metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C28).55D4, (C2×C28).40Q8, (C2×C4).13Dic14, (C22×C4).103D14, C14.60(C22⋊Q8), C2.9(C28.3Q8), C2.5(C28.23D4), C14.50(C4.4D4), (C22×C28).67C22, C14.25(C42.C2), C22.49(C2×Dic14), C23.379(C22×D7), C14.27(C42⋊2C2), C2.11(C28.48D4), C22.107(C4○D28), C14.C42.38C2, (C22×C14).350C23, C7⋊5(C23.83C23), C22.50(Q8⋊2D7), C22.103(D4⋊2D7), C14.76(C22.D4), (C22×Dic7).57C22, C2.10(C23.18D14), (C2×C4⋊C4).23D7, (C14×C4⋊C4).24C2, (C2×C14).39(C2×Q8), (C2×C14).450(C2×D4), (C2×C4).40(C7⋊D4), (C2×C4⋊Dic7).20C2, C2.13(C4⋊C4⋊D7), C22.139(C2×C7⋊D4), (C2×C14).188(C4○D4), SmallGroup(448,520)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C28).55D4
G = < a,b,c,d | a2=b28=c4=1, d2=ab14, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=b14c-1 >
Subgroups: 548 in 134 conjugacy classes, 59 normal (27 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.83C23, C4⋊Dic7, C7×C4⋊C4, C22×Dic7, C22×C28, C14.C42, C14.C42, C2×C4⋊Dic7, C14×C4⋊C4, (C2×C28).55D4
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, Dic14, C7⋊D4, C22×D7, C23.83C23, C2×Dic14, C4○D28, D4⋊2D7, Q8⋊2D7, C2×C7⋊D4, C28.3Q8, C4⋊C4⋊D7, C28.48D4, C23.18D14, C28.23D4, (C2×C28).55D4
►Regular action on 448 points
Generators in S448
(1 241)(2 242)(3 243)(4 244)(5 245)(6 246)(7 247)(8 248)(9 249)(10 250)(11 251)(12 252)(13 225)(14 226)(15 227)(16 228)(17 229)(18 230)(19 231)(20 232)(21 233)(22 234)(23 235)(24 236)(25 237)(26 238)(27 239)(28 240)(29 341)(30 342)(31 343)(32 344)(33 345)(34 346)(35 347)(36 348)(37 349)(38 350)(39 351)(40 352)(41 353)(42 354)(43 355)(44 356)(45 357)(46 358)(47 359)(48 360)(49 361)(50 362)(51 363)(52 364)(53 337)(54 338)(55 339)(56 340)(57 90)(58 91)(59 92)(60 93)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 111)(79 112)(80 85)(81 86)(82 87)(83 88)(84 89)(113 151)(114 152)(115 153)(116 154)(117 155)(118 156)(119 157)(120 158)(121 159)(122 160)(123 161)(124 162)(125 163)(126 164)(127 165)(128 166)(129 167)(130 168)(131 141)(132 142)(133 143)(134 144)(135 145)(136 146)(137 147)(138 148)(139 149)(140 150)(169 411)(170 412)(171 413)(172 414)(173 415)(174 416)(175 417)(176 418)(177 419)(178 420)(179 393)(180 394)(181 395)(182 396)(183 397)(184 398)(185 399)(186 400)(187 401)(188 402)(189 403)(190 404)(191 405)(192 406)(193 407)(194 408)(195 409)(196 410)(197 448)(198 421)(199 422)(200 423)(201 424)(202 425)(203 426)(204 427)(205 428)(206 429)(207 430)(208 431)(209 432)(210 433)(211 434)(212 435)(213 436)(214 437)(215 438)(216 439)(217 440)(218 441)(219 442)(220 443)(221 444)(222 445)(223 446)(224 447)(253 385)(254 386)(255 387)(256 388)(257 389)(258 390)(259 391)(260 392)(261 365)(262 366)(263 367)(264 368)(265 369)(266 370)(267 371)(268 372)(269 373)(270 374)(271 375)(272 376)(273 377)(274 378)(275 379)(276 380)(277 381)(278 382)(279 383)(280 384)(281 332)(282 333)(283 334)(284 335)(285 336)(286 309)(287 310)(288 311)(289 312)(290 313)(291 314)(292 315)(293 316)(294 317)(295 318)(296 319)(297 320)(298 321)(299 322)(300 323)(301 324)(302 325)(303 326)(304 327)(305 328)(306 329)(307 330)(308 331)
(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)
(1 168 348 208)(2 129 349 430)(3 166 350 206)(4 127 351 428)(5 164 352 204)(6 125 353 426)(7 162 354 202)(8 123 355 424)(9 160 356 200)(10 121 357 422)(11 158 358 198)(12 119 359 448)(13 156 360 224)(14 117 361 446)(15 154 362 222)(16 115 363 444)(17 152 364 220)(18 113 337 442)(19 150 338 218)(20 139 339 440)(21 148 340 216)(22 137 341 438)(23 146 342 214)(24 135 343 436)(25 144 344 212)(26 133 345 434)(27 142 346 210)(28 131 347 432)(29 215 234 147)(30 437 235 136)(31 213 236 145)(32 435 237 134)(33 211 238 143)(34 433 239 132)(35 209 240 141)(36 431 241 130)(37 207 242 167)(38 429 243 128)(39 205 244 165)(40 427 245 126)(41 203 246 163)(42 425 247 124)(43 201 248 161)(44 423 249 122)(45 199 250 159)(46 421 251 120)(47 197 252 157)(48 447 225 118)(49 223 226 155)(50 445 227 116)(51 221 228 153)(52 443 229 114)(53 219 230 151)(54 441 231 140)(55 217 232 149)(56 439 233 138)(57 285 279 195)(58 335 280 408)(59 283 253 193)(60 333 254 406)(61 281 255 191)(62 331 256 404)(63 307 257 189)(64 329 258 402)(65 305 259 187)(66 327 260 400)(67 303 261 185)(68 325 262 398)(69 301 263 183)(70 323 264 396)(71 299 265 181)(72 321 266 394)(73 297 267 179)(74 319 268 420)(75 295 269 177)(76 317 270 418)(77 293 271 175)(78 315 272 416)(79 291 273 173)(80 313 274 414)(81 289 275 171)(82 311 276 412)(83 287 277 169)(84 309 278 410)(85 290 378 172)(86 312 379 413)(87 288 380 170)(88 310 381 411)(89 286 382 196)(90 336 383 409)(91 284 384 194)(92 334 385 407)(93 282 386 192)(94 332 387 405)(95 308 388 190)(96 330 389 403)(97 306 390 188)(98 328 391 401)(99 304 392 186)(100 326 365 399)(101 302 366 184)(102 324 367 397)(103 300 368 182)(104 322 369 395)(105 298 370 180)(106 320 371 393)(107 296 372 178)(108 318 373 419)(109 294 374 176)(110 316 375 417)(111 292 376 174)(112 314 377 415)
(1 404 227 176)(2 403 228 175)(3 402 229 174)(4 401 230 173)(5 400 231 172)(6 399 232 171)(7 398 233 170)(8 397 234 169)(9 396 235 196)(10 395 236 195)(11 394 237 194)(12 393 238 193)(13 420 239 192)(14 419 240 191)(15 418 241 190)(16 417 242 189)(17 416 243 188)(18 415 244 187)(19 414 245 186)(20 413 246 185)(21 412 247 184)(22 411 248 183)(23 410 249 182)(24 409 250 181)(25 408 251 180)(26 407 252 179)(27 406 225 178)(28 405 226 177)(29 287 355 324)(30 286 356 323)(31 285 357 322)(32 284 358 321)(33 283 359 320)(34 282 360 319)(35 281 361 318)(36 308 362 317)(37 307 363 316)(38 306 364 315)(39 305 337 314)(40 304 338 313)(41 303 339 312)(42 302 340 311)(43 301 341 310)(44 300 342 309)(45 299 343 336)(46 298 344 335)(47 297 345 334)(48 296 346 333)(49 295 347 332)(50 294 348 331)(51 293 349 330)(52 292 350 329)(53 291 351 328)(54 290 352 327)(55 289 353 326)(56 288 354 325)(57 436 104 199)(58 435 105 198)(59 434 106 197)(60 433 107 224)(61 432 108 223)(62 431 109 222)(63 430 110 221)(64 429 111 220)(65 428 112 219)(66 427 85 218)(67 426 86 217)(68 425 87 216)(69 424 88 215)(70 423 89 214)(71 422 90 213)(72 421 91 212)(73 448 92 211)(74 447 93 210)(75 446 94 209)(76 445 95 208)(77 444 96 207)(78 443 97 206)(79 442 98 205)(80 441 99 204)(81 440 100 203)(82 439 101 202)(83 438 102 201)(84 437 103 200)(113 391 165 273)(114 390 166 272)(115 389 167 271)(116 388 168 270)(117 387 141 269)(118 386 142 268)(119 385 143 267)(120 384 144 266)(121 383 145 265)(122 382 146 264)(123 381 147 263)(124 380 148 262)(125 379 149 261)(126 378 150 260)(127 377 151 259)(128 376 152 258)(129 375 153 257)(130 374 154 256)(131 373 155 255)(132 372 156 254)(133 371 157 253)(134 370 158 280)(135 369 159 279)(136 368 160 278)(137 367 161 277)(138 366 162 276)(139 365 163 275)(140 392 164 274)
G:=sub<Sym(448)| (1,241)(2,242)(3,243)(4,244)(5,245)(6,246)(7,247)(8,248)(9,249)(10,250)(11,251)(12,252)(13,225)(14,226)(15,227)(16,228)(17,229)(18,230)(19,231)(20,232)(21,233)(22,234)(23,235)(24,236)(25,237)(26,238)(27,239)(28,240)(29,341)(30,342)(31,343)(32,344)(33,345)(34,346)(35,347)(36,348)(37,349)(38,350)(39,351)(40,352)(41,353)(42,354)(43,355)(44,356)(45,357)(46,358)(47,359)(48,360)(49,361)(50,362)(51,363)(52,364)(53,337)(54,338)(55,339)(56,340)(57,90)(58,91)(59,92)(60,93)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(79,112)(80,85)(81,86)(82,87)(83,88)(84,89)(113,151)(114,152)(115,153)(116,154)(117,155)(118,156)(119,157)(120,158)(121,159)(122,160)(123,161)(124,162)(125,163)(126,164)(127,165)(128,166)(129,167)(130,168)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150)(169,411)(170,412)(171,413)(172,414)(173,415)(174,416)(175,417)(176,418)(177,419)(178,420)(179,393)(180,394)(181,395)(182,396)(183,397)(184,398)(185,399)(186,400)(187,401)(188,402)(189,403)(190,404)(191,405)(192,406)(193,407)(194,408)(195,409)(196,410)(197,448)(198,421)(199,422)(200,423)(201,424)(202,425)(203,426)(204,427)(205,428)(206,429)(207,430)(208,431)(209,432)(210,433)(211,434)(212,435)(213,436)(214,437)(215,438)(216,439)(217,440)(218,441)(219,442)(220,443)(221,444)(222,445)(223,446)(224,447)(253,385)(254,386)(255,387)(256,388)(257,389)(258,390)(259,391)(260,392)(261,365)(262,366)(263,367)(264,368)(265,369)(266,370)(267,371)(268,372)(269,373)(270,374)(271,375)(272,376)(273,377)(274,378)(275,379)(276,380)(277,381)(278,382)(279,383)(280,384)(281,332)(282,333)(283,334)(284,335)(285,336)(286,309)(287,310)(288,311)(289,312)(290,313)(291,314)(292,315)(293,316)(294,317)(295,318)(296,319)(297,320)(298,321)(299,322)(300,323)(301,324)(302,325)(303,326)(304,327)(305,328)(306,329)(307,330)(308,331), (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), (1,168,348,208)(2,129,349,430)(3,166,350,206)(4,127,351,428)(5,164,352,204)(6,125,353,426)(7,162,354,202)(8,123,355,424)(9,160,356,200)(10,121,357,422)(11,158,358,198)(12,119,359,448)(13,156,360,224)(14,117,361,446)(15,154,362,222)(16,115,363,444)(17,152,364,220)(18,113,337,442)(19,150,338,218)(20,139,339,440)(21,148,340,216)(22,137,341,438)(23,146,342,214)(24,135,343,436)(25,144,344,212)(26,133,345,434)(27,142,346,210)(28,131,347,432)(29,215,234,147)(30,437,235,136)(31,213,236,145)(32,435,237,134)(33,211,238,143)(34,433,239,132)(35,209,240,141)(36,431,241,130)(37,207,242,167)(38,429,243,128)(39,205,244,165)(40,427,245,126)(41,203,246,163)(42,425,247,124)(43,201,248,161)(44,423,249,122)(45,199,250,159)(46,421,251,120)(47,197,252,157)(48,447,225,118)(49,223,226,155)(50,445,227,116)(51,221,228,153)(52,443,229,114)(53,219,230,151)(54,441,231,140)(55,217,232,149)(56,439,233,138)(57,285,279,195)(58,335,280,408)(59,283,253,193)(60,333,254,406)(61,281,255,191)(62,331,256,404)(63,307,257,189)(64,329,258,402)(65,305,259,187)(66,327,260,400)(67,303,261,185)(68,325,262,398)(69,301,263,183)(70,323,264,396)(71,299,265,181)(72,321,266,394)(73,297,267,179)(74,319,268,420)(75,295,269,177)(76,317,270,418)(77,293,271,175)(78,315,272,416)(79,291,273,173)(80,313,274,414)(81,289,275,171)(82,311,276,412)(83,287,277,169)(84,309,278,410)(85,290,378,172)(86,312,379,413)(87,288,380,170)(88,310,381,411)(89,286,382,196)(90,336,383,409)(91,284,384,194)(92,334,385,407)(93,282,386,192)(94,332,387,405)(95,308,388,190)(96,330,389,403)(97,306,390,188)(98,328,391,401)(99,304,392,186)(100,326,365,399)(101,302,366,184)(102,324,367,397)(103,300,368,182)(104,322,369,395)(105,298,370,180)(106,320,371,393)(107,296,372,178)(108,318,373,419)(109,294,374,176)(110,316,375,417)(111,292,376,174)(112,314,377,415), (1,404,227,176)(2,403,228,175)(3,402,229,174)(4,401,230,173)(5,400,231,172)(6,399,232,171)(7,398,233,170)(8,397,234,169)(9,396,235,196)(10,395,236,195)(11,394,237,194)(12,393,238,193)(13,420,239,192)(14,419,240,191)(15,418,241,190)(16,417,242,189)(17,416,243,188)(18,415,244,187)(19,414,245,186)(20,413,246,185)(21,412,247,184)(22,411,248,183)(23,410,249,182)(24,409,250,181)(25,408,251,180)(26,407,252,179)(27,406,225,178)(28,405,226,177)(29,287,355,324)(30,286,356,323)(31,285,357,322)(32,284,358,321)(33,283,359,320)(34,282,360,319)(35,281,361,318)(36,308,362,317)(37,307,363,316)(38,306,364,315)(39,305,337,314)(40,304,338,313)(41,303,339,312)(42,302,340,311)(43,301,341,310)(44,300,342,309)(45,299,343,336)(46,298,344,335)(47,297,345,334)(48,296,346,333)(49,295,347,332)(50,294,348,331)(51,293,349,330)(52,292,350,329)(53,291,351,328)(54,290,352,327)(55,289,353,326)(56,288,354,325)(57,436,104,199)(58,435,105,198)(59,434,106,197)(60,433,107,224)(61,432,108,223)(62,431,109,222)(63,430,110,221)(64,429,111,220)(65,428,112,219)(66,427,85,218)(67,426,86,217)(68,425,87,216)(69,424,88,215)(70,423,89,214)(71,422,90,213)(72,421,91,212)(73,448,92,211)(74,447,93,210)(75,446,94,209)(76,445,95,208)(77,444,96,207)(78,443,97,206)(79,442,98,205)(80,441,99,204)(81,440,100,203)(82,439,101,202)(83,438,102,201)(84,437,103,200)(113,391,165,273)(114,390,166,272)(115,389,167,271)(116,388,168,270)(117,387,141,269)(118,386,142,268)(119,385,143,267)(120,384,144,266)(121,383,145,265)(122,382,146,264)(123,381,147,263)(124,380,148,262)(125,379,149,261)(126,378,150,260)(127,377,151,259)(128,376,152,258)(129,375,153,257)(130,374,154,256)(131,373,155,255)(132,372,156,254)(133,371,157,253)(134,370,158,280)(135,369,159,279)(136,368,160,278)(137,367,161,277)(138,366,162,276)(139,365,163,275)(140,392,164,274)>;
G:=Group( (1,241)(2,242)(3,243)(4,244)(5,245)(6,246)(7,247)(8,248)(9,249)(10,250)(11,251)(12,252)(13,225)(14,226)(15,227)(16,228)(17,229)(18,230)(19,231)(20,232)(21,233)(22,234)(23,235)(24,236)(25,237)(26,238)(27,239)(28,240)(29,341)(30,342)(31,343)(32,344)(33,345)(34,346)(35,347)(36,348)(37,349)(38,350)(39,351)(40,352)(41,353)(42,354)(43,355)(44,356)(45,357)(46,358)(47,359)(48,360)(49,361)(50,362)(51,363)(52,364)(53,337)(54,338)(55,339)(56,340)(57,90)(58,91)(59,92)(60,93)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(79,112)(80,85)(81,86)(82,87)(83,88)(84,89)(113,151)(114,152)(115,153)(116,154)(117,155)(118,156)(119,157)(120,158)(121,159)(122,160)(123,161)(124,162)(125,163)(126,164)(127,165)(128,166)(129,167)(130,168)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150)(169,411)(170,412)(171,413)(172,414)(173,415)(174,416)(175,417)(176,418)(177,419)(178,420)(179,393)(180,394)(181,395)(182,396)(183,397)(184,398)(185,399)(186,400)(187,401)(188,402)(189,403)(190,404)(191,405)(192,406)(193,407)(194,408)(195,409)(196,410)(197,448)(198,421)(199,422)(200,423)(201,424)(202,425)(203,426)(204,427)(205,428)(206,429)(207,430)(208,431)(209,432)(210,433)(211,434)(212,435)(213,436)(214,437)(215,438)(216,439)(217,440)(218,441)(219,442)(220,443)(221,444)(222,445)(223,446)(224,447)(253,385)(254,386)(255,387)(256,388)(257,389)(258,390)(259,391)(260,392)(261,365)(262,366)(263,367)(264,368)(265,369)(266,370)(267,371)(268,372)(269,373)(270,374)(271,375)(272,376)(273,377)(274,378)(275,379)(276,380)(277,381)(278,382)(279,383)(280,384)(281,332)(282,333)(283,334)(284,335)(285,336)(286,309)(287,310)(288,311)(289,312)(290,313)(291,314)(292,315)(293,316)(294,317)(295,318)(296,319)(297,320)(298,321)(299,322)(300,323)(301,324)(302,325)(303,326)(304,327)(305,328)(306,329)(307,330)(308,331), (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), (1,168,348,208)(2,129,349,430)(3,166,350,206)(4,127,351,428)(5,164,352,204)(6,125,353,426)(7,162,354,202)(8,123,355,424)(9,160,356,200)(10,121,357,422)(11,158,358,198)(12,119,359,448)(13,156,360,224)(14,117,361,446)(15,154,362,222)(16,115,363,444)(17,152,364,220)(18,113,337,442)(19,150,338,218)(20,139,339,440)(21,148,340,216)(22,137,341,438)(23,146,342,214)(24,135,343,436)(25,144,344,212)(26,133,345,434)(27,142,346,210)(28,131,347,432)(29,215,234,147)(30,437,235,136)(31,213,236,145)(32,435,237,134)(33,211,238,143)(34,433,239,132)(35,209,240,141)(36,431,241,130)(37,207,242,167)(38,429,243,128)(39,205,244,165)(40,427,245,126)(41,203,246,163)(42,425,247,124)(43,201,248,161)(44,423,249,122)(45,199,250,159)(46,421,251,120)(47,197,252,157)(48,447,225,118)(49,223,226,155)(50,445,227,116)(51,221,228,153)(52,443,229,114)(53,219,230,151)(54,441,231,140)(55,217,232,149)(56,439,233,138)(57,285,279,195)(58,335,280,408)(59,283,253,193)(60,333,254,406)(61,281,255,191)(62,331,256,404)(63,307,257,189)(64,329,258,402)(65,305,259,187)(66,327,260,400)(67,303,261,185)(68,325,262,398)(69,301,263,183)(70,323,264,396)(71,299,265,181)(72,321,266,394)(73,297,267,179)(74,319,268,420)(75,295,269,177)(76,317,270,418)(77,293,271,175)(78,315,272,416)(79,291,273,173)(80,313,274,414)(81,289,275,171)(82,311,276,412)(83,287,277,169)(84,309,278,410)(85,290,378,172)(86,312,379,413)(87,288,380,170)(88,310,381,411)(89,286,382,196)(90,336,383,409)(91,284,384,194)(92,334,385,407)(93,282,386,192)(94,332,387,405)(95,308,388,190)(96,330,389,403)(97,306,390,188)(98,328,391,401)(99,304,392,186)(100,326,365,399)(101,302,366,184)(102,324,367,397)(103,300,368,182)(104,322,369,395)(105,298,370,180)(106,320,371,393)(107,296,372,178)(108,318,373,419)(109,294,374,176)(110,316,375,417)(111,292,376,174)(112,314,377,415), (1,404,227,176)(2,403,228,175)(3,402,229,174)(4,401,230,173)(5,400,231,172)(6,399,232,171)(7,398,233,170)(8,397,234,169)(9,396,235,196)(10,395,236,195)(11,394,237,194)(12,393,238,193)(13,420,239,192)(14,419,240,191)(15,418,241,190)(16,417,242,189)(17,416,243,188)(18,415,244,187)(19,414,245,186)(20,413,246,185)(21,412,247,184)(22,411,248,183)(23,410,249,182)(24,409,250,181)(25,408,251,180)(26,407,252,179)(27,406,225,178)(28,405,226,177)(29,287,355,324)(30,286,356,323)(31,285,357,322)(32,284,358,321)(33,283,359,320)(34,282,360,319)(35,281,361,318)(36,308,362,317)(37,307,363,316)(38,306,364,315)(39,305,337,314)(40,304,338,313)(41,303,339,312)(42,302,340,311)(43,301,341,310)(44,300,342,309)(45,299,343,336)(46,298,344,335)(47,297,345,334)(48,296,346,333)(49,295,347,332)(50,294,348,331)(51,293,349,330)(52,292,350,329)(53,291,351,328)(54,290,352,327)(55,289,353,326)(56,288,354,325)(57,436,104,199)(58,435,105,198)(59,434,106,197)(60,433,107,224)(61,432,108,223)(62,431,109,222)(63,430,110,221)(64,429,111,220)(65,428,112,219)(66,427,85,218)(67,426,86,217)(68,425,87,216)(69,424,88,215)(70,423,89,214)(71,422,90,213)(72,421,91,212)(73,448,92,211)(74,447,93,210)(75,446,94,209)(76,445,95,208)(77,444,96,207)(78,443,97,206)(79,442,98,205)(80,441,99,204)(81,440,100,203)(82,439,101,202)(83,438,102,201)(84,437,103,200)(113,391,165,273)(114,390,166,272)(115,389,167,271)(116,388,168,270)(117,387,141,269)(118,386,142,268)(119,385,143,267)(120,384,144,266)(121,383,145,265)(122,382,146,264)(123,381,147,263)(124,380,148,262)(125,379,149,261)(126,378,150,260)(127,377,151,259)(128,376,152,258)(129,375,153,257)(130,374,154,256)(131,373,155,255)(132,372,156,254)(133,371,157,253)(134,370,158,280)(135,369,159,279)(136,368,160,278)(137,367,161,277)(138,366,162,276)(139,365,163,275)(140,392,164,274) );
G=PermutationGroup([[(1,241),(2,242),(3,243),(4,244),(5,245),(6,246),(7,247),(8,248),(9,249),(10,250),(11,251),(12,252),(13,225),(14,226),(15,227),(16,228),(17,229),(18,230),(19,231),(20,232),(21,233),(22,234),(23,235),(24,236),(25,237),(26,238),(27,239),(28,240),(29,341),(30,342),(31,343),(32,344),(33,345),(34,346),(35,347),(36,348),(37,349),(38,350),(39,351),(40,352),(41,353),(42,354),(43,355),(44,356),(45,357),(46,358),(47,359),(48,360),(49,361),(50,362),(51,363),(52,364),(53,337),(54,338),(55,339),(56,340),(57,90),(58,91),(59,92),(60,93),(61,94),(62,95),(63,96),(64,97),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,105),(73,106),(74,107),(75,108),(76,109),(77,110),(78,111),(79,112),(80,85),(81,86),(82,87),(83,88),(84,89),(113,151),(114,152),(115,153),(116,154),(117,155),(118,156),(119,157),(120,158),(121,159),(122,160),(123,161),(124,162),(125,163),(126,164),(127,165),(128,166),(129,167),(130,168),(131,141),(132,142),(133,143),(134,144),(135,145),(136,146),(137,147),(138,148),(139,149),(140,150),(169,411),(170,412),(171,413),(172,414),(173,415),(174,416),(175,417),(176,418),(177,419),(178,420),(179,393),(180,394),(181,395),(182,396),(183,397),(184,398),(185,399),(186,400),(187,401),(188,402),(189,403),(190,404),(191,405),(192,406),(193,407),(194,408),(195,409),(196,410),(197,448),(198,421),(199,422),(200,423),(201,424),(202,425),(203,426),(204,427),(205,428),(206,429),(207,430),(208,431),(209,432),(210,433),(211,434),(212,435),(213,436),(214,437),(215,438),(216,439),(217,440),(218,441),(219,442),(220,443),(221,444),(222,445),(223,446),(224,447),(253,385),(254,386),(255,387),(256,388),(257,389),(258,390),(259,391),(260,392),(261,365),(262,366),(263,367),(264,368),(265,369),(266,370),(267,371),(268,372),(269,373),(270,374),(271,375),(272,376),(273,377),(274,378),(275,379),(276,380),(277,381),(278,382),(279,383),(280,384),(281,332),(282,333),(283,334),(284,335),(285,336),(286,309),(287,310),(288,311),(289,312),(290,313),(291,314),(292,315),(293,316),(294,317),(295,318),(296,319),(297,320),(298,321),(299,322),(300,323),(301,324),(302,325),(303,326),(304,327),(305,328),(306,329),(307,330),(308,331)], [(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)], [(1,168,348,208),(2,129,349,430),(3,166,350,206),(4,127,351,428),(5,164,352,204),(6,125,353,426),(7,162,354,202),(8,123,355,424),(9,160,356,200),(10,121,357,422),(11,158,358,198),(12,119,359,448),(13,156,360,224),(14,117,361,446),(15,154,362,222),(16,115,363,444),(17,152,364,220),(18,113,337,442),(19,150,338,218),(20,139,339,440),(21,148,340,216),(22,137,341,438),(23,146,342,214),(24,135,343,436),(25,144,344,212),(26,133,345,434),(27,142,346,210),(28,131,347,432),(29,215,234,147),(30,437,235,136),(31,213,236,145),(32,435,237,134),(33,211,238,143),(34,433,239,132),(35,209,240,141),(36,431,241,130),(37,207,242,167),(38,429,243,128),(39,205,244,165),(40,427,245,126),(41,203,246,163),(42,425,247,124),(43,201,248,161),(44,423,249,122),(45,199,250,159),(46,421,251,120),(47,197,252,157),(48,447,225,118),(49,223,226,155),(50,445,227,116),(51,221,228,153),(52,443,229,114),(53,219,230,151),(54,441,231,140),(55,217,232,149),(56,439,233,138),(57,285,279,195),(58,335,280,408),(59,283,253,193),(60,333,254,406),(61,281,255,191),(62,331,256,404),(63,307,257,189),(64,329,258,402),(65,305,259,187),(66,327,260,400),(67,303,261,185),(68,325,262,398),(69,301,263,183),(70,323,264,396),(71,299,265,181),(72,321,266,394),(73,297,267,179),(74,319,268,420),(75,295,269,177),(76,317,270,418),(77,293,271,175),(78,315,272,416),(79,291,273,173),(80,313,274,414),(81,289,275,171),(82,311,276,412),(83,287,277,169),(84,309,278,410),(85,290,378,172),(86,312,379,413),(87,288,380,170),(88,310,381,411),(89,286,382,196),(90,336,383,409),(91,284,384,194),(92,334,385,407),(93,282,386,192),(94,332,387,405),(95,308,388,190),(96,330,389,403),(97,306,390,188),(98,328,391,401),(99,304,392,186),(100,326,365,399),(101,302,366,184),(102,324,367,397),(103,300,368,182),(104,322,369,395),(105,298,370,180),(106,320,371,393),(107,296,372,178),(108,318,373,419),(109,294,374,176),(110,316,375,417),(111,292,376,174),(112,314,377,415)], [(1,404,227,176),(2,403,228,175),(3,402,229,174),(4,401,230,173),(5,400,231,172),(6,399,232,171),(7,398,233,170),(8,397,234,169),(9,396,235,196),(10,395,236,195),(11,394,237,194),(12,393,238,193),(13,420,239,192),(14,419,240,191),(15,418,241,190),(16,417,242,189),(17,416,243,188),(18,415,244,187),(19,414,245,186),(20,413,246,185),(21,412,247,184),(22,411,248,183),(23,410,249,182),(24,409,250,181),(25,408,251,180),(26,407,252,179),(27,406,225,178),(28,405,226,177),(29,287,355,324),(30,286,356,323),(31,285,357,322),(32,284,358,321),(33,283,359,320),(34,282,360,319),(35,281,361,318),(36,308,362,317),(37,307,363,316),(38,306,364,315),(39,305,337,314),(40,304,338,313),(41,303,339,312),(42,302,340,311),(43,301,341,310),(44,300,342,309),(45,299,343,336),(46,298,344,335),(47,297,345,334),(48,296,346,333),(49,295,347,332),(50,294,348,331),(51,293,349,330),(52,292,350,329),(53,291,351,328),(54,290,352,327),(55,289,353,326),(56,288,354,325),(57,436,104,199),(58,435,105,198),(59,434,106,197),(60,433,107,224),(61,432,108,223),(62,431,109,222),(63,430,110,221),(64,429,111,220),(65,428,112,219),(66,427,85,218),(67,426,86,217),(68,425,87,216),(69,424,88,215),(70,423,89,214),(71,422,90,213),(72,421,91,212),(73,448,92,211),(74,447,93,210),(75,446,94,209),(76,445,95,208),(77,444,96,207),(78,443,97,206),(79,442,98,205),(80,441,99,204),(81,440,100,203),(82,439,101,202),(83,438,102,201),(84,437,103,200),(113,391,165,273),(114,390,166,272),(115,389,167,271),(116,388,168,270),(117,387,141,269),(118,386,142,268),(119,385,143,267),(120,384,144,266),(121,383,145,265),(122,382,146,264),(123,381,147,263),(124,380,148,262),(125,379,149,261),(126,378,150,260),(127,377,151,259),(128,376,152,258),(129,375,153,257),(130,374,154,256),(131,373,155,255),(132,372,156,254),(133,371,157,253),(134,370,158,280),(135,369,159,279),(136,368,160,278),(137,367,161,277),(138,366,162,276),(139,365,163,275),(140,392,164,274)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | D4 | Q8 | D7 | C4○D4 | D14 | Dic14 | C7⋊D4 | C4○D28 | D4⋊2D7 | Q8⋊2D7 |
| kernel | (C2×C28).55D4 | C14.C42 | C2×C4⋊Dic7 | C14×C4⋊C4 | C2×C28 | C2×C28 | C2×C4⋊C4 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 5 | 1 | 1 | 2 | 2 | 3 | 10 | 9 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of (C2×C28).55D4 ►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 |
| 27 | 24 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 18 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 11 | 12 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 27 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 22 | 0 | 0 |
| 0 | 0 | 21 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 13 | 1 |
| 7 | 20 | 0 | 0 | 0 | 0 |
| 12 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 4 | 0 | 0 |
| 0 | 0 | 6 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 1 |
| 0 | 0 | 0 | 0 | 24 | 8 |
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],[27,5,0,0,0,0,24,12,0,0,0,0,0,0,3,8,0,0,0,0,18,0,0,0,0,0,0,0,17,11,0,0,0,0,0,12],[5,27,0,0,0,0,12,24,0,0,0,0,0,0,28,21,0,0,0,0,22,1,0,0,0,0,0,0,28,13,0,0,0,0,0,1],[7,12,0,0,0,0,20,22,0,0,0,0,0,0,21,6,0,0,0,0,4,8,0,0,0,0,0,0,21,24,0,0,0,0,1,8] >;
(C2×C28).55D4 in GAP, Magma, Sage, TeX
(C_2\times C_{28})._{55}D_4 % in TeX
G:=Group("(C2xC28).55D4"); // GroupNames label
G:=SmallGroup(448,520);
// by ID
G=gap.SmallGroup(448,520);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,701,344,254,387,184,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^4=1,d^2=a*b^14,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=b^14*c^-1>;
// generators/relations