 181 171)(87 311 182 160)(88 330 183 179)(89 319 184 168)(90 308 185 157)(241 450 343 381)(242 439 344 370)(243 428 345 389)(244 447 346 378)(245 436 347 367)(246 425 348 386)(247 444 349 375)(248 433 350 364)(249 422 351 383)(250 441 352 372)(251 430 353 361)(252 449 354 380)(253 438 355 369)(254 427 356 388)(255 446 357 377)(256 435 358 366)(257 424 359 385)(258 443 360 374)(259 432 331 363)(260 421 332 382)(261 440 333 371)(262 429 334 390)(263 448 335 379)(264 437 336 368)(265 426 337 387)(266 445 338 376)(267 434 339 365)(268 423 340 384)(269 442 341 373)(270 431 342 362)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + - + + + + - + - + - + - + image C1 C2 C2 C2 C2 C4 S3 D4 Q8 D5 Dic3 D6 D6 D10 D10 Dic6 D12 Dic10 C4×D5 C5⋊D4 S3×D5 D5×Dic3 C5⋊D12 C15⋊Q8 C2×S3×D5 kernel C2×C30.Q8 C30.Q8 C2×C6×Dic5 Dic3×C2×C10 C22×Dic15 C6×Dic5 C22×Dic5 C2×C30 C2×C30 C22×Dic3 C2×Dic5 C2×Dic5 C22×C10 C2×Dic3 C22×C6 C2×C10 C2×C10 C2×C6 C2×C6 C2×C6 C23 C22 C22 C22 C22 # reps 1 4 1 1 1 8 1 2 2 2 4 2 1 4 2 4 4 8 8 8 2 4 4 4 2

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

 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 60 0 0 0 0 1 1 0 0 0 0 0 0 0 60 0 0 0 0 1 1 0 0 0 0 0 0 44 60 0 0 0 0 45 60
,
 38 59 0 0 0 0 21 23 0 0 0 0 0 0 25 51 0 0 0 0 26 36 0 0 0 0 0 0 29 57 0 0 0 0 58 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 58 0 0 0 0 29 56

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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