metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C28)⋊10Q8, C14.9(C4×Q8), (C2×C4)⋊8Dic14, (C2×C4).89D28, C14.7(C4⋊Q8), (C2×Dic14)⋊7C4, (C2×C28).470D4, (C2×C42).14D7, C4.22(D14⋊C4), C2.2(C28⋊2Q8), C2.11(C4×Dic14), C22.34(C2×D28), C28.46(C22⋊C4), C2.1(C4.D28), (C22×C4).395D14, C14.53(C22⋊Q8), C2.2(C28.48D4), C14.10(C4.4D4), C22.40(C4○D28), (C22×Dic14).4C2, C22.17(C2×Dic14), C23.262(C22×D7), C14.C42.10C2, (C22×C28).470C22, (C22×C14).304C23, C7⋊2(C23.67C23), (C22×Dic7).27C22, (C2×C4×C28).10C2, C2.5(C2×D14⋊C4), (C2×C4).109(C4×D7), (C2×C14).24(C2×Q8), C22.117(C2×C4×D7), (C2×C28).224(C2×C4), (C2×C14).424(C2×D4), C14.31(C2×C22⋊C4), (C2×C4⋊Dic7).14C2, C22.39(C2×C7⋊D4), (C2×C14).65(C4○D4), (C2×C4).238(C7⋊D4), (C2×C14).94(C22×C4), (C2×Dic7).25(C2×C4), SmallGroup(448,463)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C28)⋊10Q8
G = < a,b,c,d | a2=b28=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=ab13, dcd-1=c-1 >
Subgroups: 772 in 186 conjugacy classes, 87 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.67C23, C4⋊Dic7, C4×C28, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, C22×C28, C14.C42, C2×C4⋊Dic7, C2×C4×C28, C22×Dic14, (C2×C28)⋊10Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D14, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, Dic14, C4×D7, D28, C7⋊D4, C22×D7, C23.67C23, D14⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C4×Dic14, C28⋊2Q8, C4.D28, C28.48D4, C2×D14⋊C4, (C2×C28)⋊10Q8
►Regular action on 448 points
Generators in S448
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(57 353)(58 354)(59 355)(60 356)(61 357)(62 358)(63 359)(64 360)(65 361)(66 362)(67 363)(68 364)(69 337)(70 338)(71 339)(72 340)(73 341)(74 342)(75 343)(76 344)(77 345)(78 346)(79 347)(80 348)(81 349)(82 350)(83 351)(84 352)(85 277)(86 278)(87 279)(88 280)(89 253)(90 254)(91 255)(92 256)(93 257)(94 258)(95 259)(96 260)(97 261)(98 262)(99 263)(100 264)(101 265)(102 266)(103 267)(104 268)(105 269)(106 270)(107 271)(108 272)(109 273)(110 274)(111 275)(112 276)(113 372)(114 373)(115 374)(116 375)(117 376)(118 377)(119 378)(120 379)(121 380)(122 381)(123 382)(124 383)(125 384)(126 385)(127 386)(128 387)(129 388)(130 389)(131 390)(132 391)(133 392)(134 365)(135 366)(136 367)(137 368)(138 369)(139 370)(140 371)(141 285)(142 286)(143 287)(144 288)(145 289)(146 290)(147 291)(148 292)(149 293)(150 294)(151 295)(152 296)(153 297)(154 298)(155 299)(156 300)(157 301)(158 302)(159 303)(160 304)(161 305)(162 306)(163 307)(164 308)(165 281)(166 282)(167 283)(168 284)(169 425)(170 426)(171 427)(172 428)(173 429)(174 430)(175 431)(176 432)(177 433)(178 434)(179 435)(180 436)(181 437)(182 438)(183 439)(184 440)(185 441)(186 442)(187 443)(188 444)(189 445)(190 446)(191 447)(192 448)(193 421)(194 422)(195 423)(196 424)(197 319)(198 320)(199 321)(200 322)(201 323)(202 324)(203 325)(204 326)(205 327)(206 328)(207 329)(208 330)(209 331)(210 332)(211 333)(212 334)(213 335)(214 336)(215 309)(216 310)(217 311)(218 312)(219 313)(220 314)(221 315)(222 316)(223 317)(224 318)(225 393)(226 394)(227 395)(228 396)(229 397)(230 398)(231 399)(232 400)(233 401)(234 402)(235 403)(236 404)(237 405)(238 406)(239 407)(240 408)(241 409)(242 410)(243 411)(244 412)(245 413)(246 414)(247 415)(248 416)(249 417)(250 418)(251 419)(252 420)
(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)
(1 117 193 63)(2 118 194 64)(3 119 195 65)(4 120 196 66)(5 121 169 67)(6 122 170 68)(7 123 171 69)(8 124 172 70)(9 125 173 71)(10 126 174 72)(11 127 175 73)(12 128 176 74)(13 129 177 75)(14 130 178 76)(15 131 179 77)(16 132 180 78)(17 133 181 79)(18 134 182 80)(19 135 183 81)(20 136 184 82)(21 137 185 83)(22 138 186 84)(23 139 187 57)(24 140 188 58)(25 113 189 59)(26 114 190 60)(27 115 191 61)(28 116 192 62)(29 383 428 338)(30 384 429 339)(31 385 430 340)(32 386 431 341)(33 387 432 342)(34 388 433 343)(35 389 434 344)(36 390 435 345)(37 391 436 346)(38 392 437 347)(39 365 438 348)(40 366 439 349)(41 367 440 350)(42 368 441 351)(43 369 442 352)(44 370 443 353)(45 371 444 354)(46 372 445 355)(47 373 446 356)(48 374 447 357)(49 375 448 358)(50 376 421 359)(51 377 422 360)(52 378 423 361)(53 379 424 362)(54 380 425 363)(55 381 426 364)(56 382 427 337)(85 226 303 327)(86 227 304 328)(87 228 305 329)(88 229 306 330)(89 230 307 331)(90 231 308 332)(91 232 281 333)(92 233 282 334)(93 234 283 335)(94 235 284 336)(95 236 285 309)(96 237 286 310)(97 238 287 311)(98 239 288 312)(99 240 289 313)(100 241 290 314)(101 242 291 315)(102 243 292 316)(103 244 293 317)(104 245 294 318)(105 246 295 319)(106 247 296 320)(107 248 297 321)(108 249 298 322)(109 250 299 323)(110 251 300 324)(111 252 301 325)(112 225 302 326)(141 215 259 404)(142 216 260 405)(143 217 261 406)(144 218 262 407)(145 219 263 408)(146 220 264 409)(147 221 265 410)(148 222 266 411)(149 223 267 412)(150 224 268 413)(151 197 269 414)(152 198 270 415)(153 199 271 416)(154 200 272 417)(155 201 273 418)(156 202 274 419)(157 203 275 420)(158 204 276 393)(159 205 277 394)(160 206 278 395)(161 207 279 396)(162 208 280 397)(163 209 253 398)(164 210 254 399)(165 211 255 400)(166 212 256 401)(167 213 257 402)(168 214 258 403)
(1 280 193 162)(2 101 194 291)(3 278 195 160)(4 99 196 289)(5 276 169 158)(6 97 170 287)(7 274 171 156)(8 95 172 285)(9 272 173 154)(10 93 174 283)(11 270 175 152)(12 91 176 281)(13 268 177 150)(14 89 178 307)(15 266 179 148)(16 87 180 305)(17 264 181 146)(18 85 182 303)(19 262 183 144)(20 111 184 301)(21 260 185 142)(22 109 186 299)(23 258 187 168)(24 107 188 297)(25 256 189 166)(26 105 190 295)(27 254 191 164)(28 103 192 293)(29 259 428 141)(30 108 429 298)(31 257 430 167)(32 106 431 296)(33 255 432 165)(34 104 433 294)(35 253 434 163)(36 102 435 292)(37 279 436 161)(38 100 437 290)(39 277 438 159)(40 98 439 288)(41 275 440 157)(42 96 441 286)(43 273 442 155)(44 94 443 284)(45 271 444 153)(46 92 445 282)(47 269 446 151)(48 90 447 308)(49 267 448 149)(50 88 421 306)(51 265 422 147)(52 86 423 304)(53 263 424 145)(54 112 425 302)(55 261 426 143)(56 110 427 300)(57 403 139 214)(58 248 140 321)(59 401 113 212)(60 246 114 319)(61 399 115 210)(62 244 116 317)(63 397 117 208)(64 242 118 315)(65 395 119 206)(66 240 120 313)(67 393 121 204)(68 238 122 311)(69 419 123 202)(70 236 124 309)(71 417 125 200)(72 234 126 335)(73 415 127 198)(74 232 128 333)(75 413 129 224)(76 230 130 331)(77 411 131 222)(78 228 132 329)(79 409 133 220)(80 226 134 327)(81 407 135 218)(82 252 136 325)(83 405 137 216)(84 250 138 323)(197 356 414 373)(199 354 416 371)(201 352 418 369)(203 350 420 367)(205 348 394 365)(207 346 396 391)(209 344 398 389)(211 342 400 387)(213 340 402 385)(215 338 404 383)(217 364 406 381)(219 362 408 379)(221 360 410 377)(223 358 412 375)(225 380 326 363)(227 378 328 361)(229 376 330 359)(231 374 332 357)(233 372 334 355)(235 370 336 353)(237 368 310 351)(239 366 312 349)(241 392 314 347)(243 390 316 345)(245 388 318 343)(247 386 320 341)(249 384 322 339)(251 382 324 337)
G:=sub<Sym(448)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,353)(58,354)(59,355)(60,356)(61,357)(62,358)(63,359)(64,360)(65,361)(66,362)(67,363)(68,364)(69,337)(70,338)(71,339)(72,340)(73,341)(74,342)(75,343)(76,344)(77,345)(78,346)(79,347)(80,348)(81,349)(82,350)(83,351)(84,352)(85,277)(86,278)(87,279)(88,280)(89,253)(90,254)(91,255)(92,256)(93,257)(94,258)(95,259)(96,260)(97,261)(98,262)(99,263)(100,264)(101,265)(102,266)(103,267)(104,268)(105,269)(106,270)(107,271)(108,272)(109,273)(110,274)(111,275)(112,276)(113,372)(114,373)(115,374)(116,375)(117,376)(118,377)(119,378)(120,379)(121,380)(122,381)(123,382)(124,383)(125,384)(126,385)(127,386)(128,387)(129,388)(130,389)(131,390)(132,391)(133,392)(134,365)(135,366)(136,367)(137,368)(138,369)(139,370)(140,371)(141,285)(142,286)(143,287)(144,288)(145,289)(146,290)(147,291)(148,292)(149,293)(150,294)(151,295)(152,296)(153,297)(154,298)(155,299)(156,300)(157,301)(158,302)(159,303)(160,304)(161,305)(162,306)(163,307)(164,308)(165,281)(166,282)(167,283)(168,284)(169,425)(170,426)(171,427)(172,428)(173,429)(174,430)(175,431)(176,432)(177,433)(178,434)(179,435)(180,436)(181,437)(182,438)(183,439)(184,440)(185,441)(186,442)(187,443)(188,444)(189,445)(190,446)(191,447)(192,448)(193,421)(194,422)(195,423)(196,424)(197,319)(198,320)(199,321)(200,322)(201,323)(202,324)(203,325)(204,326)(205,327)(206,328)(207,329)(208,330)(209,331)(210,332)(211,333)(212,334)(213,335)(214,336)(215,309)(216,310)(217,311)(218,312)(219,313)(220,314)(221,315)(222,316)(223,317)(224,318)(225,393)(226,394)(227,395)(228,396)(229,397)(230,398)(231,399)(232,400)(233,401)(234,402)(235,403)(236,404)(237,405)(238,406)(239,407)(240,408)(241,409)(242,410)(243,411)(244,412)(245,413)(246,414)(247,415)(248,416)(249,417)(250,418)(251,419)(252,420), (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), (1,117,193,63)(2,118,194,64)(3,119,195,65)(4,120,196,66)(5,121,169,67)(6,122,170,68)(7,123,171,69)(8,124,172,70)(9,125,173,71)(10,126,174,72)(11,127,175,73)(12,128,176,74)(13,129,177,75)(14,130,178,76)(15,131,179,77)(16,132,180,78)(17,133,181,79)(18,134,182,80)(19,135,183,81)(20,136,184,82)(21,137,185,83)(22,138,186,84)(23,139,187,57)(24,140,188,58)(25,113,189,59)(26,114,190,60)(27,115,191,61)(28,116,192,62)(29,383,428,338)(30,384,429,339)(31,385,430,340)(32,386,431,341)(33,387,432,342)(34,388,433,343)(35,389,434,344)(36,390,435,345)(37,391,436,346)(38,392,437,347)(39,365,438,348)(40,366,439,349)(41,367,440,350)(42,368,441,351)(43,369,442,352)(44,370,443,353)(45,371,444,354)(46,372,445,355)(47,373,446,356)(48,374,447,357)(49,375,448,358)(50,376,421,359)(51,377,422,360)(52,378,423,361)(53,379,424,362)(54,380,425,363)(55,381,426,364)(56,382,427,337)(85,226,303,327)(86,227,304,328)(87,228,305,329)(88,229,306,330)(89,230,307,331)(90,231,308,332)(91,232,281,333)(92,233,282,334)(93,234,283,335)(94,235,284,336)(95,236,285,309)(96,237,286,310)(97,238,287,311)(98,239,288,312)(99,240,289,313)(100,241,290,314)(101,242,291,315)(102,243,292,316)(103,244,293,317)(104,245,294,318)(105,246,295,319)(106,247,296,320)(107,248,297,321)(108,249,298,322)(109,250,299,323)(110,251,300,324)(111,252,301,325)(112,225,302,326)(141,215,259,404)(142,216,260,405)(143,217,261,406)(144,218,262,407)(145,219,263,408)(146,220,264,409)(147,221,265,410)(148,222,266,411)(149,223,267,412)(150,224,268,413)(151,197,269,414)(152,198,270,415)(153,199,271,416)(154,200,272,417)(155,201,273,418)(156,202,274,419)(157,203,275,420)(158,204,276,393)(159,205,277,394)(160,206,278,395)(161,207,279,396)(162,208,280,397)(163,209,253,398)(164,210,254,399)(165,211,255,400)(166,212,256,401)(167,213,257,402)(168,214,258,403), (1,280,193,162)(2,101,194,291)(3,278,195,160)(4,99,196,289)(5,276,169,158)(6,97,170,287)(7,274,171,156)(8,95,172,285)(9,272,173,154)(10,93,174,283)(11,270,175,152)(12,91,176,281)(13,268,177,150)(14,89,178,307)(15,266,179,148)(16,87,180,305)(17,264,181,146)(18,85,182,303)(19,262,183,144)(20,111,184,301)(21,260,185,142)(22,109,186,299)(23,258,187,168)(24,107,188,297)(25,256,189,166)(26,105,190,295)(27,254,191,164)(28,103,192,293)(29,259,428,141)(30,108,429,298)(31,257,430,167)(32,106,431,296)(33,255,432,165)(34,104,433,294)(35,253,434,163)(36,102,435,292)(37,279,436,161)(38,100,437,290)(39,277,438,159)(40,98,439,288)(41,275,440,157)(42,96,441,286)(43,273,442,155)(44,94,443,284)(45,271,444,153)(46,92,445,282)(47,269,446,151)(48,90,447,308)(49,267,448,149)(50,88,421,306)(51,265,422,147)(52,86,423,304)(53,263,424,145)(54,112,425,302)(55,261,426,143)(56,110,427,300)(57,403,139,214)(58,248,140,321)(59,401,113,212)(60,246,114,319)(61,399,115,210)(62,244,116,317)(63,397,117,208)(64,242,118,315)(65,395,119,206)(66,240,120,313)(67,393,121,204)(68,238,122,311)(69,419,123,202)(70,236,124,309)(71,417,125,200)(72,234,126,335)(73,415,127,198)(74,232,128,333)(75,413,129,224)(76,230,130,331)(77,411,131,222)(78,228,132,329)(79,409,133,220)(80,226,134,327)(81,407,135,218)(82,252,136,325)(83,405,137,216)(84,250,138,323)(197,356,414,373)(199,354,416,371)(201,352,418,369)(203,350,420,367)(205,348,394,365)(207,346,396,391)(209,344,398,389)(211,342,400,387)(213,340,402,385)(215,338,404,383)(217,364,406,381)(219,362,408,379)(221,360,410,377)(223,358,412,375)(225,380,326,363)(227,378,328,361)(229,376,330,359)(231,374,332,357)(233,372,334,355)(235,370,336,353)(237,368,310,351)(239,366,312,349)(241,392,314,347)(243,390,316,345)(245,388,318,343)(247,386,320,341)(249,384,322,339)(251,382,324,337)>;
G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,353)(58,354)(59,355)(60,356)(61,357)(62,358)(63,359)(64,360)(65,361)(66,362)(67,363)(68,364)(69,337)(70,338)(71,339)(72,340)(73,341)(74,342)(75,343)(76,344)(77,345)(78,346)(79,347)(80,348)(81,349)(82,350)(83,351)(84,352)(85,277)(86,278)(87,279)(88,280)(89,253)(90,254)(91,255)(92,256)(93,257)(94,258)(95,259)(96,260)(97,261)(98,262)(99,263)(100,264)(101,265)(102,266)(103,267)(104,268)(105,269)(106,270)(107,271)(108,272)(109,273)(110,274)(111,275)(112,276)(113,372)(114,373)(115,374)(116,375)(117,376)(118,377)(119,378)(120,379)(121,380)(122,381)(123,382)(124,383)(125,384)(126,385)(127,386)(128,387)(129,388)(130,389)(131,390)(132,391)(133,392)(134,365)(135,366)(136,367)(137,368)(138,369)(139,370)(140,371)(141,285)(142,286)(143,287)(144,288)(145,289)(146,290)(147,291)(148,292)(149,293)(150,294)(151,295)(152,296)(153,297)(154,298)(155,299)(156,300)(157,301)(158,302)(159,303)(160,304)(161,305)(162,306)(163,307)(164,308)(165,281)(166,282)(167,283)(168,284)(169,425)(170,426)(171,427)(172,428)(173,429)(174,430)(175,431)(176,432)(177,433)(178,434)(179,435)(180,436)(181,437)(182,438)(183,439)(184,440)(185,441)(186,442)(187,443)(188,444)(189,445)(190,446)(191,447)(192,448)(193,421)(194,422)(195,423)(196,424)(197,319)(198,320)(199,321)(200,322)(201,323)(202,324)(203,325)(204,326)(205,327)(206,328)(207,329)(208,330)(209,331)(210,332)(211,333)(212,334)(213,335)(214,336)(215,309)(216,310)(217,311)(218,312)(219,313)(220,314)(221,315)(222,316)(223,317)(224,318)(225,393)(226,394)(227,395)(228,396)(229,397)(230,398)(231,399)(232,400)(233,401)(234,402)(235,403)(236,404)(237,405)(238,406)(239,407)(240,408)(241,409)(242,410)(243,411)(244,412)(245,413)(246,414)(247,415)(248,416)(249,417)(250,418)(251,419)(252,420), (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), (1,117,193,63)(2,118,194,64)(3,119,195,65)(4,120,196,66)(5,121,169,67)(6,122,170,68)(7,123,171,69)(8,124,172,70)(9,125,173,71)(10,126,174,72)(11,127,175,73)(12,128,176,74)(13,129,177,75)(14,130,178,76)(15,131,179,77)(16,132,180,78)(17,133,181,79)(18,134,182,80)(19,135,183,81)(20,136,184,82)(21,137,185,83)(22,138,186,84)(23,139,187,57)(24,140,188,58)(25,113,189,59)(26,114,190,60)(27,115,191,61)(28,116,192,62)(29,383,428,338)(30,384,429,339)(31,385,430,340)(32,386,431,341)(33,387,432,342)(34,388,433,343)(35,389,434,344)(36,390,435,345)(37,391,436,346)(38,392,437,347)(39,365,438,348)(40,366,439,349)(41,367,440,350)(42,368,441,351)(43,369,442,352)(44,370,443,353)(45,371,444,354)(46,372,445,355)(47,373,446,356)(48,374,447,357)(49,375,448,358)(50,376,421,359)(51,377,422,360)(52,378,423,361)(53,379,424,362)(54,380,425,363)(55,381,426,364)(56,382,427,337)(85,226,303,327)(86,227,304,328)(87,228,305,329)(88,229,306,330)(89,230,307,331)(90,231,308,332)(91,232,281,333)(92,233,282,334)(93,234,283,335)(94,235,284,336)(95,236,285,309)(96,237,286,310)(97,238,287,311)(98,239,288,312)(99,240,289,313)(100,241,290,314)(101,242,291,315)(102,243,292,316)(103,244,293,317)(104,245,294,318)(105,246,295,319)(106,247,296,320)(107,248,297,321)(108,249,298,322)(109,250,299,323)(110,251,300,324)(111,252,301,325)(112,225,302,326)(141,215,259,404)(142,216,260,405)(143,217,261,406)(144,218,262,407)(145,219,263,408)(146,220,264,409)(147,221,265,410)(148,222,266,411)(149,223,267,412)(150,224,268,413)(151,197,269,414)(152,198,270,415)(153,199,271,416)(154,200,272,417)(155,201,273,418)(156,202,274,419)(157,203,275,420)(158,204,276,393)(159,205,277,394)(160,206,278,395)(161,207,279,396)(162,208,280,397)(163,209,253,398)(164,210,254,399)(165,211,255,400)(166,212,256,401)(167,213,257,402)(168,214,258,403), (1,280,193,162)(2,101,194,291)(3,278,195,160)(4,99,196,289)(5,276,169,158)(6,97,170,287)(7,274,171,156)(8,95,172,285)(9,272,173,154)(10,93,174,283)(11,270,175,152)(12,91,176,281)(13,268,177,150)(14,89,178,307)(15,266,179,148)(16,87,180,305)(17,264,181,146)(18,85,182,303)(19,262,183,144)(20,111,184,301)(21,260,185,142)(22,109,186,299)(23,258,187,168)(24,107,188,297)(25,256,189,166)(26,105,190,295)(27,254,191,164)(28,103,192,293)(29,259,428,141)(30,108,429,298)(31,257,430,167)(32,106,431,296)(33,255,432,165)(34,104,433,294)(35,253,434,163)(36,102,435,292)(37,279,436,161)(38,100,437,290)(39,277,438,159)(40,98,439,288)(41,275,440,157)(42,96,441,286)(43,273,442,155)(44,94,443,284)(45,271,444,153)(46,92,445,282)(47,269,446,151)(48,90,447,308)(49,267,448,149)(50,88,421,306)(51,265,422,147)(52,86,423,304)(53,263,424,145)(54,112,425,302)(55,261,426,143)(56,110,427,300)(57,403,139,214)(58,248,140,321)(59,401,113,212)(60,246,114,319)(61,399,115,210)(62,244,116,317)(63,397,117,208)(64,242,118,315)(65,395,119,206)(66,240,120,313)(67,393,121,204)(68,238,122,311)(69,419,123,202)(70,236,124,309)(71,417,125,200)(72,234,126,335)(73,415,127,198)(74,232,128,333)(75,413,129,224)(76,230,130,331)(77,411,131,222)(78,228,132,329)(79,409,133,220)(80,226,134,327)(81,407,135,218)(82,252,136,325)(83,405,137,216)(84,250,138,323)(197,356,414,373)(199,354,416,371)(201,352,418,369)(203,350,420,367)(205,348,394,365)(207,346,396,391)(209,344,398,389)(211,342,400,387)(213,340,402,385)(215,338,404,383)(217,364,406,381)(219,362,408,379)(221,360,410,377)(223,358,412,375)(225,380,326,363)(227,378,328,361)(229,376,330,359)(231,374,332,357)(233,372,334,355)(235,370,336,353)(237,368,310,351)(239,366,312,349)(241,392,314,347)(243,390,316,345)(245,388,318,343)(247,386,320,341)(249,384,322,339)(251,382,324,337) );
G=PermutationGroup([[(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(57,353),(58,354),(59,355),(60,356),(61,357),(62,358),(63,359),(64,360),(65,361),(66,362),(67,363),(68,364),(69,337),(70,338),(71,339),(72,340),(73,341),(74,342),(75,343),(76,344),(77,345),(78,346),(79,347),(80,348),(81,349),(82,350),(83,351),(84,352),(85,277),(86,278),(87,279),(88,280),(89,253),(90,254),(91,255),(92,256),(93,257),(94,258),(95,259),(96,260),(97,261),(98,262),(99,263),(100,264),(101,265),(102,266),(103,267),(104,268),(105,269),(106,270),(107,271),(108,272),(109,273),(110,274),(111,275),(112,276),(113,372),(114,373),(115,374),(116,375),(117,376),(118,377),(119,378),(120,379),(121,380),(122,381),(123,382),(124,383),(125,384),(126,385),(127,386),(128,387),(129,388),(130,389),(131,390),(132,391),(133,392),(134,365),(135,366),(136,367),(137,368),(138,369),(139,370),(140,371),(141,285),(142,286),(143,287),(144,288),(145,289),(146,290),(147,291),(148,292),(149,293),(150,294),(151,295),(152,296),(153,297),(154,298),(155,299),(156,300),(157,301),(158,302),(159,303),(160,304),(161,305),(162,306),(163,307),(164,308),(165,281),(166,282),(167,283),(168,284),(169,425),(170,426),(171,427),(172,428),(173,429),(174,430),(175,431),(176,432),(177,433),(178,434),(179,435),(180,436),(181,437),(182,438),(183,439),(184,440),(185,441),(186,442),(187,443),(188,444),(189,445),(190,446),(191,447),(192,448),(193,421),(194,422),(195,423),(196,424),(197,319),(198,320),(199,321),(200,322),(201,323),(202,324),(203,325),(204,326),(205,327),(206,328),(207,329),(208,330),(209,331),(210,332),(211,333),(212,334),(213,335),(214,336),(215,309),(216,310),(217,311),(218,312),(219,313),(220,314),(221,315),(222,316),(223,317),(224,318),(225,393),(226,394),(227,395),(228,396),(229,397),(230,398),(231,399),(232,400),(233,401),(234,402),(235,403),(236,404),(237,405),(238,406),(239,407),(240,408),(241,409),(242,410),(243,411),(244,412),(245,413),(246,414),(247,415),(248,416),(249,417),(250,418),(251,419),(252,420)], [(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)], [(1,117,193,63),(2,118,194,64),(3,119,195,65),(4,120,196,66),(5,121,169,67),(6,122,170,68),(7,123,171,69),(8,124,172,70),(9,125,173,71),(10,126,174,72),(11,127,175,73),(12,128,176,74),(13,129,177,75),(14,130,178,76),(15,131,179,77),(16,132,180,78),(17,133,181,79),(18,134,182,80),(19,135,183,81),(20,136,184,82),(21,137,185,83),(22,138,186,84),(23,139,187,57),(24,140,188,58),(25,113,189,59),(26,114,190,60),(27,115,191,61),(28,116,192,62),(29,383,428,338),(30,384,429,339),(31,385,430,340),(32,386,431,341),(33,387,432,342),(34,388,433,343),(35,389,434,344),(36,390,435,345),(37,391,436,346),(38,392,437,347),(39,365,438,348),(40,366,439,349),(41,367,440,350),(42,368,441,351),(43,369,442,352),(44,370,443,353),(45,371,444,354),(46,372,445,355),(47,373,446,356),(48,374,447,357),(49,375,448,358),(50,376,421,359),(51,377,422,360),(52,378,423,361),(53,379,424,362),(54,380,425,363),(55,381,426,364),(56,382,427,337),(85,226,303,327),(86,227,304,328),(87,228,305,329),(88,229,306,330),(89,230,307,331),(90,231,308,332),(91,232,281,333),(92,233,282,334),(93,234,283,335),(94,235,284,336),(95,236,285,309),(96,237,286,310),(97,238,287,311),(98,239,288,312),(99,240,289,313),(100,241,290,314),(101,242,291,315),(102,243,292,316),(103,244,293,317),(104,245,294,318),(105,246,295,319),(106,247,296,320),(107,248,297,321),(108,249,298,322),(109,250,299,323),(110,251,300,324),(111,252,301,325),(112,225,302,326),(141,215,259,404),(142,216,260,405),(143,217,261,406),(144,218,262,407),(145,219,263,408),(146,220,264,409),(147,221,265,410),(148,222,266,411),(149,223,267,412),(150,224,268,413),(151,197,269,414),(152,198,270,415),(153,199,271,416),(154,200,272,417),(155,201,273,418),(156,202,274,419),(157,203,275,420),(158,204,276,393),(159,205,277,394),(160,206,278,395),(161,207,279,396),(162,208,280,397),(163,209,253,398),(164,210,254,399),(165,211,255,400),(166,212,256,401),(167,213,257,402),(168,214,258,403)], [(1,280,193,162),(2,101,194,291),(3,278,195,160),(4,99,196,289),(5,276,169,158),(6,97,170,287),(7,274,171,156),(8,95,172,285),(9,272,173,154),(10,93,174,283),(11,270,175,152),(12,91,176,281),(13,268,177,150),(14,89,178,307),(15,266,179,148),(16,87,180,305),(17,264,181,146),(18,85,182,303),(19,262,183,144),(20,111,184,301),(21,260,185,142),(22,109,186,299),(23,258,187,168),(24,107,188,297),(25,256,189,166),(26,105,190,295),(27,254,191,164),(28,103,192,293),(29,259,428,141),(30,108,429,298),(31,257,430,167),(32,106,431,296),(33,255,432,165),(34,104,433,294),(35,253,434,163),(36,102,435,292),(37,279,436,161),(38,100,437,290),(39,277,438,159),(40,98,439,288),(41,275,440,157),(42,96,441,286),(43,273,442,155),(44,94,443,284),(45,271,444,153),(46,92,445,282),(47,269,446,151),(48,90,447,308),(49,267,448,149),(50,88,421,306),(51,265,422,147),(52,86,423,304),(53,263,424,145),(54,112,425,302),(55,261,426,143),(56,110,427,300),(57,403,139,214),(58,248,140,321),(59,401,113,212),(60,246,114,319),(61,399,115,210),(62,244,116,317),(63,397,117,208),(64,242,118,315),(65,395,119,206),(66,240,120,313),(67,393,121,204),(68,238,122,311),(69,419,123,202),(70,236,124,309),(71,417,125,200),(72,234,126,335),(73,415,127,198),(74,232,128,333),(75,413,129,224),(76,230,130,331),(77,411,131,222),(78,228,132,329),(79,409,133,220),(80,226,134,327),(81,407,135,218),(82,252,136,325),(83,405,137,216),(84,250,138,323),(197,356,414,373),(199,354,416,371),(201,352,418,369),(203,350,420,367),(205,348,394,365),(207,346,396,391),(209,344,398,389),(211,342,400,387),(213,340,402,385),(215,338,404,383),(217,364,406,381),(219,362,408,379),(221,360,410,377),(223,358,412,375),(225,380,326,363),(227,378,328,361),(229,376,330,359),(231,374,332,357),(233,372,334,355),(235,370,336,353),(237,368,310,351),(239,366,312,349),(241,392,314,347),(243,390,316,345),(245,388,318,343),(247,386,320,341),(249,384,322,339),(251,382,324,337)]])
124 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D7 | C4○D4 | D14 | Dic14 | C4×D7 | D28 | C7⋊D4 | C4○D28 |
| kernel | (C2×C28)⋊10Q8 | C14.C42 | C2×C4⋊Dic7 | C2×C4×C28 | C22×Dic14 | C2×Dic14 | C2×C28 | C2×C28 | C2×C42 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 4 | 3 | 4 | 9 | 24 | 12 | 12 | 12 | 24 |
Matrix representation of (C2×C28)⋊10Q8 ►in GL5(𝔽29)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 21 | 4 | 0 | 0 |
| 0 | 25 | 20 | 0 | 0 |
| 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 23 | 3 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 8 | 6 | 0 | 0 |
| 0 | 23 | 21 | 0 | 0 |
| 0 | 0 | 0 | 3 | 18 |
| 0 | 0 | 0 | 22 | 26 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 |
| 0 | 3 | 12 | 0 | 0 |
| 0 | 0 | 0 | 13 | 24 |
| 0 | 0 | 0 | 5 | 16 |
G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,21,25,0,0,0,4,20,0,0,0,0,0,4,23,0,0,0,3,3],[28,0,0,0,0,0,8,23,0,0,0,6,21,0,0,0,0,0,3,22,0,0,0,18,26],[28,0,0,0,0,0,17,3,0,0,0,0,12,0,0,0,0,0,13,5,0,0,0,24,16] >;
(C2×C28)⋊10Q8 in GAP, Magma, Sage, TeX
(C_2\times C_{28})\rtimes_{10}Q_8 % in TeX
G:=Group("(C2xC28):10Q8"); // GroupNames label
G:=SmallGroup(448,463);
// by ID
G=gap.SmallGroup(448,463);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,758,58,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a*b^13,d*c*d^-1=c^-1>;
// generators/relations