metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C28).53D4, (C2×C4).44D28, C14.27(C4⋊Q8), (C2×Dic7).7Q8, C22.48(Q8×D7), C2.9(C28⋊7D4), C14.61(C4⋊D4), (C22×C4).100D14, C22.128(C2×D28), C14.47(C22⋊Q8), C2.16(D14⋊2Q8), C2.7(Dic7⋊Q8), C2.13(Dic7.Q8), (C22×C28).64C22, C14.22(C42.C2), C23.377(C22×D7), C22.105(C4○D28), C14.C42.28C2, (C22×C14).347C23, C7⋊4(C23.81C23), C22.101(D4⋊2D7), C14.75(C22.D4), C2.9(C23.18D14), (C22×Dic7).55C22, (C2×C4⋊C4).20D7, (C14×C4⋊C4).22C2, (C2×C14).82(C2×Q8), (C2×C14).151(C2×D4), (C2×C4).38(C7⋊D4), (C2×C4⋊Dic7).18C2, (C2×C14).84(C4○D4), (C2×Dic7⋊C4).32C2, C22.137(C2×C7⋊D4), SmallGroup(448,517)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C28).53D4
G = < a,b,c,d | a2=b28=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 612 in 150 conjugacy classes, 63 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×11], C22 [×3], C22 [×4], C7, C2×C4 [×4], C2×C4 [×21], C23, C14 [×3], C14 [×4], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×4], Dic7 [×6], C28 [×5], C2×C14 [×3], C2×C14 [×4], C2.C42 [×3], C2×C4⋊C4, C2×C4⋊C4 [×3], C2×Dic7 [×4], C2×Dic7 [×10], C2×C28 [×4], C2×C28 [×7], C22×C14, C23.81C23, Dic7⋊C4 [×4], C4⋊Dic7 [×2], C7×C4⋊C4 [×2], C22×Dic7 [×4], C22×C28 [×3], C14.C42, C14.C42 [×2], C2×Dic7⋊C4 [×2], C2×C4⋊Dic7, C14×C4⋊C4, (C2×C28).53D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, D7, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], D14 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, D28 [×2], C7⋊D4 [×2], C22×D7, C23.81C23, C2×D28, C4○D28, D4⋊2D7 [×2], Q8×D7 [×2], C2×C7⋊D4, Dic7.Q8 [×2], D14⋊2Q8 [×2], C28⋊7D4, C23.18D14, Dic7⋊Q8, (C2×C28).53D4
►Regular action on 448 points
Generators in S448
(1 182)(2 183)(3 184)(4 185)(5 186)(6 187)(7 188)(8 189)(9 190)(10 191)(11 192)(12 193)(13 194)(14 195)(15 196)(16 169)(17 170)(18 171)(19 172)(20 173)(21 174)(22 175)(23 176)(24 177)(25 178)(26 179)(27 180)(28 181)(29 371)(30 372)(31 373)(32 374)(33 375)(34 376)(35 377)(36 378)(37 379)(38 380)(39 381)(40 382)(41 383)(42 384)(43 385)(44 386)(45 387)(46 388)(47 389)(48 390)(49 391)(50 392)(51 365)(52 366)(53 367)(54 368)(55 369)(56 370)(57 342)(58 343)(59 344)(60 345)(61 346)(62 347)(63 348)(64 349)(65 350)(66 351)(67 352)(68 353)(69 354)(70 355)(71 356)(72 357)(73 358)(74 359)(75 360)(76 361)(77 362)(78 363)(79 364)(80 337)(81 338)(82 339)(83 340)(84 341)(85 166)(86 167)(87 168)(88 141)(89 142)(90 143)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 151)(99 152)(100 153)(101 154)(102 155)(103 156)(104 157)(105 158)(106 159)(107 160)(108 161)(109 162)(110 163)(111 164)(112 165)(113 320)(114 321)(115 322)(116 323)(117 324)(118 325)(119 326)(120 327)(121 328)(122 329)(123 330)(124 331)(125 332)(126 333)(127 334)(128 335)(129 336)(130 309)(131 310)(132 311)(133 312)(134 313)(135 314)(136 315)(137 316)(138 317)(139 318)(140 319)(197 279)(198 280)(199 253)(200 254)(201 255)(202 256)(203 257)(204 258)(205 259)(206 260)(207 261)(208 262)(209 263)(210 264)(211 265)(212 266)(213 267)(214 268)(215 269)(216 270)(217 271)(218 272)(219 273)(220 274)(221 275)(222 276)(223 277)(224 278)(225 302)(226 303)(227 304)(228 305)(229 306)(230 307)(231 308)(232 281)(233 282)(234 283)(235 284)(236 285)(237 286)(238 287)(239 288)(240 289)(241 290)(242 291)(243 292)(244 293)(245 294)(246 295)(247 296)(248 297)(249 298)(250 299)(251 300)(252 301)(393 431)(394 432)(395 433)(396 434)(397 435)(398 436)(399 437)(400 438)(401 439)(402 440)(403 441)(404 442)(405 443)(406 444)(407 445)(408 446)(409 447)(410 448)(411 421)(412 422)(413 423)(414 424)(415 425)(416 426)(417 427)(418 428)(419 429)(420 430)
(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 34 141 295)(2 375 142 245)(3 32 143 293)(4 373 144 243)(5 30 145 291)(6 371 146 241)(7 56 147 289)(8 369 148 239)(9 54 149 287)(10 367 150 237)(11 52 151 285)(12 365 152 235)(13 50 153 283)(14 391 154 233)(15 48 155 281)(16 389 156 231)(17 46 157 307)(18 387 158 229)(19 44 159 305)(20 385 160 227)(21 42 161 303)(22 383 162 225)(23 40 163 301)(24 381 164 251)(25 38 165 299)(26 379 166 249)(27 36 167 297)(28 377 168 247)(29 93 290 187)(31 91 292 185)(33 89 294 183)(35 87 296 181)(37 85 298 179)(39 111 300 177)(41 109 302 175)(43 107 304 173)(45 105 306 171)(47 103 308 169)(49 101 282 195)(51 99 284 193)(53 97 286 191)(55 95 288 189)(57 398 133 266)(58 435 134 211)(59 396 135 264)(60 433 136 209)(61 394 137 262)(62 431 138 207)(63 420 139 260)(64 429 140 205)(65 418 113 258)(66 427 114 203)(67 416 115 256)(68 425 116 201)(69 414 117 254)(70 423 118 199)(71 412 119 280)(72 421 120 197)(73 410 121 278)(74 447 122 223)(75 408 123 276)(76 445 124 221)(77 406 125 274)(78 443 126 219)(79 404 127 272)(80 441 128 217)(81 402 129 270)(82 439 130 215)(83 400 131 268)(84 437 132 213)(86 248 180 378)(88 246 182 376)(90 244 184 374)(92 242 186 372)(94 240 188 370)(96 238 190 368)(98 236 192 366)(100 234 194 392)(102 232 196 390)(104 230 170 388)(106 228 172 386)(108 226 174 384)(110 252 176 382)(112 250 178 380)(198 356 422 326)(200 354 424 324)(202 352 426 322)(204 350 428 320)(206 348 430 318)(208 346 432 316)(210 344 434 314)(212 342 436 312)(214 340 438 310)(216 338 440 336)(218 364 442 334)(220 362 444 332)(222 360 446 330)(224 358 448 328)(253 355 413 325)(255 353 415 323)(257 351 417 321)(259 349 419 319)(261 347 393 317)(263 345 395 315)(265 343 397 313)(267 341 399 311)(269 339 401 309)(271 337 403 335)(273 363 405 333)(275 361 407 331)(277 359 409 329)(279 357 411 327)
(1 117 182 324)(2 116 183 323)(3 115 184 322)(4 114 185 321)(5 113 186 320)(6 140 187 319)(7 139 188 318)(8 138 189 317)(9 137 190 316)(10 136 191 315)(11 135 192 314)(12 134 193 313)(13 133 194 312)(14 132 195 311)(15 131 196 310)(16 130 169 309)(17 129 170 336)(18 128 171 335)(19 127 172 334)(20 126 173 333)(21 125 174 332)(22 124 175 331)(23 123 176 330)(24 122 177 329)(25 121 178 328)(26 120 179 327)(27 119 180 326)(28 118 181 325)(29 419 371 429)(30 418 372 428)(31 417 373 427)(32 416 374 426)(33 415 375 425)(34 414 376 424)(35 413 377 423)(36 412 378 422)(37 411 379 421)(38 410 380 448)(39 409 381 447)(40 408 382 446)(41 407 383 445)(42 406 384 444)(43 405 385 443)(44 404 386 442)(45 403 387 441)(46 402 388 440)(47 401 389 439)(48 400 390 438)(49 399 391 437)(50 398 392 436)(51 397 365 435)(52 396 366 434)(53 395 367 433)(54 394 368 432)(55 393 369 431)(56 420 370 430)(57 100 342 153)(58 99 343 152)(59 98 344 151)(60 97 345 150)(61 96 346 149)(62 95 347 148)(63 94 348 147)(64 93 349 146)(65 92 350 145)(66 91 351 144)(67 90 352 143)(68 89 353 142)(69 88 354 141)(70 87 355 168)(71 86 356 167)(72 85 357 166)(73 112 358 165)(74 111 359 164)(75 110 360 163)(76 109 361 162)(77 108 362 161)(78 107 363 160)(79 106 364 159)(80 105 337 158)(81 104 338 157)(82 103 339 156)(83 102 340 155)(84 101 341 154)(197 298 279 249)(198 297 280 248)(199 296 253 247)(200 295 254 246)(201 294 255 245)(202 293 256 244)(203 292 257 243)(204 291 258 242)(205 290 259 241)(206 289 260 240)(207 288 261 239)(208 287 262 238)(209 286 263 237)(210 285 264 236)(211 284 265 235)(212 283 266 234)(213 282 267 233)(214 281 268 232)(215 308 269 231)(216 307 270 230)(217 306 271 229)(218 305 272 228)(219 304 273 227)(220 303 274 226)(221 302 275 225)(222 301 276 252)(223 300 277 251)(224 299 278 250)
G:=sub<Sym(448)| (1,182)(2,183)(3,184)(4,185)(5,186)(6,187)(7,188)(8,189)(9,190)(10,191)(11,192)(12,193)(13,194)(14,195)(15,196)(16,169)(17,170)(18,171)(19,172)(20,173)(21,174)(22,175)(23,176)(24,177)(25,178)(26,179)(27,180)(28,181)(29,371)(30,372)(31,373)(32,374)(33,375)(34,376)(35,377)(36,378)(37,379)(38,380)(39,381)(40,382)(41,383)(42,384)(43,385)(44,386)(45,387)(46,388)(47,389)(48,390)(49,391)(50,392)(51,365)(52,366)(53,367)(54,368)(55,369)(56,370)(57,342)(58,343)(59,344)(60,345)(61,346)(62,347)(63,348)(64,349)(65,350)(66,351)(67,352)(68,353)(69,354)(70,355)(71,356)(72,357)(73,358)(74,359)(75,360)(76,361)(77,362)(78,363)(79,364)(80,337)(81,338)(82,339)(83,340)(84,341)(85,166)(86,167)(87,168)(88,141)(89,142)(90,143)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,151)(99,152)(100,153)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,161)(109,162)(110,163)(111,164)(112,165)(113,320)(114,321)(115,322)(116,323)(117,324)(118,325)(119,326)(120,327)(121,328)(122,329)(123,330)(124,331)(125,332)(126,333)(127,334)(128,335)(129,336)(130,309)(131,310)(132,311)(133,312)(134,313)(135,314)(136,315)(137,316)(138,317)(139,318)(140,319)(197,279)(198,280)(199,253)(200,254)(201,255)(202,256)(203,257)(204,258)(205,259)(206,260)(207,261)(208,262)(209,263)(210,264)(211,265)(212,266)(213,267)(214,268)(215,269)(216,270)(217,271)(218,272)(219,273)(220,274)(221,275)(222,276)(223,277)(224,278)(225,302)(226,303)(227,304)(228,305)(229,306)(230,307)(231,308)(232,281)(233,282)(234,283)(235,284)(236,285)(237,286)(238,287)(239,288)(240,289)(241,290)(242,291)(243,292)(244,293)(245,294)(246,295)(247,296)(248,297)(249,298)(250,299)(251,300)(252,301)(393,431)(394,432)(395,433)(396,434)(397,435)(398,436)(399,437)(400,438)(401,439)(402,440)(403,441)(404,442)(405,443)(406,444)(407,445)(408,446)(409,447)(410,448)(411,421)(412,422)(413,423)(414,424)(415,425)(416,426)(417,427)(418,428)(419,429)(420,430), (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), (1,34,141,295)(2,375,142,245)(3,32,143,293)(4,373,144,243)(5,30,145,291)(6,371,146,241)(7,56,147,289)(8,369,148,239)(9,54,149,287)(10,367,150,237)(11,52,151,285)(12,365,152,235)(13,50,153,283)(14,391,154,233)(15,48,155,281)(16,389,156,231)(17,46,157,307)(18,387,158,229)(19,44,159,305)(20,385,160,227)(21,42,161,303)(22,383,162,225)(23,40,163,301)(24,381,164,251)(25,38,165,299)(26,379,166,249)(27,36,167,297)(28,377,168,247)(29,93,290,187)(31,91,292,185)(33,89,294,183)(35,87,296,181)(37,85,298,179)(39,111,300,177)(41,109,302,175)(43,107,304,173)(45,105,306,171)(47,103,308,169)(49,101,282,195)(51,99,284,193)(53,97,286,191)(55,95,288,189)(57,398,133,266)(58,435,134,211)(59,396,135,264)(60,433,136,209)(61,394,137,262)(62,431,138,207)(63,420,139,260)(64,429,140,205)(65,418,113,258)(66,427,114,203)(67,416,115,256)(68,425,116,201)(69,414,117,254)(70,423,118,199)(71,412,119,280)(72,421,120,197)(73,410,121,278)(74,447,122,223)(75,408,123,276)(76,445,124,221)(77,406,125,274)(78,443,126,219)(79,404,127,272)(80,441,128,217)(81,402,129,270)(82,439,130,215)(83,400,131,268)(84,437,132,213)(86,248,180,378)(88,246,182,376)(90,244,184,374)(92,242,186,372)(94,240,188,370)(96,238,190,368)(98,236,192,366)(100,234,194,392)(102,232,196,390)(104,230,170,388)(106,228,172,386)(108,226,174,384)(110,252,176,382)(112,250,178,380)(198,356,422,326)(200,354,424,324)(202,352,426,322)(204,350,428,320)(206,348,430,318)(208,346,432,316)(210,344,434,314)(212,342,436,312)(214,340,438,310)(216,338,440,336)(218,364,442,334)(220,362,444,332)(222,360,446,330)(224,358,448,328)(253,355,413,325)(255,353,415,323)(257,351,417,321)(259,349,419,319)(261,347,393,317)(263,345,395,315)(265,343,397,313)(267,341,399,311)(269,339,401,309)(271,337,403,335)(273,363,405,333)(275,361,407,331)(277,359,409,329)(279,357,411,327), (1,117,182,324)(2,116,183,323)(3,115,184,322)(4,114,185,321)(5,113,186,320)(6,140,187,319)(7,139,188,318)(8,138,189,317)(9,137,190,316)(10,136,191,315)(11,135,192,314)(12,134,193,313)(13,133,194,312)(14,132,195,311)(15,131,196,310)(16,130,169,309)(17,129,170,336)(18,128,171,335)(19,127,172,334)(20,126,173,333)(21,125,174,332)(22,124,175,331)(23,123,176,330)(24,122,177,329)(25,121,178,328)(26,120,179,327)(27,119,180,326)(28,118,181,325)(29,419,371,429)(30,418,372,428)(31,417,373,427)(32,416,374,426)(33,415,375,425)(34,414,376,424)(35,413,377,423)(36,412,378,422)(37,411,379,421)(38,410,380,448)(39,409,381,447)(40,408,382,446)(41,407,383,445)(42,406,384,444)(43,405,385,443)(44,404,386,442)(45,403,387,441)(46,402,388,440)(47,401,389,439)(48,400,390,438)(49,399,391,437)(50,398,392,436)(51,397,365,435)(52,396,366,434)(53,395,367,433)(54,394,368,432)(55,393,369,431)(56,420,370,430)(57,100,342,153)(58,99,343,152)(59,98,344,151)(60,97,345,150)(61,96,346,149)(62,95,347,148)(63,94,348,147)(64,93,349,146)(65,92,350,145)(66,91,351,144)(67,90,352,143)(68,89,353,142)(69,88,354,141)(70,87,355,168)(71,86,356,167)(72,85,357,166)(73,112,358,165)(74,111,359,164)(75,110,360,163)(76,109,361,162)(77,108,362,161)(78,107,363,160)(79,106,364,159)(80,105,337,158)(81,104,338,157)(82,103,339,156)(83,102,340,155)(84,101,341,154)(197,298,279,249)(198,297,280,248)(199,296,253,247)(200,295,254,246)(201,294,255,245)(202,293,256,244)(203,292,257,243)(204,291,258,242)(205,290,259,241)(206,289,260,240)(207,288,261,239)(208,287,262,238)(209,286,263,237)(210,285,264,236)(211,284,265,235)(212,283,266,234)(213,282,267,233)(214,281,268,232)(215,308,269,231)(216,307,270,230)(217,306,271,229)(218,305,272,228)(219,304,273,227)(220,303,274,226)(221,302,275,225)(222,301,276,252)(223,300,277,251)(224,299,278,250)>;
G:=Group( (1,182)(2,183)(3,184)(4,185)(5,186)(6,187)(7,188)(8,189)(9,190)(10,191)(11,192)(12,193)(13,194)(14,195)(15,196)(16,169)(17,170)(18,171)(19,172)(20,173)(21,174)(22,175)(23,176)(24,177)(25,178)(26,179)(27,180)(28,181)(29,371)(30,372)(31,373)(32,374)(33,375)(34,376)(35,377)(36,378)(37,379)(38,380)(39,381)(40,382)(41,383)(42,384)(43,385)(44,386)(45,387)(46,388)(47,389)(48,390)(49,391)(50,392)(51,365)(52,366)(53,367)(54,368)(55,369)(56,370)(57,342)(58,343)(59,344)(60,345)(61,346)(62,347)(63,348)(64,349)(65,350)(66,351)(67,352)(68,353)(69,354)(70,355)(71,356)(72,357)(73,358)(74,359)(75,360)(76,361)(77,362)(78,363)(79,364)(80,337)(81,338)(82,339)(83,340)(84,341)(85,166)(86,167)(87,168)(88,141)(89,142)(90,143)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,151)(99,152)(100,153)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,161)(109,162)(110,163)(111,164)(112,165)(113,320)(114,321)(115,322)(116,323)(117,324)(118,325)(119,326)(120,327)(121,328)(122,329)(123,330)(124,331)(125,332)(126,333)(127,334)(128,335)(129,336)(130,309)(131,310)(132,311)(133,312)(134,313)(135,314)(136,315)(137,316)(138,317)(139,318)(140,319)(197,279)(198,280)(199,253)(200,254)(201,255)(202,256)(203,257)(204,258)(205,259)(206,260)(207,261)(208,262)(209,263)(210,264)(211,265)(212,266)(213,267)(214,268)(215,269)(216,270)(217,271)(218,272)(219,273)(220,274)(221,275)(222,276)(223,277)(224,278)(225,302)(226,303)(227,304)(228,305)(229,306)(230,307)(231,308)(232,281)(233,282)(234,283)(235,284)(236,285)(237,286)(238,287)(239,288)(240,289)(241,290)(242,291)(243,292)(244,293)(245,294)(246,295)(247,296)(248,297)(249,298)(250,299)(251,300)(252,301)(393,431)(394,432)(395,433)(396,434)(397,435)(398,436)(399,437)(400,438)(401,439)(402,440)(403,441)(404,442)(405,443)(406,444)(407,445)(408,446)(409,447)(410,448)(411,421)(412,422)(413,423)(414,424)(415,425)(416,426)(417,427)(418,428)(419,429)(420,430), (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), (1,34,141,295)(2,375,142,245)(3,32,143,293)(4,373,144,243)(5,30,145,291)(6,371,146,241)(7,56,147,289)(8,369,148,239)(9,54,149,287)(10,367,150,237)(11,52,151,285)(12,365,152,235)(13,50,153,283)(14,391,154,233)(15,48,155,281)(16,389,156,231)(17,46,157,307)(18,387,158,229)(19,44,159,305)(20,385,160,227)(21,42,161,303)(22,383,162,225)(23,40,163,301)(24,381,164,251)(25,38,165,299)(26,379,166,249)(27,36,167,297)(28,377,168,247)(29,93,290,187)(31,91,292,185)(33,89,294,183)(35,87,296,181)(37,85,298,179)(39,111,300,177)(41,109,302,175)(43,107,304,173)(45,105,306,171)(47,103,308,169)(49,101,282,195)(51,99,284,193)(53,97,286,191)(55,95,288,189)(57,398,133,266)(58,435,134,211)(59,396,135,264)(60,433,136,209)(61,394,137,262)(62,431,138,207)(63,420,139,260)(64,429,140,205)(65,418,113,258)(66,427,114,203)(67,416,115,256)(68,425,116,201)(69,414,117,254)(70,423,118,199)(71,412,119,280)(72,421,120,197)(73,410,121,278)(74,447,122,223)(75,408,123,276)(76,445,124,221)(77,406,125,274)(78,443,126,219)(79,404,127,272)(80,441,128,217)(81,402,129,270)(82,439,130,215)(83,400,131,268)(84,437,132,213)(86,248,180,378)(88,246,182,376)(90,244,184,374)(92,242,186,372)(94,240,188,370)(96,238,190,368)(98,236,192,366)(100,234,194,392)(102,232,196,390)(104,230,170,388)(106,228,172,386)(108,226,174,384)(110,252,176,382)(112,250,178,380)(198,356,422,326)(200,354,424,324)(202,352,426,322)(204,350,428,320)(206,348,430,318)(208,346,432,316)(210,344,434,314)(212,342,436,312)(214,340,438,310)(216,338,440,336)(218,364,442,334)(220,362,444,332)(222,360,446,330)(224,358,448,328)(253,355,413,325)(255,353,415,323)(257,351,417,321)(259,349,419,319)(261,347,393,317)(263,345,395,315)(265,343,397,313)(267,341,399,311)(269,339,401,309)(271,337,403,335)(273,363,405,333)(275,361,407,331)(277,359,409,329)(279,357,411,327), (1,117,182,324)(2,116,183,323)(3,115,184,322)(4,114,185,321)(5,113,186,320)(6,140,187,319)(7,139,188,318)(8,138,189,317)(9,137,190,316)(10,136,191,315)(11,135,192,314)(12,134,193,313)(13,133,194,312)(14,132,195,311)(15,131,196,310)(16,130,169,309)(17,129,170,336)(18,128,171,335)(19,127,172,334)(20,126,173,333)(21,125,174,332)(22,124,175,331)(23,123,176,330)(24,122,177,329)(25,121,178,328)(26,120,179,327)(27,119,180,326)(28,118,181,325)(29,419,371,429)(30,418,372,428)(31,417,373,427)(32,416,374,426)(33,415,375,425)(34,414,376,424)(35,413,377,423)(36,412,378,422)(37,411,379,421)(38,410,380,448)(39,409,381,447)(40,408,382,446)(41,407,383,445)(42,406,384,444)(43,405,385,443)(44,404,386,442)(45,403,387,441)(46,402,388,440)(47,401,389,439)(48,400,390,438)(49,399,391,437)(50,398,392,436)(51,397,365,435)(52,396,366,434)(53,395,367,433)(54,394,368,432)(55,393,369,431)(56,420,370,430)(57,100,342,153)(58,99,343,152)(59,98,344,151)(60,97,345,150)(61,96,346,149)(62,95,347,148)(63,94,348,147)(64,93,349,146)(65,92,350,145)(66,91,351,144)(67,90,352,143)(68,89,353,142)(69,88,354,141)(70,87,355,168)(71,86,356,167)(72,85,357,166)(73,112,358,165)(74,111,359,164)(75,110,360,163)(76,109,361,162)(77,108,362,161)(78,107,363,160)(79,106,364,159)(80,105,337,158)(81,104,338,157)(82,103,339,156)(83,102,340,155)(84,101,341,154)(197,298,279,249)(198,297,280,248)(199,296,253,247)(200,295,254,246)(201,294,255,245)(202,293,256,244)(203,292,257,243)(204,291,258,242)(205,290,259,241)(206,289,260,240)(207,288,261,239)(208,287,262,238)(209,286,263,237)(210,285,264,236)(211,284,265,235)(212,283,266,234)(213,282,267,233)(214,281,268,232)(215,308,269,231)(216,307,270,230)(217,306,271,229)(218,305,272,228)(219,304,273,227)(220,303,274,226)(221,302,275,225)(222,301,276,252)(223,300,277,251)(224,299,278,250) );
G=PermutationGroup([(1,182),(2,183),(3,184),(4,185),(5,186),(6,187),(7,188),(8,189),(9,190),(10,191),(11,192),(12,193),(13,194),(14,195),(15,196),(16,169),(17,170),(18,171),(19,172),(20,173),(21,174),(22,175),(23,176),(24,177),(25,178),(26,179),(27,180),(28,181),(29,371),(30,372),(31,373),(32,374),(33,375),(34,376),(35,377),(36,378),(37,379),(38,380),(39,381),(40,382),(41,383),(42,384),(43,385),(44,386),(45,387),(46,388),(47,389),(48,390),(49,391),(50,392),(51,365),(52,366),(53,367),(54,368),(55,369),(56,370),(57,342),(58,343),(59,344),(60,345),(61,346),(62,347),(63,348),(64,349),(65,350),(66,351),(67,352),(68,353),(69,354),(70,355),(71,356),(72,357),(73,358),(74,359),(75,360),(76,361),(77,362),(78,363),(79,364),(80,337),(81,338),(82,339),(83,340),(84,341),(85,166),(86,167),(87,168),(88,141),(89,142),(90,143),(91,144),(92,145),(93,146),(94,147),(95,148),(96,149),(97,150),(98,151),(99,152),(100,153),(101,154),(102,155),(103,156),(104,157),(105,158),(106,159),(107,160),(108,161),(109,162),(110,163),(111,164),(112,165),(113,320),(114,321),(115,322),(116,323),(117,324),(118,325),(119,326),(120,327),(121,328),(122,329),(123,330),(124,331),(125,332),(126,333),(127,334),(128,335),(129,336),(130,309),(131,310),(132,311),(133,312),(134,313),(135,314),(136,315),(137,316),(138,317),(139,318),(140,319),(197,279),(198,280),(199,253),(200,254),(201,255),(202,256),(203,257),(204,258),(205,259),(206,260),(207,261),(208,262),(209,263),(210,264),(211,265),(212,266),(213,267),(214,268),(215,269),(216,270),(217,271),(218,272),(219,273),(220,274),(221,275),(222,276),(223,277),(224,278),(225,302),(226,303),(227,304),(228,305),(229,306),(230,307),(231,308),(232,281),(233,282),(234,283),(235,284),(236,285),(237,286),(238,287),(239,288),(240,289),(241,290),(242,291),(243,292),(244,293),(245,294),(246,295),(247,296),(248,297),(249,298),(250,299),(251,300),(252,301),(393,431),(394,432),(395,433),(396,434),(397,435),(398,436),(399,437),(400,438),(401,439),(402,440),(403,441),(404,442),(405,443),(406,444),(407,445),(408,446),(409,447),(410,448),(411,421),(412,422),(413,423),(414,424),(415,425),(416,426),(417,427),(418,428),(419,429),(420,430)], [(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)], [(1,34,141,295),(2,375,142,245),(3,32,143,293),(4,373,144,243),(5,30,145,291),(6,371,146,241),(7,56,147,289),(8,369,148,239),(9,54,149,287),(10,367,150,237),(11,52,151,285),(12,365,152,235),(13,50,153,283),(14,391,154,233),(15,48,155,281),(16,389,156,231),(17,46,157,307),(18,387,158,229),(19,44,159,305),(20,385,160,227),(21,42,161,303),(22,383,162,225),(23,40,163,301),(24,381,164,251),(25,38,165,299),(26,379,166,249),(27,36,167,297),(28,377,168,247),(29,93,290,187),(31,91,292,185),(33,89,294,183),(35,87,296,181),(37,85,298,179),(39,111,300,177),(41,109,302,175),(43,107,304,173),(45,105,306,171),(47,103,308,169),(49,101,282,195),(51,99,284,193),(53,97,286,191),(55,95,288,189),(57,398,133,266),(58,435,134,211),(59,396,135,264),(60,433,136,209),(61,394,137,262),(62,431,138,207),(63,420,139,260),(64,429,140,205),(65,418,113,258),(66,427,114,203),(67,416,115,256),(68,425,116,201),(69,414,117,254),(70,423,118,199),(71,412,119,280),(72,421,120,197),(73,410,121,278),(74,447,122,223),(75,408,123,276),(76,445,124,221),(77,406,125,274),(78,443,126,219),(79,404,127,272),(80,441,128,217),(81,402,129,270),(82,439,130,215),(83,400,131,268),(84,437,132,213),(86,248,180,378),(88,246,182,376),(90,244,184,374),(92,242,186,372),(94,240,188,370),(96,238,190,368),(98,236,192,366),(100,234,194,392),(102,232,196,390),(104,230,170,388),(106,228,172,386),(108,226,174,384),(110,252,176,382),(112,250,178,380),(198,356,422,326),(200,354,424,324),(202,352,426,322),(204,350,428,320),(206,348,430,318),(208,346,432,316),(210,344,434,314),(212,342,436,312),(214,340,438,310),(216,338,440,336),(218,364,442,334),(220,362,444,332),(222,360,446,330),(224,358,448,328),(253,355,413,325),(255,353,415,323),(257,351,417,321),(259,349,419,319),(261,347,393,317),(263,345,395,315),(265,343,397,313),(267,341,399,311),(269,339,401,309),(271,337,403,335),(273,363,405,333),(275,361,407,331),(277,359,409,329),(279,357,411,327)], [(1,117,182,324),(2,116,183,323),(3,115,184,322),(4,114,185,321),(5,113,186,320),(6,140,187,319),(7,139,188,318),(8,138,189,317),(9,137,190,316),(10,136,191,315),(11,135,192,314),(12,134,193,313),(13,133,194,312),(14,132,195,311),(15,131,196,310),(16,130,169,309),(17,129,170,336),(18,128,171,335),(19,127,172,334),(20,126,173,333),(21,125,174,332),(22,124,175,331),(23,123,176,330),(24,122,177,329),(25,121,178,328),(26,120,179,327),(27,119,180,326),(28,118,181,325),(29,419,371,429),(30,418,372,428),(31,417,373,427),(32,416,374,426),(33,415,375,425),(34,414,376,424),(35,413,377,423),(36,412,378,422),(37,411,379,421),(38,410,380,448),(39,409,381,447),(40,408,382,446),(41,407,383,445),(42,406,384,444),(43,405,385,443),(44,404,386,442),(45,403,387,441),(46,402,388,440),(47,401,389,439),(48,400,390,438),(49,399,391,437),(50,398,392,436),(51,397,365,435),(52,396,366,434),(53,395,367,433),(54,394,368,432),(55,393,369,431),(56,420,370,430),(57,100,342,153),(58,99,343,152),(59,98,344,151),(60,97,345,150),(61,96,346,149),(62,95,347,148),(63,94,348,147),(64,93,349,146),(65,92,350,145),(66,91,351,144),(67,90,352,143),(68,89,353,142),(69,88,354,141),(70,87,355,168),(71,86,356,167),(72,85,357,166),(73,112,358,165),(74,111,359,164),(75,110,360,163),(76,109,361,162),(77,108,362,161),(78,107,363,160),(79,106,364,159),(80,105,337,158),(81,104,338,157),(82,103,339,156),(83,102,340,155),(84,101,341,154),(197,298,279,249),(198,297,280,248),(199,296,253,247),(200,295,254,246),(201,294,255,245),(202,293,256,244),(203,292,257,243),(204,291,258,242),(205,290,259,241),(206,289,260,240),(207,288,261,239),(208,287,262,238),(209,286,263,237),(210,285,264,236),(211,284,265,235),(212,283,266,234),(213,282,267,233),(214,281,268,232),(215,308,269,231),(216,307,270,230),(217,306,271,229),(218,305,272,228),(219,304,273,227),(220,303,274,226),(221,302,275,225),(222,301,276,252),(223,300,277,251),(224,299,278,250)])
82 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4F | 4G | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | C4○D4 | D14 | D28 | C7⋊D4 | C4○D28 | D4⋊2D7 | Q8×D7 |
| kernel | (C2×C28).53D4 | C14.C42 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C14×C4⋊C4 | C2×Dic7 | C2×C28 | C2×C4⋊C4 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 3 | 2 | 1 | 1 | 4 | 4 | 3 | 6 | 9 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of (C2×C28).53D4 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 24 | 0 | 0 |
| 0 | 0 | 26 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[4,0,0,0,0,0,0,22,0,0,0,0,0,0,8,26,0,0,0,0,24,2,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,24,5,0,0,0,0,1,5,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,24,5,0,0,0,0,1,5,0,0,0,0,0,0,12,0,0,0,0,0,0,17] >;
(C2×C28).53D4 in GAP, Magma, Sage, TeX
(C_2\times C_{28})._{53}D_4 % in TeX
G:=Group("(C2xC28).53D4"); // GroupNames label
G:=SmallGroup(448,517);
// by ID
G=gap.SmallGroup(448,517);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,254,387,184,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^4=1,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations