metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C28).54D4, (C2×C28).39Q8, C14.22(C4⋊Q8), C2.15(C28⋊Q8), (C2×Dic7).8Q8, C22.49(Q8×D7), (C2×Dic7).60D4, (C2×C4).12Dic14, C22.246(D4×D7), C14.90(C4⋊D4), C2.7(D14⋊3Q8), (C22×C4).101D14, C14.59(C22⋊Q8), C2.8(C28.3Q8), C2.14(Dic7.Q8), (C22×C28).65C22, C14.23(C42.C2), C22.48(C2×Dic14), C23.378(C22×D7), C2.12(Dic7⋊D4), C2.19(D14.5D4), C2.10(C28.48D4), C22.106(C4○D28), C14.C42.20C2, (C22×C14).348C23, C7⋊5(C23.81C23), C22.49(Q8⋊2D7), C22.102(D4⋊2D7), C14.50(C22.D4), (C22×Dic7).56C22, (C2×C4⋊C4).21D7, (C14×C4⋊C4).23C2, (C2×C14).37(C2×Q8), (C2×C14).332(C2×D4), (C2×C4).39(C7⋊D4), (C2×C4⋊Dic7).19C2, (C2×C14).85(C4○D4), (C2×Dic7⋊C4).15C2, C22.138(C2×C7⋊D4), SmallGroup(448,518)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C28).54D4
G = < a,b,c,d | a2=b28=c4=1, d2=ab14, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=b14c-1 >
Subgroups: 612 in 150 conjugacy classes, 63 normal (51 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, C23, C14, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.81C23, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C22×Dic7, C22×C28, C14.C42, C2×Dic7⋊C4, C2×C4⋊Dic7, C14×C4⋊C4, (C2×C28).54D4
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, Dic14, C7⋊D4, C22×D7, C23.81C23, C2×Dic14, C4○D28, D4×D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C2×C7⋊D4, C28⋊Q8, Dic7.Q8, C28.3Q8, D14.5D4, C28.48D4, Dic7⋊D4, D14⋊3Q8, (C2×C28).54D4
►Regular action on 448 points
Generators in S448
(1 398)(2 399)(3 400)(4 401)(5 402)(6 403)(7 404)(8 405)(9 406)(10 407)(11 408)(12 409)(13 410)(14 411)(15 412)(16 413)(17 414)(18 415)(19 416)(20 417)(21 418)(22 419)(23 420)(24 393)(25 394)(26 395)(27 396)(28 397)(29 326)(30 327)(31 328)(32 329)(33 330)(34 331)(35 332)(36 333)(37 334)(38 335)(39 336)(40 309)(41 310)(42 311)(43 312)(44 313)(45 314)(46 315)(47 316)(48 317)(49 318)(50 319)(51 320)(52 321)(53 322)(54 323)(55 324)(56 325)(57 226)(58 227)(59 228)(60 229)(61 230)(62 231)(63 232)(64 233)(65 234)(66 235)(67 236)(68 237)(69 238)(70 239)(71 240)(72 241)(73 242)(74 243)(75 244)(76 245)(77 246)(78 247)(79 248)(80 249)(81 250)(82 251)(83 252)(84 225)(85 425)(86 426)(87 427)(88 428)(89 429)(90 430)(91 431)(92 432)(93 433)(94 434)(95 435)(96 436)(97 437)(98 438)(99 439)(100 440)(101 441)(102 442)(103 443)(104 444)(105 445)(106 446)(107 447)(108 448)(109 421)(110 422)(111 423)(112 424)(113 337)(114 338)(115 339)(116 340)(117 341)(118 342)(119 343)(120 344)(121 345)(122 346)(123 347)(124 348)(125 349)(126 350)(127 351)(128 352)(129 353)(130 354)(131 355)(132 356)(133 357)(134 358)(135 359)(136 360)(137 361)(138 362)(139 363)(140 364)(141 208)(142 209)(143 210)(144 211)(145 212)(146 213)(147 214)(148 215)(149 216)(150 217)(151 218)(152 219)(153 220)(154 221)(155 222)(156 223)(157 224)(158 197)(159 198)(160 199)(161 200)(162 201)(163 202)(164 203)(165 204)(166 205)(167 206)(168 207)(169 268)(170 269)(171 270)(172 271)(173 272)(174 273)(175 274)(176 275)(177 276)(178 277)(179 278)(180 279)(181 280)(182 253)(183 254)(184 255)(185 256)(186 257)(187 258)(188 259)(189 260)(190 261)(191 262)(192 263)(193 264)(194 265)(195 266)(196 267)(281 386)(282 387)(283 388)(284 389)(285 390)(286 391)(287 392)(288 365)(289 366)(290 367)(291 368)(292 369)(293 370)(294 371)(295 372)(296 373)(297 374)(298 375)(299 376)(300 377)(301 378)(302 379)(303 380)(304 381)(305 382)(306 383)(307 384)(308 385)
(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 80 375 329)(2 79 376 328)(3 78 377 327)(4 77 378 326)(5 76 379 325)(6 75 380 324)(7 74 381 323)(8 73 382 322)(9 72 383 321)(10 71 384 320)(11 70 385 319)(12 69 386 318)(13 68 387 317)(14 67 388 316)(15 66 389 315)(16 65 390 314)(17 64 391 313)(18 63 392 312)(19 62 365 311)(20 61 366 310)(21 60 367 309)(22 59 368 336)(23 58 369 335)(24 57 370 334)(25 84 371 333)(26 83 372 332)(27 82 373 331)(28 81 374 330)(29 401 246 301)(30 400 247 300)(31 399 248 299)(32 398 249 298)(33 397 250 297)(34 396 251 296)(35 395 252 295)(36 394 225 294)(37 393 226 293)(38 420 227 292)(39 419 228 291)(40 418 229 290)(41 417 230 289)(42 416 231 288)(43 415 232 287)(44 414 233 286)(45 413 234 285)(46 412 235 284)(47 411 236 283)(48 410 237 282)(49 409 238 281)(50 408 239 308)(51 407 240 307)(52 406 241 306)(53 405 242 305)(54 404 243 304)(55 403 244 303)(56 402 245 302)(85 261 352 222)(86 260 353 221)(87 259 354 220)(88 258 355 219)(89 257 356 218)(90 256 357 217)(91 255 358 216)(92 254 359 215)(93 253 360 214)(94 280 361 213)(95 279 362 212)(96 278 363 211)(97 277 364 210)(98 276 337 209)(99 275 338 208)(100 274 339 207)(101 273 340 206)(102 272 341 205)(103 271 342 204)(104 270 343 203)(105 269 344 202)(106 268 345 201)(107 267 346 200)(108 266 347 199)(109 265 348 198)(110 264 349 197)(111 263 350 224)(112 262 351 223)(113 142 438 177)(114 141 439 176)(115 168 440 175)(116 167 441 174)(117 166 442 173)(118 165 443 172)(119 164 444 171)(120 163 445 170)(121 162 446 169)(122 161 447 196)(123 160 448 195)(124 159 421 194)(125 158 422 193)(126 157 423 192)(127 156 424 191)(128 155 425 190)(129 154 426 189)(130 153 427 188)(131 152 428 187)(132 151 429 186)(133 150 430 185)(134 149 431 184)(135 148 432 183)(136 147 433 182)(137 146 434 181)(138 145 435 180)(139 144 436 179)(140 143 437 178)
(1 271 412 186)(2 171 413 256)(3 269 414 184)(4 169 415 254)(5 267 416 182)(6 195 417 280)(7 265 418 180)(8 193 419 278)(9 263 420 178)(10 191 393 276)(11 261 394 176)(12 189 395 274)(13 259 396 174)(14 187 397 272)(15 257 398 172)(16 185 399 270)(17 255 400 170)(18 183 401 268)(19 253 402 196)(20 181 403 266)(21 279 404 194)(22 179 405 264)(23 277 406 192)(24 177 407 262)(25 275 408 190)(26 175 409 260)(27 273 410 188)(28 173 411 258)(29 359 312 121)(30 134 313 344)(31 357 314 119)(32 132 315 342)(33 355 316 117)(34 130 317 340)(35 353 318 115)(36 128 319 338)(37 351 320 113)(38 126 321 364)(39 349 322 139)(40 124 323 362)(41 347 324 137)(42 122 325 360)(43 345 326 135)(44 120 327 358)(45 343 328 133)(46 118 329 356)(47 341 330 131)(48 116 331 354)(49 339 332 129)(50 114 333 352)(51 337 334 127)(52 140 335 350)(53 363 336 125)(54 138 309 348)(55 361 310 123)(56 136 311 346)(57 424 240 98)(58 111 241 437)(59 422 242 96)(60 109 243 435)(61 448 244 94)(62 107 245 433)(63 446 246 92)(64 105 247 431)(65 444 248 90)(66 103 249 429)(67 442 250 88)(68 101 251 427)(69 440 252 86)(70 99 225 425)(71 438 226 112)(72 97 227 423)(73 436 228 110)(74 95 229 421)(75 434 230 108)(76 93 231 447)(77 432 232 106)(78 91 233 445)(79 430 234 104)(80 89 235 443)(81 428 236 102)(82 87 237 441)(83 426 238 100)(84 85 239 439)(141 385 222 294)(142 307 223 370)(143 383 224 292)(144 305 197 368)(145 381 198 290)(146 303 199 366)(147 379 200 288)(148 301 201 392)(149 377 202 286)(150 299 203 390)(151 375 204 284)(152 297 205 388)(153 373 206 282)(154 295 207 386)(155 371 208 308)(156 293 209 384)(157 369 210 306)(158 291 211 382)(159 367 212 304)(160 289 213 380)(161 365 214 302)(162 287 215 378)(163 391 216 300)(164 285 217 376)(165 389 218 298)(166 283 219 374)(167 387 220 296)(168 281 221 372)
G:=sub<Sym(448)| (1,398)(2,399)(3,400)(4,401)(5,402)(6,403)(7,404)(8,405)(9,406)(10,407)(11,408)(12,409)(13,410)(14,411)(15,412)(16,413)(17,414)(18,415)(19,416)(20,417)(21,418)(22,419)(23,420)(24,393)(25,394)(26,395)(27,396)(28,397)(29,326)(30,327)(31,328)(32,329)(33,330)(34,331)(35,332)(36,333)(37,334)(38,335)(39,336)(40,309)(41,310)(42,311)(43,312)(44,313)(45,314)(46,315)(47,316)(48,317)(49,318)(50,319)(51,320)(52,321)(53,322)(54,323)(55,324)(56,325)(57,226)(58,227)(59,228)(60,229)(61,230)(62,231)(63,232)(64,233)(65,234)(66,235)(67,236)(68,237)(69,238)(70,239)(71,240)(72,241)(73,242)(74,243)(75,244)(76,245)(77,246)(78,247)(79,248)(80,249)(81,250)(82,251)(83,252)(84,225)(85,425)(86,426)(87,427)(88,428)(89,429)(90,430)(91,431)(92,432)(93,433)(94,434)(95,435)(96,436)(97,437)(98,438)(99,439)(100,440)(101,441)(102,442)(103,443)(104,444)(105,445)(106,446)(107,447)(108,448)(109,421)(110,422)(111,423)(112,424)(113,337)(114,338)(115,339)(116,340)(117,341)(118,342)(119,343)(120,344)(121,345)(122,346)(123,347)(124,348)(125,349)(126,350)(127,351)(128,352)(129,353)(130,354)(131,355)(132,356)(133,357)(134,358)(135,359)(136,360)(137,361)(138,362)(139,363)(140,364)(141,208)(142,209)(143,210)(144,211)(145,212)(146,213)(147,214)(148,215)(149,216)(150,217)(151,218)(152,219)(153,220)(154,221)(155,222)(156,223)(157,224)(158,197)(159,198)(160,199)(161,200)(162,201)(163,202)(164,203)(165,204)(166,205)(167,206)(168,207)(169,268)(170,269)(171,270)(172,271)(173,272)(174,273)(175,274)(176,275)(177,276)(178,277)(179,278)(180,279)(181,280)(182,253)(183,254)(184,255)(185,256)(186,257)(187,258)(188,259)(189,260)(190,261)(191,262)(192,263)(193,264)(194,265)(195,266)(196,267)(281,386)(282,387)(283,388)(284,389)(285,390)(286,391)(287,392)(288,365)(289,366)(290,367)(291,368)(292,369)(293,370)(294,371)(295,372)(296,373)(297,374)(298,375)(299,376)(300,377)(301,378)(302,379)(303,380)(304,381)(305,382)(306,383)(307,384)(308,385), (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), (1,80,375,329)(2,79,376,328)(3,78,377,327)(4,77,378,326)(5,76,379,325)(6,75,380,324)(7,74,381,323)(8,73,382,322)(9,72,383,321)(10,71,384,320)(11,70,385,319)(12,69,386,318)(13,68,387,317)(14,67,388,316)(15,66,389,315)(16,65,390,314)(17,64,391,313)(18,63,392,312)(19,62,365,311)(20,61,366,310)(21,60,367,309)(22,59,368,336)(23,58,369,335)(24,57,370,334)(25,84,371,333)(26,83,372,332)(27,82,373,331)(28,81,374,330)(29,401,246,301)(30,400,247,300)(31,399,248,299)(32,398,249,298)(33,397,250,297)(34,396,251,296)(35,395,252,295)(36,394,225,294)(37,393,226,293)(38,420,227,292)(39,419,228,291)(40,418,229,290)(41,417,230,289)(42,416,231,288)(43,415,232,287)(44,414,233,286)(45,413,234,285)(46,412,235,284)(47,411,236,283)(48,410,237,282)(49,409,238,281)(50,408,239,308)(51,407,240,307)(52,406,241,306)(53,405,242,305)(54,404,243,304)(55,403,244,303)(56,402,245,302)(85,261,352,222)(86,260,353,221)(87,259,354,220)(88,258,355,219)(89,257,356,218)(90,256,357,217)(91,255,358,216)(92,254,359,215)(93,253,360,214)(94,280,361,213)(95,279,362,212)(96,278,363,211)(97,277,364,210)(98,276,337,209)(99,275,338,208)(100,274,339,207)(101,273,340,206)(102,272,341,205)(103,271,342,204)(104,270,343,203)(105,269,344,202)(106,268,345,201)(107,267,346,200)(108,266,347,199)(109,265,348,198)(110,264,349,197)(111,263,350,224)(112,262,351,223)(113,142,438,177)(114,141,439,176)(115,168,440,175)(116,167,441,174)(117,166,442,173)(118,165,443,172)(119,164,444,171)(120,163,445,170)(121,162,446,169)(122,161,447,196)(123,160,448,195)(124,159,421,194)(125,158,422,193)(126,157,423,192)(127,156,424,191)(128,155,425,190)(129,154,426,189)(130,153,427,188)(131,152,428,187)(132,151,429,186)(133,150,430,185)(134,149,431,184)(135,148,432,183)(136,147,433,182)(137,146,434,181)(138,145,435,180)(139,144,436,179)(140,143,437,178), (1,271,412,186)(2,171,413,256)(3,269,414,184)(4,169,415,254)(5,267,416,182)(6,195,417,280)(7,265,418,180)(8,193,419,278)(9,263,420,178)(10,191,393,276)(11,261,394,176)(12,189,395,274)(13,259,396,174)(14,187,397,272)(15,257,398,172)(16,185,399,270)(17,255,400,170)(18,183,401,268)(19,253,402,196)(20,181,403,266)(21,279,404,194)(22,179,405,264)(23,277,406,192)(24,177,407,262)(25,275,408,190)(26,175,409,260)(27,273,410,188)(28,173,411,258)(29,359,312,121)(30,134,313,344)(31,357,314,119)(32,132,315,342)(33,355,316,117)(34,130,317,340)(35,353,318,115)(36,128,319,338)(37,351,320,113)(38,126,321,364)(39,349,322,139)(40,124,323,362)(41,347,324,137)(42,122,325,360)(43,345,326,135)(44,120,327,358)(45,343,328,133)(46,118,329,356)(47,341,330,131)(48,116,331,354)(49,339,332,129)(50,114,333,352)(51,337,334,127)(52,140,335,350)(53,363,336,125)(54,138,309,348)(55,361,310,123)(56,136,311,346)(57,424,240,98)(58,111,241,437)(59,422,242,96)(60,109,243,435)(61,448,244,94)(62,107,245,433)(63,446,246,92)(64,105,247,431)(65,444,248,90)(66,103,249,429)(67,442,250,88)(68,101,251,427)(69,440,252,86)(70,99,225,425)(71,438,226,112)(72,97,227,423)(73,436,228,110)(74,95,229,421)(75,434,230,108)(76,93,231,447)(77,432,232,106)(78,91,233,445)(79,430,234,104)(80,89,235,443)(81,428,236,102)(82,87,237,441)(83,426,238,100)(84,85,239,439)(141,385,222,294)(142,307,223,370)(143,383,224,292)(144,305,197,368)(145,381,198,290)(146,303,199,366)(147,379,200,288)(148,301,201,392)(149,377,202,286)(150,299,203,390)(151,375,204,284)(152,297,205,388)(153,373,206,282)(154,295,207,386)(155,371,208,308)(156,293,209,384)(157,369,210,306)(158,291,211,382)(159,367,212,304)(160,289,213,380)(161,365,214,302)(162,287,215,378)(163,391,216,300)(164,285,217,376)(165,389,218,298)(166,283,219,374)(167,387,220,296)(168,281,221,372)>;
G:=Group( (1,398)(2,399)(3,400)(4,401)(5,402)(6,403)(7,404)(8,405)(9,406)(10,407)(11,408)(12,409)(13,410)(14,411)(15,412)(16,413)(17,414)(18,415)(19,416)(20,417)(21,418)(22,419)(23,420)(24,393)(25,394)(26,395)(27,396)(28,397)(29,326)(30,327)(31,328)(32,329)(33,330)(34,331)(35,332)(36,333)(37,334)(38,335)(39,336)(40,309)(41,310)(42,311)(43,312)(44,313)(45,314)(46,315)(47,316)(48,317)(49,318)(50,319)(51,320)(52,321)(53,322)(54,323)(55,324)(56,325)(57,226)(58,227)(59,228)(60,229)(61,230)(62,231)(63,232)(64,233)(65,234)(66,235)(67,236)(68,237)(69,238)(70,239)(71,240)(72,241)(73,242)(74,243)(75,244)(76,245)(77,246)(78,247)(79,248)(80,249)(81,250)(82,251)(83,252)(84,225)(85,425)(86,426)(87,427)(88,428)(89,429)(90,430)(91,431)(92,432)(93,433)(94,434)(95,435)(96,436)(97,437)(98,438)(99,439)(100,440)(101,441)(102,442)(103,443)(104,444)(105,445)(106,446)(107,447)(108,448)(109,421)(110,422)(111,423)(112,424)(113,337)(114,338)(115,339)(116,340)(117,341)(118,342)(119,343)(120,344)(121,345)(122,346)(123,347)(124,348)(125,349)(126,350)(127,351)(128,352)(129,353)(130,354)(131,355)(132,356)(133,357)(134,358)(135,359)(136,360)(137,361)(138,362)(139,363)(140,364)(141,208)(142,209)(143,210)(144,211)(145,212)(146,213)(147,214)(148,215)(149,216)(150,217)(151,218)(152,219)(153,220)(154,221)(155,222)(156,223)(157,224)(158,197)(159,198)(160,199)(161,200)(162,201)(163,202)(164,203)(165,204)(166,205)(167,206)(168,207)(169,268)(170,269)(171,270)(172,271)(173,272)(174,273)(175,274)(176,275)(177,276)(178,277)(179,278)(180,279)(181,280)(182,253)(183,254)(184,255)(185,256)(186,257)(187,258)(188,259)(189,260)(190,261)(191,262)(192,263)(193,264)(194,265)(195,266)(196,267)(281,386)(282,387)(283,388)(284,389)(285,390)(286,391)(287,392)(288,365)(289,366)(290,367)(291,368)(292,369)(293,370)(294,371)(295,372)(296,373)(297,374)(298,375)(299,376)(300,377)(301,378)(302,379)(303,380)(304,381)(305,382)(306,383)(307,384)(308,385), (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), (1,80,375,329)(2,79,376,328)(3,78,377,327)(4,77,378,326)(5,76,379,325)(6,75,380,324)(7,74,381,323)(8,73,382,322)(9,72,383,321)(10,71,384,320)(11,70,385,319)(12,69,386,318)(13,68,387,317)(14,67,388,316)(15,66,389,315)(16,65,390,314)(17,64,391,313)(18,63,392,312)(19,62,365,311)(20,61,366,310)(21,60,367,309)(22,59,368,336)(23,58,369,335)(24,57,370,334)(25,84,371,333)(26,83,372,332)(27,82,373,331)(28,81,374,330)(29,401,246,301)(30,400,247,300)(31,399,248,299)(32,398,249,298)(33,397,250,297)(34,396,251,296)(35,395,252,295)(36,394,225,294)(37,393,226,293)(38,420,227,292)(39,419,228,291)(40,418,229,290)(41,417,230,289)(42,416,231,288)(43,415,232,287)(44,414,233,286)(45,413,234,285)(46,412,235,284)(47,411,236,283)(48,410,237,282)(49,409,238,281)(50,408,239,308)(51,407,240,307)(52,406,241,306)(53,405,242,305)(54,404,243,304)(55,403,244,303)(56,402,245,302)(85,261,352,222)(86,260,353,221)(87,259,354,220)(88,258,355,219)(89,257,356,218)(90,256,357,217)(91,255,358,216)(92,254,359,215)(93,253,360,214)(94,280,361,213)(95,279,362,212)(96,278,363,211)(97,277,364,210)(98,276,337,209)(99,275,338,208)(100,274,339,207)(101,273,340,206)(102,272,341,205)(103,271,342,204)(104,270,343,203)(105,269,344,202)(106,268,345,201)(107,267,346,200)(108,266,347,199)(109,265,348,198)(110,264,349,197)(111,263,350,224)(112,262,351,223)(113,142,438,177)(114,141,439,176)(115,168,440,175)(116,167,441,174)(117,166,442,173)(118,165,443,172)(119,164,444,171)(120,163,445,170)(121,162,446,169)(122,161,447,196)(123,160,448,195)(124,159,421,194)(125,158,422,193)(126,157,423,192)(127,156,424,191)(128,155,425,190)(129,154,426,189)(130,153,427,188)(131,152,428,187)(132,151,429,186)(133,150,430,185)(134,149,431,184)(135,148,432,183)(136,147,433,182)(137,146,434,181)(138,145,435,180)(139,144,436,179)(140,143,437,178), (1,271,412,186)(2,171,413,256)(3,269,414,184)(4,169,415,254)(5,267,416,182)(6,195,417,280)(7,265,418,180)(8,193,419,278)(9,263,420,178)(10,191,393,276)(11,261,394,176)(12,189,395,274)(13,259,396,174)(14,187,397,272)(15,257,398,172)(16,185,399,270)(17,255,400,170)(18,183,401,268)(19,253,402,196)(20,181,403,266)(21,279,404,194)(22,179,405,264)(23,277,406,192)(24,177,407,262)(25,275,408,190)(26,175,409,260)(27,273,410,188)(28,173,411,258)(29,359,312,121)(30,134,313,344)(31,357,314,119)(32,132,315,342)(33,355,316,117)(34,130,317,340)(35,353,318,115)(36,128,319,338)(37,351,320,113)(38,126,321,364)(39,349,322,139)(40,124,323,362)(41,347,324,137)(42,122,325,360)(43,345,326,135)(44,120,327,358)(45,343,328,133)(46,118,329,356)(47,341,330,131)(48,116,331,354)(49,339,332,129)(50,114,333,352)(51,337,334,127)(52,140,335,350)(53,363,336,125)(54,138,309,348)(55,361,310,123)(56,136,311,346)(57,424,240,98)(58,111,241,437)(59,422,242,96)(60,109,243,435)(61,448,244,94)(62,107,245,433)(63,446,246,92)(64,105,247,431)(65,444,248,90)(66,103,249,429)(67,442,250,88)(68,101,251,427)(69,440,252,86)(70,99,225,425)(71,438,226,112)(72,97,227,423)(73,436,228,110)(74,95,229,421)(75,434,230,108)(76,93,231,447)(77,432,232,106)(78,91,233,445)(79,430,234,104)(80,89,235,443)(81,428,236,102)(82,87,237,441)(83,426,238,100)(84,85,239,439)(141,385,222,294)(142,307,223,370)(143,383,224,292)(144,305,197,368)(145,381,198,290)(146,303,199,366)(147,379,200,288)(148,301,201,392)(149,377,202,286)(150,299,203,390)(151,375,204,284)(152,297,205,388)(153,373,206,282)(154,295,207,386)(155,371,208,308)(156,293,209,384)(157,369,210,306)(158,291,211,382)(159,367,212,304)(160,289,213,380)(161,365,214,302)(162,287,215,378)(163,391,216,300)(164,285,217,376)(165,389,218,298)(166,283,219,374)(167,387,220,296)(168,281,221,372) );
G=PermutationGroup([[(1,398),(2,399),(3,400),(4,401),(5,402),(6,403),(7,404),(8,405),(9,406),(10,407),(11,408),(12,409),(13,410),(14,411),(15,412),(16,413),(17,414),(18,415),(19,416),(20,417),(21,418),(22,419),(23,420),(24,393),(25,394),(26,395),(27,396),(28,397),(29,326),(30,327),(31,328),(32,329),(33,330),(34,331),(35,332),(36,333),(37,334),(38,335),(39,336),(40,309),(41,310),(42,311),(43,312),(44,313),(45,314),(46,315),(47,316),(48,317),(49,318),(50,319),(51,320),(52,321),(53,322),(54,323),(55,324),(56,325),(57,226),(58,227),(59,228),(60,229),(61,230),(62,231),(63,232),(64,233),(65,234),(66,235),(67,236),(68,237),(69,238),(70,239),(71,240),(72,241),(73,242),(74,243),(75,244),(76,245),(77,246),(78,247),(79,248),(80,249),(81,250),(82,251),(83,252),(84,225),(85,425),(86,426),(87,427),(88,428),(89,429),(90,430),(91,431),(92,432),(93,433),(94,434),(95,435),(96,436),(97,437),(98,438),(99,439),(100,440),(101,441),(102,442),(103,443),(104,444),(105,445),(106,446),(107,447),(108,448),(109,421),(110,422),(111,423),(112,424),(113,337),(114,338),(115,339),(116,340),(117,341),(118,342),(119,343),(120,344),(121,345),(122,346),(123,347),(124,348),(125,349),(126,350),(127,351),(128,352),(129,353),(130,354),(131,355),(132,356),(133,357),(134,358),(135,359),(136,360),(137,361),(138,362),(139,363),(140,364),(141,208),(142,209),(143,210),(144,211),(145,212),(146,213),(147,214),(148,215),(149,216),(150,217),(151,218),(152,219),(153,220),(154,221),(155,222),(156,223),(157,224),(158,197),(159,198),(160,199),(161,200),(162,201),(163,202),(164,203),(165,204),(166,205),(167,206),(168,207),(169,268),(170,269),(171,270),(172,271),(173,272),(174,273),(175,274),(176,275),(177,276),(178,277),(179,278),(180,279),(181,280),(182,253),(183,254),(184,255),(185,256),(186,257),(187,258),(188,259),(189,260),(190,261),(191,262),(192,263),(193,264),(194,265),(195,266),(196,267),(281,386),(282,387),(283,388),(284,389),(285,390),(286,391),(287,392),(288,365),(289,366),(290,367),(291,368),(292,369),(293,370),(294,371),(295,372),(296,373),(297,374),(298,375),(299,376),(300,377),(301,378),(302,379),(303,380),(304,381),(305,382),(306,383),(307,384),(308,385)], [(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)], [(1,80,375,329),(2,79,376,328),(3,78,377,327),(4,77,378,326),(5,76,379,325),(6,75,380,324),(7,74,381,323),(8,73,382,322),(9,72,383,321),(10,71,384,320),(11,70,385,319),(12,69,386,318),(13,68,387,317),(14,67,388,316),(15,66,389,315),(16,65,390,314),(17,64,391,313),(18,63,392,312),(19,62,365,311),(20,61,366,310),(21,60,367,309),(22,59,368,336),(23,58,369,335),(24,57,370,334),(25,84,371,333),(26,83,372,332),(27,82,373,331),(28,81,374,330),(29,401,246,301),(30,400,247,300),(31,399,248,299),(32,398,249,298),(33,397,250,297),(34,396,251,296),(35,395,252,295),(36,394,225,294),(37,393,226,293),(38,420,227,292),(39,419,228,291),(40,418,229,290),(41,417,230,289),(42,416,231,288),(43,415,232,287),(44,414,233,286),(45,413,234,285),(46,412,235,284),(47,411,236,283),(48,410,237,282),(49,409,238,281),(50,408,239,308),(51,407,240,307),(52,406,241,306),(53,405,242,305),(54,404,243,304),(55,403,244,303),(56,402,245,302),(85,261,352,222),(86,260,353,221),(87,259,354,220),(88,258,355,219),(89,257,356,218),(90,256,357,217),(91,255,358,216),(92,254,359,215),(93,253,360,214),(94,280,361,213),(95,279,362,212),(96,278,363,211),(97,277,364,210),(98,276,337,209),(99,275,338,208),(100,274,339,207),(101,273,340,206),(102,272,341,205),(103,271,342,204),(104,270,343,203),(105,269,344,202),(106,268,345,201),(107,267,346,200),(108,266,347,199),(109,265,348,198),(110,264,349,197),(111,263,350,224),(112,262,351,223),(113,142,438,177),(114,141,439,176),(115,168,440,175),(116,167,441,174),(117,166,442,173),(118,165,443,172),(119,164,444,171),(120,163,445,170),(121,162,446,169),(122,161,447,196),(123,160,448,195),(124,159,421,194),(125,158,422,193),(126,157,423,192),(127,156,424,191),(128,155,425,190),(129,154,426,189),(130,153,427,188),(131,152,428,187),(132,151,429,186),(133,150,430,185),(134,149,431,184),(135,148,432,183),(136,147,433,182),(137,146,434,181),(138,145,435,180),(139,144,436,179),(140,143,437,178)], [(1,271,412,186),(2,171,413,256),(3,269,414,184),(4,169,415,254),(5,267,416,182),(6,195,417,280),(7,265,418,180),(8,193,419,278),(9,263,420,178),(10,191,393,276),(11,261,394,176),(12,189,395,274),(13,259,396,174),(14,187,397,272),(15,257,398,172),(16,185,399,270),(17,255,400,170),(18,183,401,268),(19,253,402,196),(20,181,403,266),(21,279,404,194),(22,179,405,264),(23,277,406,192),(24,177,407,262),(25,275,408,190),(26,175,409,260),(27,273,410,188),(28,173,411,258),(29,359,312,121),(30,134,313,344),(31,357,314,119),(32,132,315,342),(33,355,316,117),(34,130,317,340),(35,353,318,115),(36,128,319,338),(37,351,320,113),(38,126,321,364),(39,349,322,139),(40,124,323,362),(41,347,324,137),(42,122,325,360),(43,345,326,135),(44,120,327,358),(45,343,328,133),(46,118,329,356),(47,341,330,131),(48,116,331,354),(49,339,332,129),(50,114,333,352),(51,337,334,127),(52,140,335,350),(53,363,336,125),(54,138,309,348),(55,361,310,123),(56,136,311,346),(57,424,240,98),(58,111,241,437),(59,422,242,96),(60,109,243,435),(61,448,244,94),(62,107,245,433),(63,446,246,92),(64,105,247,431),(65,444,248,90),(66,103,249,429),(67,442,250,88),(68,101,251,427),(69,440,252,86),(70,99,225,425),(71,438,226,112),(72,97,227,423),(73,436,228,110),(74,95,229,421),(75,434,230,108),(76,93,231,447),(77,432,232,106),(78,91,233,445),(79,430,234,104),(80,89,235,443),(81,428,236,102),(82,87,237,441),(83,426,238,100),(84,85,239,439),(141,385,222,294),(142,307,223,370),(143,383,224,292),(144,305,197,368),(145,381,198,290),(146,303,199,366),(147,379,200,288),(148,301,201,392),(149,377,202,286),(150,299,203,390),(151,375,204,284),(152,297,205,388),(153,373,206,282),(154,295,207,386),(155,371,208,308),(156,293,209,384),(157,369,210,306),(158,291,211,382),(159,367,212,304),(160,289,213,380),(161,365,214,302),(162,287,215,378),(163,391,216,300),(164,285,217,376),(165,389,218,298),(166,283,219,374),(167,387,220,296),(168,281,221,372)]])
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 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | Q8 | D7 | C4○D4 | D14 | Dic14 | C7⋊D4 | C4○D28 | D4×D7 | D4⋊2D7 | Q8×D7 | Q8⋊2D7 |
| kernel | (C2×C28).54D4 | C14.C42 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C14×C4⋊C4 | C2×Dic7 | C2×Dic7 | C2×C28 | C2×C28 | C2×C4⋊C4 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 | C22 |
| # reps | 1 | 3 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 6 | 9 | 12 | 12 | 12 | 3 | 3 | 3 | 3 |
Matrix representation of (C2×C28).54D4 ►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 |
| 25 | 0 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 0 | 0 | 0 |
| 0 | 0 | 7 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 6 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 3 | 13 | 0 | 0 | 0 | 0 |
| 26 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 17 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 14 |
| 0 | 0 | 0 | 0 | 21 | 16 |
| 3 | 13 | 0 | 0 | 0 | 0 |
| 15 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 4 | 13 |
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],[25,7,0,0,0,0,0,7,0,0,0,0,0,0,19,7,0,0,0,0,0,26,0,0,0,0,0,0,18,9,0,0,0,0,6,11],[3,26,0,0,0,0,13,26,0,0,0,0,0,0,17,0,0,0,0,0,17,12,0,0,0,0,0,0,13,21,0,0,0,0,14,16],[3,15,0,0,0,0,13,26,0,0,0,0,0,0,12,0,0,0,0,0,12,17,0,0,0,0,0,0,16,4,0,0,0,0,1,13] >;
(C2×C28).54D4 in GAP, Magma, Sage, TeX
(C_2\times C_{28})._{54}D_4 % in TeX
G:=Group("(C2xC28).54D4"); // GroupNames label
G:=SmallGroup(448,518);
// by ID
G=gap.SmallGroup(448,518);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,344,254,387,184,18822]);
// 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=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=b^14*c^-1>;
// generators/relations