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G = C2×C30.4Q8  order 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, C30⋊7(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, C10⋊4(Dic3⋊C4), Dic15⋊20(C2×C4), (C2×Dic15)⋊10C4, (C2×C12).381D10, C6⋊3(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(C15⋊7D4), (C22×Dic15).4C2, C22.20(C22×D15), (C22×C30).137C22, (C2×Dic15).167C22, C15⋊17(C2×C4⋊C4), C6.74(C2×C4×D5), C5⋊5(C2×Dic3⋊C4), C2.18(C2×C4×D15), C10.106(S3×C2×C4), (C2×C6).35(C4×D5), C2.1(C2×C15⋊7D4), C3⋊4(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

Derived series
C1 — C30 — C2×C30.4Q8
Chief series
C1 — C5 — C15 — C30 — C2×C30 — C2×Dic15 — C22×Dic15 — C2×C30.4Q8
Lower central
C15 — C30 — C2×C30.4Q8
Upper central
C1 — C23 — C22×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, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×C6, C30, C30, C2×C4⋊C4, C2×Dic5, C2×C20, C2×C20, C22×C10, Dic3⋊C4, C22×Dic3, C22×C12, Dic15, Dic15, C60, C2×C30, C2×C30, C10.D4, C22×Dic5, C22×C20, C2×Dic3⋊C4, C2×Dic15, C2×Dic15, C2×C60, C2×C60, C22×C30, C2×C10.D4, C30.4Q8, C22×Dic15, C22×C60, C2×C30.4Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D5, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, Dic6, C4×S3, C3⋊D4, C22×S3, D15, C2×C4⋊C4, Dic10, C4×D5, C5⋊D4, C22×D5, Dic3⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, D30, C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×Dic3⋊C4, Dic30, C4×D15, C15⋊7D4, C22×D15, C2×C10.D4, C30.4Q8, C2×Dic30, C2×C4×D15, C2×C15⋊7D4, C2×C30.4Q8

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

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

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

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×D15C15⋊7D4
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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