direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4⋊C4×C28, C28⋊4C42, C42⋊7C28, C4⋊(C4×C28), (C4×C28)⋊17C4, C2.2(D4×C28), C2.1(Q8×C28), (C2×C28).82Q8, C14.29(C4×Q8), C14.104(C4×D4), (C2×C28).534D4, (C2×C42).3C14, C22.8(Q8×C14), C14.32(C2×C42), C22.28(D4×C14), C23.51(C22×C14), C22.15(C22×C28), C14.52(C42⋊C2), C2.C42.12C14, (C22×C28).489C22, (C22×C14).442C23, C2.4(C2×C4×C28), (C2×C4×C28).5C2, C2.2(C14×C4⋊C4), C14.58(C2×C4⋊C4), (C2×C4⋊C4).21C14, (C14×C4⋊C4).50C2, (C2×C4).24(C7×Q8), (C2×C4).43(C2×C28), (C2×C4).144(C7×D4), (C2×C28).211(C2×C4), (C2×C14).595(C2×D4), C2.3(C7×C42⋊C2), (C2×C14).100(C2×Q8), C22.14(C7×C4○D4), (C22×C4).83(C2×C14), (C2×C14).204(C4○D4), (C2×C14).214(C22×C4), (C7×C2.C42).30C2, SmallGroup(448,786)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4×C28
G = < a,b,c | a28=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 242 in 194 conjugacy classes, 146 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C42, C4⋊C4, C22×C4, C22×C4, C28, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C28, C2×C28, C22×C14, C4×C4⋊C4, C4×C28, C4×C28, C7×C4⋊C4, C22×C28, C22×C28, C7×C2.C42, C2×C4×C28, C2×C4×C28, C14×C4⋊C4, C4⋊C4×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C28, C2×C14, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×C28, C7×D4, C7×Q8, C22×C14, C4×C4⋊C4, C4×C28, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C2×C4×C28, C14×C4⋊C4, C7×C42⋊C2, D4×C28, Q8×C28, C4⋊C4×C28
►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)
(1 213 114 176)(2 214 115 177)(3 215 116 178)(4 216 117 179)(5 217 118 180)(6 218 119 181)(7 219 120 182)(8 220 121 183)(9 221 122 184)(10 222 123 185)(11 223 124 186)(12 224 125 187)(13 197 126 188)(14 198 127 189)(15 199 128 190)(16 200 129 191)(17 201 130 192)(18 202 131 193)(19 203 132 194)(20 204 133 195)(21 205 134 196)(22 206 135 169)(23 207 136 170)(24 208 137 171)(25 209 138 172)(26 210 139 173)(27 211 140 174)(28 212 113 175)(29 228 303 371)(30 229 304 372)(31 230 305 373)(32 231 306 374)(33 232 307 375)(34 233 308 376)(35 234 281 377)(36 235 282 378)(37 236 283 379)(38 237 284 380)(39 238 285 381)(40 239 286 382)(41 240 287 383)(42 241 288 384)(43 242 289 385)(44 243 290 386)(45 244 291 387)(46 245 292 388)(47 246 293 389)(48 247 294 390)(49 248 295 391)(50 249 296 392)(51 250 297 365)(52 251 298 366)(53 252 299 367)(54 225 300 368)(55 226 301 369)(56 227 302 370)(57 430 311 409)(58 431 312 410)(59 432 313 411)(60 433 314 412)(61 434 315 413)(62 435 316 414)(63 436 317 415)(64 437 318 416)(65 438 319 417)(66 439 320 418)(67 440 321 419)(68 441 322 420)(69 442 323 393)(70 443 324 394)(71 444 325 395)(72 445 326 396)(73 446 327 397)(74 447 328 398)(75 448 329 399)(76 421 330 400)(77 422 331 401)(78 423 332 402)(79 424 333 403)(80 425 334 404)(81 426 335 405)(82 427 336 406)(83 428 309 407)(84 429 310 408)(85 148 342 275)(86 149 343 276)(87 150 344 277)(88 151 345 278)(89 152 346 279)(90 153 347 280)(91 154 348 253)(92 155 349 254)(93 156 350 255)(94 157 351 256)(95 158 352 257)(96 159 353 258)(97 160 354 259)(98 161 355 260)(99 162 356 261)(100 163 357 262)(101 164 358 263)(102 165 359 264)(103 166 360 265)(104 167 361 266)(105 168 362 267)(106 141 363 268)(107 142 364 269)(108 143 337 270)(109 144 338 271)(110 145 339 272)(111 146 340 273)(112 147 341 274)
(1 51 341 397)(2 52 342 398)(3 53 343 399)(4 54 344 400)(5 55 345 401)(6 56 346 402)(7 29 347 403)(8 30 348 404)(9 31 349 405)(10 32 350 406)(11 33 351 407)(12 34 352 408)(13 35 353 409)(14 36 354 410)(15 37 355 411)(16 38 356 412)(17 39 357 413)(18 40 358 414)(19 41 359 415)(20 42 360 416)(21 43 361 417)(22 44 362 418)(23 45 363 419)(24 46 364 420)(25 47 337 393)(26 48 338 394)(27 49 339 395)(28 50 340 396)(57 188 234 159)(58 189 235 160)(59 190 236 161)(60 191 237 162)(61 192 238 163)(62 193 239 164)(63 194 240 165)(64 195 241 166)(65 196 242 167)(66 169 243 168)(67 170 244 141)(68 171 245 142)(69 172 246 143)(70 173 247 144)(71 174 248 145)(72 175 249 146)(73 176 250 147)(74 177 251 148)(75 178 252 149)(76 179 225 150)(77 180 226 151)(78 181 227 152)(79 182 228 153)(80 183 229 154)(81 184 230 155)(82 185 231 156)(83 186 232 157)(84 187 233 158)(85 447 115 298)(86 448 116 299)(87 421 117 300)(88 422 118 301)(89 423 119 302)(90 424 120 303)(91 425 121 304)(92 426 122 305)(93 427 123 306)(94 428 124 307)(95 429 125 308)(96 430 126 281)(97 431 127 282)(98 432 128 283)(99 433 129 284)(100 434 130 285)(101 435 131 286)(102 436 132 287)(103 437 133 288)(104 438 134 289)(105 439 135 290)(106 440 136 291)(107 441 137 292)(108 442 138 293)(109 443 139 294)(110 444 140 295)(111 445 113 296)(112 446 114 297)(197 377 258 311)(198 378 259 312)(199 379 260 313)(200 380 261 314)(201 381 262 315)(202 382 263 316)(203 383 264 317)(204 384 265 318)(205 385 266 319)(206 386 267 320)(207 387 268 321)(208 388 269 322)(209 389 270 323)(210 390 271 324)(211 391 272 325)(212 392 273 326)(213 365 274 327)(214 366 275 328)(215 367 276 329)(216 368 277 330)(217 369 278 331)(218 370 279 332)(219 371 280 333)(220 372 253 334)(221 373 254 335)(222 374 255 336)(223 375 256 309)(224 376 257 310)
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), (1,213,114,176)(2,214,115,177)(3,215,116,178)(4,216,117,179)(5,217,118,180)(6,218,119,181)(7,219,120,182)(8,220,121,183)(9,221,122,184)(10,222,123,185)(11,223,124,186)(12,224,125,187)(13,197,126,188)(14,198,127,189)(15,199,128,190)(16,200,129,191)(17,201,130,192)(18,202,131,193)(19,203,132,194)(20,204,133,195)(21,205,134,196)(22,206,135,169)(23,207,136,170)(24,208,137,171)(25,209,138,172)(26,210,139,173)(27,211,140,174)(28,212,113,175)(29,228,303,371)(30,229,304,372)(31,230,305,373)(32,231,306,374)(33,232,307,375)(34,233,308,376)(35,234,281,377)(36,235,282,378)(37,236,283,379)(38,237,284,380)(39,238,285,381)(40,239,286,382)(41,240,287,383)(42,241,288,384)(43,242,289,385)(44,243,290,386)(45,244,291,387)(46,245,292,388)(47,246,293,389)(48,247,294,390)(49,248,295,391)(50,249,296,392)(51,250,297,365)(52,251,298,366)(53,252,299,367)(54,225,300,368)(55,226,301,369)(56,227,302,370)(57,430,311,409)(58,431,312,410)(59,432,313,411)(60,433,314,412)(61,434,315,413)(62,435,316,414)(63,436,317,415)(64,437,318,416)(65,438,319,417)(66,439,320,418)(67,440,321,419)(68,441,322,420)(69,442,323,393)(70,443,324,394)(71,444,325,395)(72,445,326,396)(73,446,327,397)(74,447,328,398)(75,448,329,399)(76,421,330,400)(77,422,331,401)(78,423,332,402)(79,424,333,403)(80,425,334,404)(81,426,335,405)(82,427,336,406)(83,428,309,407)(84,429,310,408)(85,148,342,275)(86,149,343,276)(87,150,344,277)(88,151,345,278)(89,152,346,279)(90,153,347,280)(91,154,348,253)(92,155,349,254)(93,156,350,255)(94,157,351,256)(95,158,352,257)(96,159,353,258)(97,160,354,259)(98,161,355,260)(99,162,356,261)(100,163,357,262)(101,164,358,263)(102,165,359,264)(103,166,360,265)(104,167,361,266)(105,168,362,267)(106,141,363,268)(107,142,364,269)(108,143,337,270)(109,144,338,271)(110,145,339,272)(111,146,340,273)(112,147,341,274), (1,51,341,397)(2,52,342,398)(3,53,343,399)(4,54,344,400)(5,55,345,401)(6,56,346,402)(7,29,347,403)(8,30,348,404)(9,31,349,405)(10,32,350,406)(11,33,351,407)(12,34,352,408)(13,35,353,409)(14,36,354,410)(15,37,355,411)(16,38,356,412)(17,39,357,413)(18,40,358,414)(19,41,359,415)(20,42,360,416)(21,43,361,417)(22,44,362,418)(23,45,363,419)(24,46,364,420)(25,47,337,393)(26,48,338,394)(27,49,339,395)(28,50,340,396)(57,188,234,159)(58,189,235,160)(59,190,236,161)(60,191,237,162)(61,192,238,163)(62,193,239,164)(63,194,240,165)(64,195,241,166)(65,196,242,167)(66,169,243,168)(67,170,244,141)(68,171,245,142)(69,172,246,143)(70,173,247,144)(71,174,248,145)(72,175,249,146)(73,176,250,147)(74,177,251,148)(75,178,252,149)(76,179,225,150)(77,180,226,151)(78,181,227,152)(79,182,228,153)(80,183,229,154)(81,184,230,155)(82,185,231,156)(83,186,232,157)(84,187,233,158)(85,447,115,298)(86,448,116,299)(87,421,117,300)(88,422,118,301)(89,423,119,302)(90,424,120,303)(91,425,121,304)(92,426,122,305)(93,427,123,306)(94,428,124,307)(95,429,125,308)(96,430,126,281)(97,431,127,282)(98,432,128,283)(99,433,129,284)(100,434,130,285)(101,435,131,286)(102,436,132,287)(103,437,133,288)(104,438,134,289)(105,439,135,290)(106,440,136,291)(107,441,137,292)(108,442,138,293)(109,443,139,294)(110,444,140,295)(111,445,113,296)(112,446,114,297)(197,377,258,311)(198,378,259,312)(199,379,260,313)(200,380,261,314)(201,381,262,315)(202,382,263,316)(203,383,264,317)(204,384,265,318)(205,385,266,319)(206,386,267,320)(207,387,268,321)(208,388,269,322)(209,389,270,323)(210,390,271,324)(211,391,272,325)(212,392,273,326)(213,365,274,327)(214,366,275,328)(215,367,276,329)(216,368,277,330)(217,369,278,331)(218,370,279,332)(219,371,280,333)(220,372,253,334)(221,373,254,335)(222,374,255,336)(223,375,256,309)(224,376,257,310)>;
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), (1,213,114,176)(2,214,115,177)(3,215,116,178)(4,216,117,179)(5,217,118,180)(6,218,119,181)(7,219,120,182)(8,220,121,183)(9,221,122,184)(10,222,123,185)(11,223,124,186)(12,224,125,187)(13,197,126,188)(14,198,127,189)(15,199,128,190)(16,200,129,191)(17,201,130,192)(18,202,131,193)(19,203,132,194)(20,204,133,195)(21,205,134,196)(22,206,135,169)(23,207,136,170)(24,208,137,171)(25,209,138,172)(26,210,139,173)(27,211,140,174)(28,212,113,175)(29,228,303,371)(30,229,304,372)(31,230,305,373)(32,231,306,374)(33,232,307,375)(34,233,308,376)(35,234,281,377)(36,235,282,378)(37,236,283,379)(38,237,284,380)(39,238,285,381)(40,239,286,382)(41,240,287,383)(42,241,288,384)(43,242,289,385)(44,243,290,386)(45,244,291,387)(46,245,292,388)(47,246,293,389)(48,247,294,390)(49,248,295,391)(50,249,296,392)(51,250,297,365)(52,251,298,366)(53,252,299,367)(54,225,300,368)(55,226,301,369)(56,227,302,370)(57,430,311,409)(58,431,312,410)(59,432,313,411)(60,433,314,412)(61,434,315,413)(62,435,316,414)(63,436,317,415)(64,437,318,416)(65,438,319,417)(66,439,320,418)(67,440,321,419)(68,441,322,420)(69,442,323,393)(70,443,324,394)(71,444,325,395)(72,445,326,396)(73,446,327,397)(74,447,328,398)(75,448,329,399)(76,421,330,400)(77,422,331,401)(78,423,332,402)(79,424,333,403)(80,425,334,404)(81,426,335,405)(82,427,336,406)(83,428,309,407)(84,429,310,408)(85,148,342,275)(86,149,343,276)(87,150,344,277)(88,151,345,278)(89,152,346,279)(90,153,347,280)(91,154,348,253)(92,155,349,254)(93,156,350,255)(94,157,351,256)(95,158,352,257)(96,159,353,258)(97,160,354,259)(98,161,355,260)(99,162,356,261)(100,163,357,262)(101,164,358,263)(102,165,359,264)(103,166,360,265)(104,167,361,266)(105,168,362,267)(106,141,363,268)(107,142,364,269)(108,143,337,270)(109,144,338,271)(110,145,339,272)(111,146,340,273)(112,147,341,274), (1,51,341,397)(2,52,342,398)(3,53,343,399)(4,54,344,400)(5,55,345,401)(6,56,346,402)(7,29,347,403)(8,30,348,404)(9,31,349,405)(10,32,350,406)(11,33,351,407)(12,34,352,408)(13,35,353,409)(14,36,354,410)(15,37,355,411)(16,38,356,412)(17,39,357,413)(18,40,358,414)(19,41,359,415)(20,42,360,416)(21,43,361,417)(22,44,362,418)(23,45,363,419)(24,46,364,420)(25,47,337,393)(26,48,338,394)(27,49,339,395)(28,50,340,396)(57,188,234,159)(58,189,235,160)(59,190,236,161)(60,191,237,162)(61,192,238,163)(62,193,239,164)(63,194,240,165)(64,195,241,166)(65,196,242,167)(66,169,243,168)(67,170,244,141)(68,171,245,142)(69,172,246,143)(70,173,247,144)(71,174,248,145)(72,175,249,146)(73,176,250,147)(74,177,251,148)(75,178,252,149)(76,179,225,150)(77,180,226,151)(78,181,227,152)(79,182,228,153)(80,183,229,154)(81,184,230,155)(82,185,231,156)(83,186,232,157)(84,187,233,158)(85,447,115,298)(86,448,116,299)(87,421,117,300)(88,422,118,301)(89,423,119,302)(90,424,120,303)(91,425,121,304)(92,426,122,305)(93,427,123,306)(94,428,124,307)(95,429,125,308)(96,430,126,281)(97,431,127,282)(98,432,128,283)(99,433,129,284)(100,434,130,285)(101,435,131,286)(102,436,132,287)(103,437,133,288)(104,438,134,289)(105,439,135,290)(106,440,136,291)(107,441,137,292)(108,442,138,293)(109,443,139,294)(110,444,140,295)(111,445,113,296)(112,446,114,297)(197,377,258,311)(198,378,259,312)(199,379,260,313)(200,380,261,314)(201,381,262,315)(202,382,263,316)(203,383,264,317)(204,384,265,318)(205,385,266,319)(206,386,267,320)(207,387,268,321)(208,388,269,322)(209,389,270,323)(210,390,271,324)(211,391,272,325)(212,392,273,326)(213,365,274,327)(214,366,275,328)(215,367,276,329)(216,368,277,330)(217,369,278,331)(218,370,279,332)(219,371,280,333)(220,372,253,334)(221,373,254,335)(222,374,255,336)(223,375,256,309)(224,376,257,310) );
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)], [(1,213,114,176),(2,214,115,177),(3,215,116,178),(4,216,117,179),(5,217,118,180),(6,218,119,181),(7,219,120,182),(8,220,121,183),(9,221,122,184),(10,222,123,185),(11,223,124,186),(12,224,125,187),(13,197,126,188),(14,198,127,189),(15,199,128,190),(16,200,129,191),(17,201,130,192),(18,202,131,193),(19,203,132,194),(20,204,133,195),(21,205,134,196),(22,206,135,169),(23,207,136,170),(24,208,137,171),(25,209,138,172),(26,210,139,173),(27,211,140,174),(28,212,113,175),(29,228,303,371),(30,229,304,372),(31,230,305,373),(32,231,306,374),(33,232,307,375),(34,233,308,376),(35,234,281,377),(36,235,282,378),(37,236,283,379),(38,237,284,380),(39,238,285,381),(40,239,286,382),(41,240,287,383),(42,241,288,384),(43,242,289,385),(44,243,290,386),(45,244,291,387),(46,245,292,388),(47,246,293,389),(48,247,294,390),(49,248,295,391),(50,249,296,392),(51,250,297,365),(52,251,298,366),(53,252,299,367),(54,225,300,368),(55,226,301,369),(56,227,302,370),(57,430,311,409),(58,431,312,410),(59,432,313,411),(60,433,314,412),(61,434,315,413),(62,435,316,414),(63,436,317,415),(64,437,318,416),(65,438,319,417),(66,439,320,418),(67,440,321,419),(68,441,322,420),(69,442,323,393),(70,443,324,394),(71,444,325,395),(72,445,326,396),(73,446,327,397),(74,447,328,398),(75,448,329,399),(76,421,330,400),(77,422,331,401),(78,423,332,402),(79,424,333,403),(80,425,334,404),(81,426,335,405),(82,427,336,406),(83,428,309,407),(84,429,310,408),(85,148,342,275),(86,149,343,276),(87,150,344,277),(88,151,345,278),(89,152,346,279),(90,153,347,280),(91,154,348,253),(92,155,349,254),(93,156,350,255),(94,157,351,256),(95,158,352,257),(96,159,353,258),(97,160,354,259),(98,161,355,260),(99,162,356,261),(100,163,357,262),(101,164,358,263),(102,165,359,264),(103,166,360,265),(104,167,361,266),(105,168,362,267),(106,141,363,268),(107,142,364,269),(108,143,337,270),(109,144,338,271),(110,145,339,272),(111,146,340,273),(112,147,341,274)], [(1,51,341,397),(2,52,342,398),(3,53,343,399),(4,54,344,400),(5,55,345,401),(6,56,346,402),(7,29,347,403),(8,30,348,404),(9,31,349,405),(10,32,350,406),(11,33,351,407),(12,34,352,408),(13,35,353,409),(14,36,354,410),(15,37,355,411),(16,38,356,412),(17,39,357,413),(18,40,358,414),(19,41,359,415),(20,42,360,416),(21,43,361,417),(22,44,362,418),(23,45,363,419),(24,46,364,420),(25,47,337,393),(26,48,338,394),(27,49,339,395),(28,50,340,396),(57,188,234,159),(58,189,235,160),(59,190,236,161),(60,191,237,162),(61,192,238,163),(62,193,239,164),(63,194,240,165),(64,195,241,166),(65,196,242,167),(66,169,243,168),(67,170,244,141),(68,171,245,142),(69,172,246,143),(70,173,247,144),(71,174,248,145),(72,175,249,146),(73,176,250,147),(74,177,251,148),(75,178,252,149),(76,179,225,150),(77,180,226,151),(78,181,227,152),(79,182,228,153),(80,183,229,154),(81,184,230,155),(82,185,231,156),(83,186,232,157),(84,187,233,158),(85,447,115,298),(86,448,116,299),(87,421,117,300),(88,422,118,301),(89,423,119,302),(90,424,120,303),(91,425,121,304),(92,426,122,305),(93,427,123,306),(94,428,124,307),(95,429,125,308),(96,430,126,281),(97,431,127,282),(98,432,128,283),(99,433,129,284),(100,434,130,285),(101,435,131,286),(102,436,132,287),(103,437,133,288),(104,438,134,289),(105,439,135,290),(106,440,136,291),(107,441,137,292),(108,442,138,293),(109,443,139,294),(110,444,140,295),(111,445,113,296),(112,446,114,297),(197,377,258,311),(198,378,259,312),(199,379,260,313),(200,380,261,314),(201,381,262,315),(202,382,263,316),(203,383,264,317),(204,384,265,318),(205,385,266,319),(206,386,267,320),(207,387,268,321),(208,388,269,322),(209,389,270,323),(210,390,271,324),(211,391,272,325),(212,392,273,326),(213,365,274,327),(214,366,275,328),(215,367,276,329),(216,368,277,330),(217,369,278,331),(218,370,279,332),(219,371,280,333),(220,372,253,334),(221,373,254,335),(222,374,255,336),(223,375,256,309),(224,376,257,310)]])
280 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF | 7A | ··· | 7F | 14A | ··· | 14AP | 28A | ··· | 28AV | 28AW | ··· | 28GJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C14 | C28 | C28 | D4 | Q8 | C4○D4 | C7×D4 | C7×Q8 | C7×C4○D4 |
| kernel | C4⋊C4×C28 | C7×C2.C42 | C2×C4×C28 | C14×C4⋊C4 | C4×C28 | C7×C4⋊C4 | C4×C4⋊C4 | C2.C42 | C2×C42 | C2×C4⋊C4 | C42 | C4⋊C4 | C2×C28 | C2×C28 | C2×C14 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 3 | 2 | 8 | 16 | 6 | 12 | 18 | 12 | 48 | 96 | 2 | 2 | 4 | 12 | 12 | 24 |
Matrix representation of C4⋊C4×C28 ►in GL4(𝔽29) generated by
| 17 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 17 |
| 17 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 |
G:=sub<GL(4,GF(29))| [17,0,0,0,0,1,0,0,0,0,5,0,0,0,0,5],[28,0,0,0,0,28,0,0,0,0,12,0,0,0,0,17],[17,0,0,0,0,17,0,0,0,0,0,28,0,0,1,0] >;
C4⋊C4×C28 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times C_{28} % in TeX
G:=Group("C4:C4xC28"); // GroupNames label
G:=SmallGroup(448,786);
// by ID
G=gap.SmallGroup(448,786);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,1576,898]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations