metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊4(C4⋊C4), C4⋊Dic7⋊9C4, C14.8(C4×Q8), (C2×C4).65D28, C2.18(C4×D28), C14.15(C4×D4), C14.6(C4⋊Q8), (C2×C28).52Q8, C4⋊3(Dic7⋊C4), (C2×C28).469D4, (C2×C42).13D7, C2.1(C28⋊7D4), C2.1(C28⋊2Q8), (C2×C4).40Dic14, C2.10(C4×Dic14), C22.33(C2×D28), C14.55(C4⋊D4), (C22×C4).394D14, C14.52(C22⋊Q8), C2.1(C28.6Q8), C14.2(C42.C2), C2.1(C28.48D4), C22.39(C4○D28), C14.C42.9C2, C22.16(C2×Dic14), C23.261(C22×D7), (C22×C28).469C22, (C22×C14).303C23, C7⋊2(C23.65C23), (C22×Dic7).26C22, (C2×C4×C28).9C2, C14.24(C2×C4⋊C4), (C2×C4).108(C4×D7), C2.4(C2×Dic7⋊C4), (C2×C14).23(C2×Q8), C22.116(C2×C4×D7), (C2×C28).223(C2×C4), (C2×C14).423(C2×D4), (C2×Dic7⋊C4).9C2, (C2×C4⋊Dic7).13C2, C22.38(C2×C7⋊D4), (C2×C14).64(C4○D4), (C2×C4).237(C7⋊D4), (C2×C14).93(C22×C4), (C2×Dic7).24(C2×C4), SmallGroup(448,462)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊4(C4⋊C4)
G = < a,b,c | a28=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=b-1 >
Subgroups: 644 in 170 conjugacy classes, 87 normal (43 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, C23, C14, C42, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.65C23, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C4×C28, C22×Dic7, C22×C28, C14.C42, C2×Dic7⋊C4, C2×C4⋊Dic7, C2×C4×C28, C28⋊4(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, D28, C7⋊D4, C22×D7, C23.65C23, Dic7⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C4×Dic14, C28⋊2Q8, C28.6Q8, C4×D28, C2×Dic7⋊C4, C28.48D4, C28⋊7D4, C28⋊4(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)
(1 382 136 288)(2 381 137 287)(3 380 138 286)(4 379 139 285)(5 378 140 284)(6 377 113 283)(7 376 114 282)(8 375 115 281)(9 374 116 308)(10 373 117 307)(11 372 118 306)(12 371 119 305)(13 370 120 304)(14 369 121 303)(15 368 122 302)(16 367 123 301)(17 366 124 300)(18 365 125 299)(19 392 126 298)(20 391 127 297)(21 390 128 296)(22 389 129 295)(23 388 130 294)(24 387 131 293)(25 386 132 292)(26 385 133 291)(27 384 134 290)(28 383 135 289)(29 262 57 158)(30 261 58 157)(31 260 59 156)(32 259 60 155)(33 258 61 154)(34 257 62 153)(35 256 63 152)(36 255 64 151)(37 254 65 150)(38 253 66 149)(39 280 67 148)(40 279 68 147)(41 278 69 146)(42 277 70 145)(43 276 71 144)(44 275 72 143)(45 274 73 142)(46 273 74 141)(47 272 75 168)(48 271 76 167)(49 270 77 166)(50 269 78 165)(51 268 79 164)(52 267 80 163)(53 266 81 162)(54 265 82 161)(55 264 83 160)(56 263 84 159)(85 333 196 241)(86 332 169 240)(87 331 170 239)(88 330 171 238)(89 329 172 237)(90 328 173 236)(91 327 174 235)(92 326 175 234)(93 325 176 233)(94 324 177 232)(95 323 178 231)(96 322 179 230)(97 321 180 229)(98 320 181 228)(99 319 182 227)(100 318 183 226)(101 317 184 225)(102 316 185 252)(103 315 186 251)(104 314 187 250)(105 313 188 249)(106 312 189 248)(107 311 190 247)(108 310 191 246)(109 309 192 245)(110 336 193 244)(111 335 194 243)(112 334 195 242)(197 360 393 435)(198 359 394 434)(199 358 395 433)(200 357 396 432)(201 356 397 431)(202 355 398 430)(203 354 399 429)(204 353 400 428)(205 352 401 427)(206 351 402 426)(207 350 403 425)(208 349 404 424)(209 348 405 423)(210 347 406 422)(211 346 407 421)(212 345 408 448)(213 344 409 447)(214 343 410 446)(215 342 411 445)(216 341 412 444)(217 340 413 443)(218 339 414 442)(219 338 415 441)(220 337 416 440)(221 364 417 439)(222 363 418 438)(223 362 419 437)(224 361 420 436)
(1 413 88 30)(2 414 89 31)(3 415 90 32)(4 416 91 33)(5 417 92 34)(6 418 93 35)(7 419 94 36)(8 420 95 37)(9 393 96 38)(10 394 97 39)(11 395 98 40)(12 396 99 41)(13 397 100 42)(14 398 101 43)(15 399 102 44)(16 400 103 45)(17 401 104 46)(18 402 105 47)(19 403 106 48)(20 404 107 49)(21 405 108 50)(22 406 109 51)(23 407 110 52)(24 408 111 53)(25 409 112 54)(26 410 85 55)(27 411 86 56)(28 412 87 29)(57 135 216 170)(58 136 217 171)(59 137 218 172)(60 138 219 173)(61 139 220 174)(62 140 221 175)(63 113 222 176)(64 114 223 177)(65 115 224 178)(66 116 197 179)(67 117 198 180)(68 118 199 181)(69 119 200 182)(70 120 201 183)(71 121 202 184)(72 122 203 185)(73 123 204 186)(74 124 205 187)(75 125 206 188)(76 126 207 189)(77 127 208 190)(78 128 209 191)(79 129 210 192)(80 130 211 193)(81 131 212 194)(82 132 213 195)(83 133 214 196)(84 134 215 169)(141 366 352 314)(142 367 353 315)(143 368 354 316)(144 369 355 317)(145 370 356 318)(146 371 357 319)(147 372 358 320)(148 373 359 321)(149 374 360 322)(150 375 361 323)(151 376 362 324)(152 377 363 325)(153 378 364 326)(154 379 337 327)(155 380 338 328)(156 381 339 329)(157 382 340 330)(158 383 341 331)(159 384 342 332)(160 385 343 333)(161 386 344 334)(162 387 345 335)(163 388 346 336)(164 389 347 309)(165 390 348 310)(166 391 349 311)(167 392 350 312)(168 365 351 313)(225 276 303 430)(226 277 304 431)(227 278 305 432)(228 279 306 433)(229 280 307 434)(230 253 308 435)(231 254 281 436)(232 255 282 437)(233 256 283 438)(234 257 284 439)(235 258 285 440)(236 259 286 441)(237 260 287 442)(238 261 288 443)(239 262 289 444)(240 263 290 445)(241 264 291 446)(242 265 292 447)(243 266 293 448)(244 267 294 421)(245 268 295 422)(246 269 296 423)(247 270 297 424)(248 271 298 425)(249 272 299 426)(250 273 300 427)(251 274 301 428)(252 275 302 429)
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), (1,382,136,288)(2,381,137,287)(3,380,138,286)(4,379,139,285)(5,378,140,284)(6,377,113,283)(7,376,114,282)(8,375,115,281)(9,374,116,308)(10,373,117,307)(11,372,118,306)(12,371,119,305)(13,370,120,304)(14,369,121,303)(15,368,122,302)(16,367,123,301)(17,366,124,300)(18,365,125,299)(19,392,126,298)(20,391,127,297)(21,390,128,296)(22,389,129,295)(23,388,130,294)(24,387,131,293)(25,386,132,292)(26,385,133,291)(27,384,134,290)(28,383,135,289)(29,262,57,158)(30,261,58,157)(31,260,59,156)(32,259,60,155)(33,258,61,154)(34,257,62,153)(35,256,63,152)(36,255,64,151)(37,254,65,150)(38,253,66,149)(39,280,67,148)(40,279,68,147)(41,278,69,146)(42,277,70,145)(43,276,71,144)(44,275,72,143)(45,274,73,142)(46,273,74,141)(47,272,75,168)(48,271,76,167)(49,270,77,166)(50,269,78,165)(51,268,79,164)(52,267,80,163)(53,266,81,162)(54,265,82,161)(55,264,83,160)(56,263,84,159)(85,333,196,241)(86,332,169,240)(87,331,170,239)(88,330,171,238)(89,329,172,237)(90,328,173,236)(91,327,174,235)(92,326,175,234)(93,325,176,233)(94,324,177,232)(95,323,178,231)(96,322,179,230)(97,321,180,229)(98,320,181,228)(99,319,182,227)(100,318,183,226)(101,317,184,225)(102,316,185,252)(103,315,186,251)(104,314,187,250)(105,313,188,249)(106,312,189,248)(107,311,190,247)(108,310,191,246)(109,309,192,245)(110,336,193,244)(111,335,194,243)(112,334,195,242)(197,360,393,435)(198,359,394,434)(199,358,395,433)(200,357,396,432)(201,356,397,431)(202,355,398,430)(203,354,399,429)(204,353,400,428)(205,352,401,427)(206,351,402,426)(207,350,403,425)(208,349,404,424)(209,348,405,423)(210,347,406,422)(211,346,407,421)(212,345,408,448)(213,344,409,447)(214,343,410,446)(215,342,411,445)(216,341,412,444)(217,340,413,443)(218,339,414,442)(219,338,415,441)(220,337,416,440)(221,364,417,439)(222,363,418,438)(223,362,419,437)(224,361,420,436), (1,413,88,30)(2,414,89,31)(3,415,90,32)(4,416,91,33)(5,417,92,34)(6,418,93,35)(7,419,94,36)(8,420,95,37)(9,393,96,38)(10,394,97,39)(11,395,98,40)(12,396,99,41)(13,397,100,42)(14,398,101,43)(15,399,102,44)(16,400,103,45)(17,401,104,46)(18,402,105,47)(19,403,106,48)(20,404,107,49)(21,405,108,50)(22,406,109,51)(23,407,110,52)(24,408,111,53)(25,409,112,54)(26,410,85,55)(27,411,86,56)(28,412,87,29)(57,135,216,170)(58,136,217,171)(59,137,218,172)(60,138,219,173)(61,139,220,174)(62,140,221,175)(63,113,222,176)(64,114,223,177)(65,115,224,178)(66,116,197,179)(67,117,198,180)(68,118,199,181)(69,119,200,182)(70,120,201,183)(71,121,202,184)(72,122,203,185)(73,123,204,186)(74,124,205,187)(75,125,206,188)(76,126,207,189)(77,127,208,190)(78,128,209,191)(79,129,210,192)(80,130,211,193)(81,131,212,194)(82,132,213,195)(83,133,214,196)(84,134,215,169)(141,366,352,314)(142,367,353,315)(143,368,354,316)(144,369,355,317)(145,370,356,318)(146,371,357,319)(147,372,358,320)(148,373,359,321)(149,374,360,322)(150,375,361,323)(151,376,362,324)(152,377,363,325)(153,378,364,326)(154,379,337,327)(155,380,338,328)(156,381,339,329)(157,382,340,330)(158,383,341,331)(159,384,342,332)(160,385,343,333)(161,386,344,334)(162,387,345,335)(163,388,346,336)(164,389,347,309)(165,390,348,310)(166,391,349,311)(167,392,350,312)(168,365,351,313)(225,276,303,430)(226,277,304,431)(227,278,305,432)(228,279,306,433)(229,280,307,434)(230,253,308,435)(231,254,281,436)(232,255,282,437)(233,256,283,438)(234,257,284,439)(235,258,285,440)(236,259,286,441)(237,260,287,442)(238,261,288,443)(239,262,289,444)(240,263,290,445)(241,264,291,446)(242,265,292,447)(243,266,293,448)(244,267,294,421)(245,268,295,422)(246,269,296,423)(247,270,297,424)(248,271,298,425)(249,272,299,426)(250,273,300,427)(251,274,301,428)(252,275,302,429)>;
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), (1,382,136,288)(2,381,137,287)(3,380,138,286)(4,379,139,285)(5,378,140,284)(6,377,113,283)(7,376,114,282)(8,375,115,281)(9,374,116,308)(10,373,117,307)(11,372,118,306)(12,371,119,305)(13,370,120,304)(14,369,121,303)(15,368,122,302)(16,367,123,301)(17,366,124,300)(18,365,125,299)(19,392,126,298)(20,391,127,297)(21,390,128,296)(22,389,129,295)(23,388,130,294)(24,387,131,293)(25,386,132,292)(26,385,133,291)(27,384,134,290)(28,383,135,289)(29,262,57,158)(30,261,58,157)(31,260,59,156)(32,259,60,155)(33,258,61,154)(34,257,62,153)(35,256,63,152)(36,255,64,151)(37,254,65,150)(38,253,66,149)(39,280,67,148)(40,279,68,147)(41,278,69,146)(42,277,70,145)(43,276,71,144)(44,275,72,143)(45,274,73,142)(46,273,74,141)(47,272,75,168)(48,271,76,167)(49,270,77,166)(50,269,78,165)(51,268,79,164)(52,267,80,163)(53,266,81,162)(54,265,82,161)(55,264,83,160)(56,263,84,159)(85,333,196,241)(86,332,169,240)(87,331,170,239)(88,330,171,238)(89,329,172,237)(90,328,173,236)(91,327,174,235)(92,326,175,234)(93,325,176,233)(94,324,177,232)(95,323,178,231)(96,322,179,230)(97,321,180,229)(98,320,181,228)(99,319,182,227)(100,318,183,226)(101,317,184,225)(102,316,185,252)(103,315,186,251)(104,314,187,250)(105,313,188,249)(106,312,189,248)(107,311,190,247)(108,310,191,246)(109,309,192,245)(110,336,193,244)(111,335,194,243)(112,334,195,242)(197,360,393,435)(198,359,394,434)(199,358,395,433)(200,357,396,432)(201,356,397,431)(202,355,398,430)(203,354,399,429)(204,353,400,428)(205,352,401,427)(206,351,402,426)(207,350,403,425)(208,349,404,424)(209,348,405,423)(210,347,406,422)(211,346,407,421)(212,345,408,448)(213,344,409,447)(214,343,410,446)(215,342,411,445)(216,341,412,444)(217,340,413,443)(218,339,414,442)(219,338,415,441)(220,337,416,440)(221,364,417,439)(222,363,418,438)(223,362,419,437)(224,361,420,436), (1,413,88,30)(2,414,89,31)(3,415,90,32)(4,416,91,33)(5,417,92,34)(6,418,93,35)(7,419,94,36)(8,420,95,37)(9,393,96,38)(10,394,97,39)(11,395,98,40)(12,396,99,41)(13,397,100,42)(14,398,101,43)(15,399,102,44)(16,400,103,45)(17,401,104,46)(18,402,105,47)(19,403,106,48)(20,404,107,49)(21,405,108,50)(22,406,109,51)(23,407,110,52)(24,408,111,53)(25,409,112,54)(26,410,85,55)(27,411,86,56)(28,412,87,29)(57,135,216,170)(58,136,217,171)(59,137,218,172)(60,138,219,173)(61,139,220,174)(62,140,221,175)(63,113,222,176)(64,114,223,177)(65,115,224,178)(66,116,197,179)(67,117,198,180)(68,118,199,181)(69,119,200,182)(70,120,201,183)(71,121,202,184)(72,122,203,185)(73,123,204,186)(74,124,205,187)(75,125,206,188)(76,126,207,189)(77,127,208,190)(78,128,209,191)(79,129,210,192)(80,130,211,193)(81,131,212,194)(82,132,213,195)(83,133,214,196)(84,134,215,169)(141,366,352,314)(142,367,353,315)(143,368,354,316)(144,369,355,317)(145,370,356,318)(146,371,357,319)(147,372,358,320)(148,373,359,321)(149,374,360,322)(150,375,361,323)(151,376,362,324)(152,377,363,325)(153,378,364,326)(154,379,337,327)(155,380,338,328)(156,381,339,329)(157,382,340,330)(158,383,341,331)(159,384,342,332)(160,385,343,333)(161,386,344,334)(162,387,345,335)(163,388,346,336)(164,389,347,309)(165,390,348,310)(166,391,349,311)(167,392,350,312)(168,365,351,313)(225,276,303,430)(226,277,304,431)(227,278,305,432)(228,279,306,433)(229,280,307,434)(230,253,308,435)(231,254,281,436)(232,255,282,437)(233,256,283,438)(234,257,284,439)(235,258,285,440)(236,259,286,441)(237,260,287,442)(238,261,288,443)(239,262,289,444)(240,263,290,445)(241,264,291,446)(242,265,292,447)(243,266,293,448)(244,267,294,421)(245,268,295,422)(246,269,296,423)(247,270,297,424)(248,271,298,425)(249,272,299,426)(250,273,300,427)(251,274,301,428)(252,275,302,429) );
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)], [(1,382,136,288),(2,381,137,287),(3,380,138,286),(4,379,139,285),(5,378,140,284),(6,377,113,283),(7,376,114,282),(8,375,115,281),(9,374,116,308),(10,373,117,307),(11,372,118,306),(12,371,119,305),(13,370,120,304),(14,369,121,303),(15,368,122,302),(16,367,123,301),(17,366,124,300),(18,365,125,299),(19,392,126,298),(20,391,127,297),(21,390,128,296),(22,389,129,295),(23,388,130,294),(24,387,131,293),(25,386,132,292),(26,385,133,291),(27,384,134,290),(28,383,135,289),(29,262,57,158),(30,261,58,157),(31,260,59,156),(32,259,60,155),(33,258,61,154),(34,257,62,153),(35,256,63,152),(36,255,64,151),(37,254,65,150),(38,253,66,149),(39,280,67,148),(40,279,68,147),(41,278,69,146),(42,277,70,145),(43,276,71,144),(44,275,72,143),(45,274,73,142),(46,273,74,141),(47,272,75,168),(48,271,76,167),(49,270,77,166),(50,269,78,165),(51,268,79,164),(52,267,80,163),(53,266,81,162),(54,265,82,161),(55,264,83,160),(56,263,84,159),(85,333,196,241),(86,332,169,240),(87,331,170,239),(88,330,171,238),(89,329,172,237),(90,328,173,236),(91,327,174,235),(92,326,175,234),(93,325,176,233),(94,324,177,232),(95,323,178,231),(96,322,179,230),(97,321,180,229),(98,320,181,228),(99,319,182,227),(100,318,183,226),(101,317,184,225),(102,316,185,252),(103,315,186,251),(104,314,187,250),(105,313,188,249),(106,312,189,248),(107,311,190,247),(108,310,191,246),(109,309,192,245),(110,336,193,244),(111,335,194,243),(112,334,195,242),(197,360,393,435),(198,359,394,434),(199,358,395,433),(200,357,396,432),(201,356,397,431),(202,355,398,430),(203,354,399,429),(204,353,400,428),(205,352,401,427),(206,351,402,426),(207,350,403,425),(208,349,404,424),(209,348,405,423),(210,347,406,422),(211,346,407,421),(212,345,408,448),(213,344,409,447),(214,343,410,446),(215,342,411,445),(216,341,412,444),(217,340,413,443),(218,339,414,442),(219,338,415,441),(220,337,416,440),(221,364,417,439),(222,363,418,438),(223,362,419,437),(224,361,420,436)], [(1,413,88,30),(2,414,89,31),(3,415,90,32),(4,416,91,33),(5,417,92,34),(6,418,93,35),(7,419,94,36),(8,420,95,37),(9,393,96,38),(10,394,97,39),(11,395,98,40),(12,396,99,41),(13,397,100,42),(14,398,101,43),(15,399,102,44),(16,400,103,45),(17,401,104,46),(18,402,105,47),(19,403,106,48),(20,404,107,49),(21,405,108,50),(22,406,109,51),(23,407,110,52),(24,408,111,53),(25,409,112,54),(26,410,85,55),(27,411,86,56),(28,412,87,29),(57,135,216,170),(58,136,217,171),(59,137,218,172),(60,138,219,173),(61,139,220,174),(62,140,221,175),(63,113,222,176),(64,114,223,177),(65,115,224,178),(66,116,197,179),(67,117,198,180),(68,118,199,181),(69,119,200,182),(70,120,201,183),(71,121,202,184),(72,122,203,185),(73,123,204,186),(74,124,205,187),(75,125,206,188),(76,126,207,189),(77,127,208,190),(78,128,209,191),(79,129,210,192),(80,130,211,193),(81,131,212,194),(82,132,213,195),(83,133,214,196),(84,134,215,169),(141,366,352,314),(142,367,353,315),(143,368,354,316),(144,369,355,317),(145,370,356,318),(146,371,357,319),(147,372,358,320),(148,373,359,321),(149,374,360,322),(150,375,361,323),(151,376,362,324),(152,377,363,325),(153,378,364,326),(154,379,337,327),(155,380,338,328),(156,381,339,329),(157,382,340,330),(158,383,341,331),(159,384,342,332),(160,385,343,333),(161,386,344,334),(162,387,345,335),(163,388,346,336),(164,389,347,309),(165,390,348,310),(166,391,349,311),(167,392,350,312),(168,365,351,313),(225,276,303,430),(226,277,304,431),(227,278,305,432),(228,279,306,433),(229,280,307,434),(230,253,308,435),(231,254,281,436),(232,255,282,437),(233,256,283,438),(234,257,284,439),(235,258,285,440),(236,259,286,441),(237,260,287,442),(238,261,288,443),(239,262,289,444),(240,263,290,445),(241,264,291,446),(242,265,292,447),(243,266,293,448),(244,267,294,421),(245,268,295,422),(246,269,296,423),(247,270,297,424),(248,271,298,425),(249,272,299,426),(250,273,300,427),(251,274,301,428),(252,275,302,429)]])
124 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D7 | C4○D4 | D14 | Dic14 | C4×D7 | D28 | C7⋊D4 | C4○D28 |
| kernel | C28⋊4(C4⋊C4) | C14.C42 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C2×C4×C28 | C4⋊Dic7 | C2×C28 | C2×C28 | C2×C42 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 4 | 4 | 3 | 4 | 9 | 24 | 12 | 12 | 12 | 24 |
Matrix representation of C28⋊4(C4⋊C4) ►in GL5(𝔽29)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 11 | 0 | 0 |
| 0 | 18 | 25 | 0 | 0 |
| 0 | 0 | 0 | 28 | 5 |
| 0 | 0 | 0 | 19 | 20 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 10 | 10 | 0 | 0 |
| 0 | 16 | 19 | 0 | 0 |
| 0 | 0 | 0 | 5 | 24 |
| 0 | 0 | 0 | 11 | 24 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 13 | 5 | 0 | 0 |
| 0 | 24 | 16 | 0 | 0 |
| 0 | 0 | 0 | 22 | 13 |
| 0 | 0 | 0 | 3 | 7 |
G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,1,18,0,0,0,11,25,0,0,0,0,0,28,19,0,0,0,5,20],[28,0,0,0,0,0,10,16,0,0,0,10,19,0,0,0,0,0,5,11,0,0,0,24,24],[17,0,0,0,0,0,13,24,0,0,0,5,16,0,0,0,0,0,22,3,0,0,0,13,7] >;
C28⋊4(C4⋊C4) in GAP, Magma, Sage, TeX
C_{28}\rtimes_4(C_4\rtimes C_4) % in TeX
G:=Group("C28:4(C4:C4)"); // GroupNames label
G:=SmallGroup(448,462);
// by ID
G=gap.SmallGroup(448,462);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,758,58,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=c^4=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations