metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊1(C4⋊C4), C4⋊Dic7⋊12C4, C14.57(C4×D4), C2.3(C28⋊Q8), (C2×C28).17Q8, C14.20(C4×Q8), C4⋊1(Dic7⋊C4), (C2×C28).136D4, C14.19(C4⋊Q8), C22.19(Q8×D7), C2.3(C28⋊2D4), (C2×Dic7).15Q8, (C2×C4).28Dic14, C22.105(D4×D7), C14.88(C4⋊D4), C2.1(D14⋊3Q8), (C2×Dic7).107D4, (C22×C4).331D14, C14.69(C22⋊Q8), C2.3(C28.3Q8), C2.17(D28⋊C4), C14.16(C42.C2), C22.26(C2×Dic14), C23.287(C22×D7), C2.10(Dic7⋊3Q8), C22.52(D4⋊2D7), C14.C42.15C2, (C22×C28).140C22, (C22×C14).337C23, C7⋊4(C23.65C23), C22.21(Q8⋊2D7), (C22×Dic7).49C22, (C14×C4⋊C4).9C2, C14.35(C2×C4⋊C4), (C2×C4⋊C4).11D7, (C2×C4).77(C4×D7), (C2×C4×Dic7).3C2, (C2×C28).80(C2×C4), (C2×C14).73(C2×Q8), C22.131(C2×C4×D7), (C2×C14).327(C2×D4), C2.10(C2×Dic7⋊C4), (C2×C4⋊Dic7).31C2, C22.61(C2×C7⋊D4), (C2×C4).183(C7⋊D4), (C2×Dic7⋊C4).11C2, (C2×Dic7).28(C2×C4), (C2×C14).186(C4○D4), (C2×C14).113(C22×C4), SmallGroup(448,507)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊(C4⋊C4)
G = < a,b,c | a28=b4=c4=1, bab-1=a-1, cac-1=a15, cbc-1=b-1 >
Subgroups: 644 in 170 conjugacy classes, 83 normal (41 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, C23, C14, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.65C23, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C14.C42, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4⋊Dic7, C14×C4⋊C4, C28⋊(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D14, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, Dic14, C4×D7, C7⋊D4, C22×D7, C23.65C23, Dic7⋊C4, C2×Dic14, C2×C4×D7, D4×D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C2×C7⋊D4, Dic7⋊3Q8, C28⋊Q8, C28.3Q8, D28⋊C4, C2×Dic7⋊C4, C28⋊2D4, D14⋊3Q8, C28⋊(C4⋊C4)
►Regular action on 448 points
Generators in S448
(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)
(1 285 206 349)(2 284 207 348)(3 283 208 347)(4 282 209 346)(5 281 210 345)(6 308 211 344)(7 307 212 343)(8 306 213 342)(9 305 214 341)(10 304 215 340)(11 303 216 339)(12 302 217 338)(13 301 218 337)(14 300 219 364)(15 299 220 363)(16 298 221 362)(17 297 222 361)(18 296 223 360)(19 295 224 359)(20 294 197 358)(21 293 198 357)(22 292 199 356)(23 291 200 355)(24 290 201 354)(25 289 202 353)(26 288 203 352)(27 287 204 351)(28 286 205 350)(29 160 382 127)(30 159 383 126)(31 158 384 125)(32 157 385 124)(33 156 386 123)(34 155 387 122)(35 154 388 121)(36 153 389 120)(37 152 390 119)(38 151 391 118)(39 150 392 117)(40 149 365 116)(41 148 366 115)(42 147 367 114)(43 146 368 113)(44 145 369 140)(45 144 370 139)(46 143 371 138)(47 142 372 137)(48 141 373 136)(49 168 374 135)(50 167 375 134)(51 166 376 133)(52 165 377 132)(53 164 378 131)(54 163 379 130)(55 162 380 129)(56 161 381 128)(57 418 110 262)(58 417 111 261)(59 416 112 260)(60 415 85 259)(61 414 86 258)(62 413 87 257)(63 412 88 256)(64 411 89 255)(65 410 90 254)(66 409 91 253)(67 408 92 280)(68 407 93 279)(69 406 94 278)(70 405 95 277)(71 404 96 276)(72 403 97 275)(73 402 98 274)(74 401 99 273)(75 400 100 272)(76 399 101 271)(77 398 102 270)(78 397 103 269)(79 396 104 268)(80 395 105 267)(81 394 106 266)(82 393 107 265)(83 420 108 264)(84 419 109 263)(169 436 318 226)(170 435 319 225)(171 434 320 252)(172 433 321 251)(173 432 322 250)(174 431 323 249)(175 430 324 248)(176 429 325 247)(177 428 326 246)(178 427 327 245)(179 426 328 244)(180 425 329 243)(181 424 330 242)(182 423 331 241)(183 422 332 240)(184 421 333 239)(185 448 334 238)(186 447 335 237)(187 446 336 236)(188 445 309 235)(189 444 310 234)(190 443 311 233)(191 442 312 232)(192 441 313 231)(193 440 314 230)(194 439 315 229)(195 438 316 228)(196 437 317 227)
(1 231 137 82)(2 246 138 69)(3 233 139 84)(4 248 140 71)(5 235 113 58)(6 250 114 73)(7 237 115 60)(8 252 116 75)(9 239 117 62)(10 226 118 77)(11 241 119 64)(12 228 120 79)(13 243 121 66)(14 230 122 81)(15 245 123 68)(16 232 124 83)(17 247 125 70)(18 234 126 57)(19 249 127 72)(20 236 128 59)(21 251 129 74)(22 238 130 61)(23 225 131 76)(24 240 132 63)(25 227 133 78)(26 242 134 65)(27 229 135 80)(28 244 136 67)(29 275 295 323)(30 262 296 310)(31 277 297 325)(32 264 298 312)(33 279 299 327)(34 266 300 314)(35 253 301 329)(36 268 302 316)(37 255 303 331)(38 270 304 318)(39 257 305 333)(40 272 306 320)(41 259 307 335)(42 274 308 322)(43 261 281 309)(44 276 282 324)(45 263 283 311)(46 278 284 326)(47 265 285 313)(48 280 286 328)(49 267 287 315)(50 254 288 330)(51 269 289 317)(52 256 290 332)(53 271 291 319)(54 258 292 334)(55 273 293 321)(56 260 294 336)(85 212 447 148)(86 199 448 163)(87 214 421 150)(88 201 422 165)(89 216 423 152)(90 203 424 167)(91 218 425 154)(92 205 426 141)(93 220 427 156)(94 207 428 143)(95 222 429 158)(96 209 430 145)(97 224 431 160)(98 211 432 147)(99 198 433 162)(100 213 434 149)(101 200 435 164)(102 215 436 151)(103 202 437 166)(104 217 438 153)(105 204 439 168)(106 219 440 155)(107 206 441 142)(108 221 442 157)(109 208 443 144)(110 223 444 159)(111 210 445 146)(112 197 446 161)(169 391 398 340)(170 378 399 355)(171 365 400 342)(172 380 401 357)(173 367 402 344)(174 382 403 359)(175 369 404 346)(176 384 405 361)(177 371 406 348)(178 386 407 363)(179 373 408 350)(180 388 409 337)(181 375 410 352)(182 390 411 339)(183 377 412 354)(184 392 413 341)(185 379 414 356)(186 366 415 343)(187 381 416 358)(188 368 417 345)(189 383 418 360)(190 370 419 347)(191 385 420 362)(192 372 393 349)(193 387 394 364)(194 374 395 351)(195 389 396 338)(196 376 397 353)
G:=sub<Sym(448)| (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), (1,285,206,349)(2,284,207,348)(3,283,208,347)(4,282,209,346)(5,281,210,345)(6,308,211,344)(7,307,212,343)(8,306,213,342)(9,305,214,341)(10,304,215,340)(11,303,216,339)(12,302,217,338)(13,301,218,337)(14,300,219,364)(15,299,220,363)(16,298,221,362)(17,297,222,361)(18,296,223,360)(19,295,224,359)(20,294,197,358)(21,293,198,357)(22,292,199,356)(23,291,200,355)(24,290,201,354)(25,289,202,353)(26,288,203,352)(27,287,204,351)(28,286,205,350)(29,160,382,127)(30,159,383,126)(31,158,384,125)(32,157,385,124)(33,156,386,123)(34,155,387,122)(35,154,388,121)(36,153,389,120)(37,152,390,119)(38,151,391,118)(39,150,392,117)(40,149,365,116)(41,148,366,115)(42,147,367,114)(43,146,368,113)(44,145,369,140)(45,144,370,139)(46,143,371,138)(47,142,372,137)(48,141,373,136)(49,168,374,135)(50,167,375,134)(51,166,376,133)(52,165,377,132)(53,164,378,131)(54,163,379,130)(55,162,380,129)(56,161,381,128)(57,418,110,262)(58,417,111,261)(59,416,112,260)(60,415,85,259)(61,414,86,258)(62,413,87,257)(63,412,88,256)(64,411,89,255)(65,410,90,254)(66,409,91,253)(67,408,92,280)(68,407,93,279)(69,406,94,278)(70,405,95,277)(71,404,96,276)(72,403,97,275)(73,402,98,274)(74,401,99,273)(75,400,100,272)(76,399,101,271)(77,398,102,270)(78,397,103,269)(79,396,104,268)(80,395,105,267)(81,394,106,266)(82,393,107,265)(83,420,108,264)(84,419,109,263)(169,436,318,226)(170,435,319,225)(171,434,320,252)(172,433,321,251)(173,432,322,250)(174,431,323,249)(175,430,324,248)(176,429,325,247)(177,428,326,246)(178,427,327,245)(179,426,328,244)(180,425,329,243)(181,424,330,242)(182,423,331,241)(183,422,332,240)(184,421,333,239)(185,448,334,238)(186,447,335,237)(187,446,336,236)(188,445,309,235)(189,444,310,234)(190,443,311,233)(191,442,312,232)(192,441,313,231)(193,440,314,230)(194,439,315,229)(195,438,316,228)(196,437,317,227), (1,231,137,82)(2,246,138,69)(3,233,139,84)(4,248,140,71)(5,235,113,58)(6,250,114,73)(7,237,115,60)(8,252,116,75)(9,239,117,62)(10,226,118,77)(11,241,119,64)(12,228,120,79)(13,243,121,66)(14,230,122,81)(15,245,123,68)(16,232,124,83)(17,247,125,70)(18,234,126,57)(19,249,127,72)(20,236,128,59)(21,251,129,74)(22,238,130,61)(23,225,131,76)(24,240,132,63)(25,227,133,78)(26,242,134,65)(27,229,135,80)(28,244,136,67)(29,275,295,323)(30,262,296,310)(31,277,297,325)(32,264,298,312)(33,279,299,327)(34,266,300,314)(35,253,301,329)(36,268,302,316)(37,255,303,331)(38,270,304,318)(39,257,305,333)(40,272,306,320)(41,259,307,335)(42,274,308,322)(43,261,281,309)(44,276,282,324)(45,263,283,311)(46,278,284,326)(47,265,285,313)(48,280,286,328)(49,267,287,315)(50,254,288,330)(51,269,289,317)(52,256,290,332)(53,271,291,319)(54,258,292,334)(55,273,293,321)(56,260,294,336)(85,212,447,148)(86,199,448,163)(87,214,421,150)(88,201,422,165)(89,216,423,152)(90,203,424,167)(91,218,425,154)(92,205,426,141)(93,220,427,156)(94,207,428,143)(95,222,429,158)(96,209,430,145)(97,224,431,160)(98,211,432,147)(99,198,433,162)(100,213,434,149)(101,200,435,164)(102,215,436,151)(103,202,437,166)(104,217,438,153)(105,204,439,168)(106,219,440,155)(107,206,441,142)(108,221,442,157)(109,208,443,144)(110,223,444,159)(111,210,445,146)(112,197,446,161)(169,391,398,340)(170,378,399,355)(171,365,400,342)(172,380,401,357)(173,367,402,344)(174,382,403,359)(175,369,404,346)(176,384,405,361)(177,371,406,348)(178,386,407,363)(179,373,408,350)(180,388,409,337)(181,375,410,352)(182,390,411,339)(183,377,412,354)(184,392,413,341)(185,379,414,356)(186,366,415,343)(187,381,416,358)(188,368,417,345)(189,383,418,360)(190,370,419,347)(191,385,420,362)(192,372,393,349)(193,387,394,364)(194,374,395,351)(195,389,396,338)(196,376,397,353)>;
G:=Group( (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), (1,285,206,349)(2,284,207,348)(3,283,208,347)(4,282,209,346)(5,281,210,345)(6,308,211,344)(7,307,212,343)(8,306,213,342)(9,305,214,341)(10,304,215,340)(11,303,216,339)(12,302,217,338)(13,301,218,337)(14,300,219,364)(15,299,220,363)(16,298,221,362)(17,297,222,361)(18,296,223,360)(19,295,224,359)(20,294,197,358)(21,293,198,357)(22,292,199,356)(23,291,200,355)(24,290,201,354)(25,289,202,353)(26,288,203,352)(27,287,204,351)(28,286,205,350)(29,160,382,127)(30,159,383,126)(31,158,384,125)(32,157,385,124)(33,156,386,123)(34,155,387,122)(35,154,388,121)(36,153,389,120)(37,152,390,119)(38,151,391,118)(39,150,392,117)(40,149,365,116)(41,148,366,115)(42,147,367,114)(43,146,368,113)(44,145,369,140)(45,144,370,139)(46,143,371,138)(47,142,372,137)(48,141,373,136)(49,168,374,135)(50,167,375,134)(51,166,376,133)(52,165,377,132)(53,164,378,131)(54,163,379,130)(55,162,380,129)(56,161,381,128)(57,418,110,262)(58,417,111,261)(59,416,112,260)(60,415,85,259)(61,414,86,258)(62,413,87,257)(63,412,88,256)(64,411,89,255)(65,410,90,254)(66,409,91,253)(67,408,92,280)(68,407,93,279)(69,406,94,278)(70,405,95,277)(71,404,96,276)(72,403,97,275)(73,402,98,274)(74,401,99,273)(75,400,100,272)(76,399,101,271)(77,398,102,270)(78,397,103,269)(79,396,104,268)(80,395,105,267)(81,394,106,266)(82,393,107,265)(83,420,108,264)(84,419,109,263)(169,436,318,226)(170,435,319,225)(171,434,320,252)(172,433,321,251)(173,432,322,250)(174,431,323,249)(175,430,324,248)(176,429,325,247)(177,428,326,246)(178,427,327,245)(179,426,328,244)(180,425,329,243)(181,424,330,242)(182,423,331,241)(183,422,332,240)(184,421,333,239)(185,448,334,238)(186,447,335,237)(187,446,336,236)(188,445,309,235)(189,444,310,234)(190,443,311,233)(191,442,312,232)(192,441,313,231)(193,440,314,230)(194,439,315,229)(195,438,316,228)(196,437,317,227), (1,231,137,82)(2,246,138,69)(3,233,139,84)(4,248,140,71)(5,235,113,58)(6,250,114,73)(7,237,115,60)(8,252,116,75)(9,239,117,62)(10,226,118,77)(11,241,119,64)(12,228,120,79)(13,243,121,66)(14,230,122,81)(15,245,123,68)(16,232,124,83)(17,247,125,70)(18,234,126,57)(19,249,127,72)(20,236,128,59)(21,251,129,74)(22,238,130,61)(23,225,131,76)(24,240,132,63)(25,227,133,78)(26,242,134,65)(27,229,135,80)(28,244,136,67)(29,275,295,323)(30,262,296,310)(31,277,297,325)(32,264,298,312)(33,279,299,327)(34,266,300,314)(35,253,301,329)(36,268,302,316)(37,255,303,331)(38,270,304,318)(39,257,305,333)(40,272,306,320)(41,259,307,335)(42,274,308,322)(43,261,281,309)(44,276,282,324)(45,263,283,311)(46,278,284,326)(47,265,285,313)(48,280,286,328)(49,267,287,315)(50,254,288,330)(51,269,289,317)(52,256,290,332)(53,271,291,319)(54,258,292,334)(55,273,293,321)(56,260,294,336)(85,212,447,148)(86,199,448,163)(87,214,421,150)(88,201,422,165)(89,216,423,152)(90,203,424,167)(91,218,425,154)(92,205,426,141)(93,220,427,156)(94,207,428,143)(95,222,429,158)(96,209,430,145)(97,224,431,160)(98,211,432,147)(99,198,433,162)(100,213,434,149)(101,200,435,164)(102,215,436,151)(103,202,437,166)(104,217,438,153)(105,204,439,168)(106,219,440,155)(107,206,441,142)(108,221,442,157)(109,208,443,144)(110,223,444,159)(111,210,445,146)(112,197,446,161)(169,391,398,340)(170,378,399,355)(171,365,400,342)(172,380,401,357)(173,367,402,344)(174,382,403,359)(175,369,404,346)(176,384,405,361)(177,371,406,348)(178,386,407,363)(179,373,408,350)(180,388,409,337)(181,375,410,352)(182,390,411,339)(183,377,412,354)(184,392,413,341)(185,379,414,356)(186,366,415,343)(187,381,416,358)(188,368,417,345)(189,383,418,360)(190,370,419,347)(191,385,420,362)(192,372,393,349)(193,387,394,364)(194,374,395,351)(195,389,396,338)(196,376,397,353) );
G=PermutationGroup([[(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)], [(1,285,206,349),(2,284,207,348),(3,283,208,347),(4,282,209,346),(5,281,210,345),(6,308,211,344),(7,307,212,343),(8,306,213,342),(9,305,214,341),(10,304,215,340),(11,303,216,339),(12,302,217,338),(13,301,218,337),(14,300,219,364),(15,299,220,363),(16,298,221,362),(17,297,222,361),(18,296,223,360),(19,295,224,359),(20,294,197,358),(21,293,198,357),(22,292,199,356),(23,291,200,355),(24,290,201,354),(25,289,202,353),(26,288,203,352),(27,287,204,351),(28,286,205,350),(29,160,382,127),(30,159,383,126),(31,158,384,125),(32,157,385,124),(33,156,386,123),(34,155,387,122),(35,154,388,121),(36,153,389,120),(37,152,390,119),(38,151,391,118),(39,150,392,117),(40,149,365,116),(41,148,366,115),(42,147,367,114),(43,146,368,113),(44,145,369,140),(45,144,370,139),(46,143,371,138),(47,142,372,137),(48,141,373,136),(49,168,374,135),(50,167,375,134),(51,166,376,133),(52,165,377,132),(53,164,378,131),(54,163,379,130),(55,162,380,129),(56,161,381,128),(57,418,110,262),(58,417,111,261),(59,416,112,260),(60,415,85,259),(61,414,86,258),(62,413,87,257),(63,412,88,256),(64,411,89,255),(65,410,90,254),(66,409,91,253),(67,408,92,280),(68,407,93,279),(69,406,94,278),(70,405,95,277),(71,404,96,276),(72,403,97,275),(73,402,98,274),(74,401,99,273),(75,400,100,272),(76,399,101,271),(77,398,102,270),(78,397,103,269),(79,396,104,268),(80,395,105,267),(81,394,106,266),(82,393,107,265),(83,420,108,264),(84,419,109,263),(169,436,318,226),(170,435,319,225),(171,434,320,252),(172,433,321,251),(173,432,322,250),(174,431,323,249),(175,430,324,248),(176,429,325,247),(177,428,326,246),(178,427,327,245),(179,426,328,244),(180,425,329,243),(181,424,330,242),(182,423,331,241),(183,422,332,240),(184,421,333,239),(185,448,334,238),(186,447,335,237),(187,446,336,236),(188,445,309,235),(189,444,310,234),(190,443,311,233),(191,442,312,232),(192,441,313,231),(193,440,314,230),(194,439,315,229),(195,438,316,228),(196,437,317,227)], [(1,231,137,82),(2,246,138,69),(3,233,139,84),(4,248,140,71),(5,235,113,58),(6,250,114,73),(7,237,115,60),(8,252,116,75),(9,239,117,62),(10,226,118,77),(11,241,119,64),(12,228,120,79),(13,243,121,66),(14,230,122,81),(15,245,123,68),(16,232,124,83),(17,247,125,70),(18,234,126,57),(19,249,127,72),(20,236,128,59),(21,251,129,74),(22,238,130,61),(23,225,131,76),(24,240,132,63),(25,227,133,78),(26,242,134,65),(27,229,135,80),(28,244,136,67),(29,275,295,323),(30,262,296,310),(31,277,297,325),(32,264,298,312),(33,279,299,327),(34,266,300,314),(35,253,301,329),(36,268,302,316),(37,255,303,331),(38,270,304,318),(39,257,305,333),(40,272,306,320),(41,259,307,335),(42,274,308,322),(43,261,281,309),(44,276,282,324),(45,263,283,311),(46,278,284,326),(47,265,285,313),(48,280,286,328),(49,267,287,315),(50,254,288,330),(51,269,289,317),(52,256,290,332),(53,271,291,319),(54,258,292,334),(55,273,293,321),(56,260,294,336),(85,212,447,148),(86,199,448,163),(87,214,421,150),(88,201,422,165),(89,216,423,152),(90,203,424,167),(91,218,425,154),(92,205,426,141),(93,220,427,156),(94,207,428,143),(95,222,429,158),(96,209,430,145),(97,224,431,160),(98,211,432,147),(99,198,433,162),(100,213,434,149),(101,200,435,164),(102,215,436,151),(103,202,437,166),(104,217,438,153),(105,204,439,168),(106,219,440,155),(107,206,441,142),(108,221,442,157),(109,208,443,144),(110,223,444,159),(111,210,445,146),(112,197,446,161),(169,391,398,340),(170,378,399,355),(171,365,400,342),(172,380,401,357),(173,367,402,344),(174,382,403,359),(175,369,404,346),(176,384,405,361),(177,371,406,348),(178,386,407,363),(179,373,408,350),(180,388,409,337),(181,375,410,352),(182,390,411,339),(183,377,412,354),(184,392,413,341),(185,379,414,356),(186,366,415,343),(187,381,416,358),(188,368,417,345),(189,383,418,360),(190,370,419,347),(191,385,420,362),(192,372,393,349),(193,387,394,364),(194,374,395,351),(195,389,396,338),(196,376,397,353)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 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 | C2 | C4 | D4 | Q8 | D4 | Q8 | D7 | C4○D4 | D14 | Dic14 | C4×D7 | C7⋊D4 | D4×D7 | D4⋊2D7 | Q8×D7 | Q8⋊2D7 |
| kernel | C28⋊(C4⋊C4) | C14.C42 | C2×C4×Dic7 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C14×C4⋊C4 | C4⋊Dic7 | C2×Dic7 | C2×Dic7 | C2×C28 | C2×C28 | C2×C4⋊C4 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 3 | 4 | 9 | 12 | 12 | 12 | 3 | 3 | 3 | 3 |
Matrix representation of C28⋊(C4⋊C4) ►in GL8(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 26 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 18 |
| 14 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 11 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(29))| [28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,16,20,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,4,15,0,0,0,0,0,0,26,18],[14,8,0,0,0,0,0,0,8,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,28,28,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,18,14,0,0,0,0,0,0,8,11],[0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,28,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
C28⋊(C4⋊C4) in GAP, Magma, Sage, TeX
C_{28}\rtimes (C_4\rtimes C_4) % in TeX
G:=Group("C28:(C4:C4)"); // GroupNames label
G:=SmallGroup(448,507);
// by ID
G=gap.SmallGroup(448,507);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,254,219,184,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=c^4=1,b*a*b^-1=a^-1,c*a*c^-1=a^15,c*b*c^-1=b^-1>;
// generators/relations