metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.9M4(2), C42.198D14, C7⋊C8⋊10Q8, C4⋊C8.12D7, C7⋊4(C8⋊4Q8), C4.53(Q8×D7), C14.18(C4×Q8), C56⋊C4.9C2, (C2×C8).180D14, C4.3(C8⋊D7), Dic7⋊C8.9C2, C4⋊Dic7.17C4, C28.111(C2×Q8), C14.27(C8○D4), Dic7⋊C4.12C4, (C4×C28).57C22, C14.7(C2×M4(2)), C28.303(C4○D4), C2.12(D28.C4), (C2×C28).828C23, (C2×C56).207C22, (C4×Dic14).10C2, (C2×Dic14).13C4, C4.129(D4⋊2D7), C2.8(Dic7⋊3Q8), (C4×Dic7).184C22, (C4×C7⋊C8).6C2, (C7×C4⋊C8).18C2, (C2×C4).71(C4×D7), C2.11(C2×C8⋊D7), C22.109(C2×C4×D7), (C2×C28).157(C2×C4), (C2×C7⋊C8).303C22, (C2×C14).83(C22×C4), (C2×Dic7).21(C2×C4), (C2×C4).770(C22×D7), SmallGroup(448,365)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.M4(2)
G = < a,b,c | a28=b8=1, c2=a14, bab-1=a13, cac-1=a-1, cbc-1=b5 >
Subgroups: 324 in 94 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic7, C28, C28, C28, C2×C14, C4×C8, C8⋊C4, C4⋊C8, C4⋊C8, C4×Q8, C7⋊C8, C7⋊C8, C56, Dic14, C2×Dic7, C2×C28, C8⋊4Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C4×C7⋊C8, Dic7⋊C8, C56⋊C4, C7×C4⋊C8, C4×Dic14, C28.M4(2)
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D7, M4(2), C22×C4, C2×Q8, C4○D4, D14, C4×Q8, C2×M4(2), C8○D4, C4×D7, C22×D7, C8⋊4Q8, C8⋊D7, C2×C4×D7, D4⋊2D7, Q8×D7, Dic7⋊3Q8, C2×C8⋊D7, D28.C4, C28.M4(2)
►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)
(1 271 251 358 171 41 326 294)(2 256 252 343 172 54 327 307)(3 269 225 356 173 39 328 292)(4 254 226 341 174 52 329 305)(5 267 227 354 175 37 330 290)(6 280 228 339 176 50 331 303)(7 265 229 352 177 35 332 288)(8 278 230 337 178 48 333 301)(9 263 231 350 179 33 334 286)(10 276 232 363 180 46 335 299)(11 261 233 348 181 31 336 284)(12 274 234 361 182 44 309 297)(13 259 235 346 183 29 310 282)(14 272 236 359 184 42 311 295)(15 257 237 344 185 55 312 308)(16 270 238 357 186 40 313 293)(17 255 239 342 187 53 314 306)(18 268 240 355 188 38 315 291)(19 253 241 340 189 51 316 304)(20 266 242 353 190 36 317 289)(21 279 243 338 191 49 318 302)(22 264 244 351 192 34 319 287)(23 277 245 364 193 47 320 300)(24 262 246 349 194 32 321 285)(25 275 247 362 195 45 322 298)(26 260 248 347 196 30 323 283)(27 273 249 360 169 43 324 296)(28 258 250 345 170 56 325 281)(57 446 85 123 406 160 390 210)(58 431 86 136 407 145 391 223)(59 444 87 121 408 158 392 208)(60 429 88 134 409 143 365 221)(61 442 89 119 410 156 366 206)(62 427 90 132 411 141 367 219)(63 440 91 117 412 154 368 204)(64 425 92 130 413 167 369 217)(65 438 93 115 414 152 370 202)(66 423 94 128 415 165 371 215)(67 436 95 113 416 150 372 200)(68 421 96 126 417 163 373 213)(69 434 97 139 418 148 374 198)(70 447 98 124 419 161 375 211)(71 432 99 137 420 146 376 224)(72 445 100 122 393 159 377 209)(73 430 101 135 394 144 378 222)(74 443 102 120 395 157 379 207)(75 428 103 133 396 142 380 220)(76 441 104 118 397 155 381 205)(77 426 105 131 398 168 382 218)(78 439 106 116 399 153 383 203)(79 424 107 129 400 166 384 216)(80 437 108 114 401 151 385 201)(81 422 109 127 402 164 386 214)(82 435 110 140 403 149 387 199)(83 448 111 125 404 162 388 212)(84 433 112 138 405 147 389 197)
(1 413 15 399)(2 412 16 398)(3 411 17 397)(4 410 18 396)(5 409 19 395)(6 408 20 394)(7 407 21 393)(8 406 22 420)(9 405 23 419)(10 404 24 418)(11 403 25 417)(12 402 26 416)(13 401 27 415)(14 400 28 414)(29 151 43 165)(30 150 44 164)(31 149 45 163)(32 148 46 162)(33 147 47 161)(34 146 48 160)(35 145 49 159)(36 144 50 158)(37 143 51 157)(38 142 52 156)(39 141 53 155)(40 168 54 154)(41 167 55 153)(42 166 56 152)(57 192 71 178)(58 191 72 177)(59 190 73 176)(60 189 74 175)(61 188 75 174)(62 187 76 173)(63 186 77 172)(64 185 78 171)(65 184 79 170)(66 183 80 169)(67 182 81 196)(68 181 82 195)(69 180 83 194)(70 179 84 193)(85 319 99 333)(86 318 100 332)(87 317 101 331)(88 316 102 330)(89 315 103 329)(90 314 104 328)(91 313 105 327)(92 312 106 326)(93 311 107 325)(94 310 108 324)(95 309 109 323)(96 336 110 322)(97 335 111 321)(98 334 112 320)(113 361 127 347)(114 360 128 346)(115 359 129 345)(116 358 130 344)(117 357 131 343)(118 356 132 342)(119 355 133 341)(120 354 134 340)(121 353 135 339)(122 352 136 338)(123 351 137 337)(124 350 138 364)(125 349 139 363)(126 348 140 362)(197 300 211 286)(198 299 212 285)(199 298 213 284)(200 297 214 283)(201 296 215 282)(202 295 216 281)(203 294 217 308)(204 293 218 307)(205 292 219 306)(206 291 220 305)(207 290 221 304)(208 289 222 303)(209 288 223 302)(210 287 224 301)(225 367 239 381)(226 366 240 380)(227 365 241 379)(228 392 242 378)(229 391 243 377)(230 390 244 376)(231 389 245 375)(232 388 246 374)(233 387 247 373)(234 386 248 372)(235 385 249 371)(236 384 250 370)(237 383 251 369)(238 382 252 368)(253 443 267 429)(254 442 268 428)(255 441 269 427)(256 440 270 426)(257 439 271 425)(258 438 272 424)(259 437 273 423)(260 436 274 422)(261 435 275 421)(262 434 276 448)(263 433 277 447)(264 432 278 446)(265 431 279 445)(266 430 280 444)
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), (1,271,251,358,171,41,326,294)(2,256,252,343,172,54,327,307)(3,269,225,356,173,39,328,292)(4,254,226,341,174,52,329,305)(5,267,227,354,175,37,330,290)(6,280,228,339,176,50,331,303)(7,265,229,352,177,35,332,288)(8,278,230,337,178,48,333,301)(9,263,231,350,179,33,334,286)(10,276,232,363,180,46,335,299)(11,261,233,348,181,31,336,284)(12,274,234,361,182,44,309,297)(13,259,235,346,183,29,310,282)(14,272,236,359,184,42,311,295)(15,257,237,344,185,55,312,308)(16,270,238,357,186,40,313,293)(17,255,239,342,187,53,314,306)(18,268,240,355,188,38,315,291)(19,253,241,340,189,51,316,304)(20,266,242,353,190,36,317,289)(21,279,243,338,191,49,318,302)(22,264,244,351,192,34,319,287)(23,277,245,364,193,47,320,300)(24,262,246,349,194,32,321,285)(25,275,247,362,195,45,322,298)(26,260,248,347,196,30,323,283)(27,273,249,360,169,43,324,296)(28,258,250,345,170,56,325,281)(57,446,85,123,406,160,390,210)(58,431,86,136,407,145,391,223)(59,444,87,121,408,158,392,208)(60,429,88,134,409,143,365,221)(61,442,89,119,410,156,366,206)(62,427,90,132,411,141,367,219)(63,440,91,117,412,154,368,204)(64,425,92,130,413,167,369,217)(65,438,93,115,414,152,370,202)(66,423,94,128,415,165,371,215)(67,436,95,113,416,150,372,200)(68,421,96,126,417,163,373,213)(69,434,97,139,418,148,374,198)(70,447,98,124,419,161,375,211)(71,432,99,137,420,146,376,224)(72,445,100,122,393,159,377,209)(73,430,101,135,394,144,378,222)(74,443,102,120,395,157,379,207)(75,428,103,133,396,142,380,220)(76,441,104,118,397,155,381,205)(77,426,105,131,398,168,382,218)(78,439,106,116,399,153,383,203)(79,424,107,129,400,166,384,216)(80,437,108,114,401,151,385,201)(81,422,109,127,402,164,386,214)(82,435,110,140,403,149,387,199)(83,448,111,125,404,162,388,212)(84,433,112,138,405,147,389,197), (1,413,15,399)(2,412,16,398)(3,411,17,397)(4,410,18,396)(5,409,19,395)(6,408,20,394)(7,407,21,393)(8,406,22,420)(9,405,23,419)(10,404,24,418)(11,403,25,417)(12,402,26,416)(13,401,27,415)(14,400,28,414)(29,151,43,165)(30,150,44,164)(31,149,45,163)(32,148,46,162)(33,147,47,161)(34,146,48,160)(35,145,49,159)(36,144,50,158)(37,143,51,157)(38,142,52,156)(39,141,53,155)(40,168,54,154)(41,167,55,153)(42,166,56,152)(57,192,71,178)(58,191,72,177)(59,190,73,176)(60,189,74,175)(61,188,75,174)(62,187,76,173)(63,186,77,172)(64,185,78,171)(65,184,79,170)(66,183,80,169)(67,182,81,196)(68,181,82,195)(69,180,83,194)(70,179,84,193)(85,319,99,333)(86,318,100,332)(87,317,101,331)(88,316,102,330)(89,315,103,329)(90,314,104,328)(91,313,105,327)(92,312,106,326)(93,311,107,325)(94,310,108,324)(95,309,109,323)(96,336,110,322)(97,335,111,321)(98,334,112,320)(113,361,127,347)(114,360,128,346)(115,359,129,345)(116,358,130,344)(117,357,131,343)(118,356,132,342)(119,355,133,341)(120,354,134,340)(121,353,135,339)(122,352,136,338)(123,351,137,337)(124,350,138,364)(125,349,139,363)(126,348,140,362)(197,300,211,286)(198,299,212,285)(199,298,213,284)(200,297,214,283)(201,296,215,282)(202,295,216,281)(203,294,217,308)(204,293,218,307)(205,292,219,306)(206,291,220,305)(207,290,221,304)(208,289,222,303)(209,288,223,302)(210,287,224,301)(225,367,239,381)(226,366,240,380)(227,365,241,379)(228,392,242,378)(229,391,243,377)(230,390,244,376)(231,389,245,375)(232,388,246,374)(233,387,247,373)(234,386,248,372)(235,385,249,371)(236,384,250,370)(237,383,251,369)(238,382,252,368)(253,443,267,429)(254,442,268,428)(255,441,269,427)(256,440,270,426)(257,439,271,425)(258,438,272,424)(259,437,273,423)(260,436,274,422)(261,435,275,421)(262,434,276,448)(263,433,277,447)(264,432,278,446)(265,431,279,445)(266,430,280,444)>;
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), (1,271,251,358,171,41,326,294)(2,256,252,343,172,54,327,307)(3,269,225,356,173,39,328,292)(4,254,226,341,174,52,329,305)(5,267,227,354,175,37,330,290)(6,280,228,339,176,50,331,303)(7,265,229,352,177,35,332,288)(8,278,230,337,178,48,333,301)(9,263,231,350,179,33,334,286)(10,276,232,363,180,46,335,299)(11,261,233,348,181,31,336,284)(12,274,234,361,182,44,309,297)(13,259,235,346,183,29,310,282)(14,272,236,359,184,42,311,295)(15,257,237,344,185,55,312,308)(16,270,238,357,186,40,313,293)(17,255,239,342,187,53,314,306)(18,268,240,355,188,38,315,291)(19,253,241,340,189,51,316,304)(20,266,242,353,190,36,317,289)(21,279,243,338,191,49,318,302)(22,264,244,351,192,34,319,287)(23,277,245,364,193,47,320,300)(24,262,246,349,194,32,321,285)(25,275,247,362,195,45,322,298)(26,260,248,347,196,30,323,283)(27,273,249,360,169,43,324,296)(28,258,250,345,170,56,325,281)(57,446,85,123,406,160,390,210)(58,431,86,136,407,145,391,223)(59,444,87,121,408,158,392,208)(60,429,88,134,409,143,365,221)(61,442,89,119,410,156,366,206)(62,427,90,132,411,141,367,219)(63,440,91,117,412,154,368,204)(64,425,92,130,413,167,369,217)(65,438,93,115,414,152,370,202)(66,423,94,128,415,165,371,215)(67,436,95,113,416,150,372,200)(68,421,96,126,417,163,373,213)(69,434,97,139,418,148,374,198)(70,447,98,124,419,161,375,211)(71,432,99,137,420,146,376,224)(72,445,100,122,393,159,377,209)(73,430,101,135,394,144,378,222)(74,443,102,120,395,157,379,207)(75,428,103,133,396,142,380,220)(76,441,104,118,397,155,381,205)(77,426,105,131,398,168,382,218)(78,439,106,116,399,153,383,203)(79,424,107,129,400,166,384,216)(80,437,108,114,401,151,385,201)(81,422,109,127,402,164,386,214)(82,435,110,140,403,149,387,199)(83,448,111,125,404,162,388,212)(84,433,112,138,405,147,389,197), (1,413,15,399)(2,412,16,398)(3,411,17,397)(4,410,18,396)(5,409,19,395)(6,408,20,394)(7,407,21,393)(8,406,22,420)(9,405,23,419)(10,404,24,418)(11,403,25,417)(12,402,26,416)(13,401,27,415)(14,400,28,414)(29,151,43,165)(30,150,44,164)(31,149,45,163)(32,148,46,162)(33,147,47,161)(34,146,48,160)(35,145,49,159)(36,144,50,158)(37,143,51,157)(38,142,52,156)(39,141,53,155)(40,168,54,154)(41,167,55,153)(42,166,56,152)(57,192,71,178)(58,191,72,177)(59,190,73,176)(60,189,74,175)(61,188,75,174)(62,187,76,173)(63,186,77,172)(64,185,78,171)(65,184,79,170)(66,183,80,169)(67,182,81,196)(68,181,82,195)(69,180,83,194)(70,179,84,193)(85,319,99,333)(86,318,100,332)(87,317,101,331)(88,316,102,330)(89,315,103,329)(90,314,104,328)(91,313,105,327)(92,312,106,326)(93,311,107,325)(94,310,108,324)(95,309,109,323)(96,336,110,322)(97,335,111,321)(98,334,112,320)(113,361,127,347)(114,360,128,346)(115,359,129,345)(116,358,130,344)(117,357,131,343)(118,356,132,342)(119,355,133,341)(120,354,134,340)(121,353,135,339)(122,352,136,338)(123,351,137,337)(124,350,138,364)(125,349,139,363)(126,348,140,362)(197,300,211,286)(198,299,212,285)(199,298,213,284)(200,297,214,283)(201,296,215,282)(202,295,216,281)(203,294,217,308)(204,293,218,307)(205,292,219,306)(206,291,220,305)(207,290,221,304)(208,289,222,303)(209,288,223,302)(210,287,224,301)(225,367,239,381)(226,366,240,380)(227,365,241,379)(228,392,242,378)(229,391,243,377)(230,390,244,376)(231,389,245,375)(232,388,246,374)(233,387,247,373)(234,386,248,372)(235,385,249,371)(236,384,250,370)(237,383,251,369)(238,382,252,368)(253,443,267,429)(254,442,268,428)(255,441,269,427)(256,440,270,426)(257,439,271,425)(258,438,272,424)(259,437,273,423)(260,436,274,422)(261,435,275,421)(262,434,276,448)(263,433,277,447)(264,432,278,446)(265,431,279,445)(266,430,280,444) );
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)], [(1,271,251,358,171,41,326,294),(2,256,252,343,172,54,327,307),(3,269,225,356,173,39,328,292),(4,254,226,341,174,52,329,305),(5,267,227,354,175,37,330,290),(6,280,228,339,176,50,331,303),(7,265,229,352,177,35,332,288),(8,278,230,337,178,48,333,301),(9,263,231,350,179,33,334,286),(10,276,232,363,180,46,335,299),(11,261,233,348,181,31,336,284),(12,274,234,361,182,44,309,297),(13,259,235,346,183,29,310,282),(14,272,236,359,184,42,311,295),(15,257,237,344,185,55,312,308),(16,270,238,357,186,40,313,293),(17,255,239,342,187,53,314,306),(18,268,240,355,188,38,315,291),(19,253,241,340,189,51,316,304),(20,266,242,353,190,36,317,289),(21,279,243,338,191,49,318,302),(22,264,244,351,192,34,319,287),(23,277,245,364,193,47,320,300),(24,262,246,349,194,32,321,285),(25,275,247,362,195,45,322,298),(26,260,248,347,196,30,323,283),(27,273,249,360,169,43,324,296),(28,258,250,345,170,56,325,281),(57,446,85,123,406,160,390,210),(58,431,86,136,407,145,391,223),(59,444,87,121,408,158,392,208),(60,429,88,134,409,143,365,221),(61,442,89,119,410,156,366,206),(62,427,90,132,411,141,367,219),(63,440,91,117,412,154,368,204),(64,425,92,130,413,167,369,217),(65,438,93,115,414,152,370,202),(66,423,94,128,415,165,371,215),(67,436,95,113,416,150,372,200),(68,421,96,126,417,163,373,213),(69,434,97,139,418,148,374,198),(70,447,98,124,419,161,375,211),(71,432,99,137,420,146,376,224),(72,445,100,122,393,159,377,209),(73,430,101,135,394,144,378,222),(74,443,102,120,395,157,379,207),(75,428,103,133,396,142,380,220),(76,441,104,118,397,155,381,205),(77,426,105,131,398,168,382,218),(78,439,106,116,399,153,383,203),(79,424,107,129,400,166,384,216),(80,437,108,114,401,151,385,201),(81,422,109,127,402,164,386,214),(82,435,110,140,403,149,387,199),(83,448,111,125,404,162,388,212),(84,433,112,138,405,147,389,197)], [(1,413,15,399),(2,412,16,398),(3,411,17,397),(4,410,18,396),(5,409,19,395),(6,408,20,394),(7,407,21,393),(8,406,22,420),(9,405,23,419),(10,404,24,418),(11,403,25,417),(12,402,26,416),(13,401,27,415),(14,400,28,414),(29,151,43,165),(30,150,44,164),(31,149,45,163),(32,148,46,162),(33,147,47,161),(34,146,48,160),(35,145,49,159),(36,144,50,158),(37,143,51,157),(38,142,52,156),(39,141,53,155),(40,168,54,154),(41,167,55,153),(42,166,56,152),(57,192,71,178),(58,191,72,177),(59,190,73,176),(60,189,74,175),(61,188,75,174),(62,187,76,173),(63,186,77,172),(64,185,78,171),(65,184,79,170),(66,183,80,169),(67,182,81,196),(68,181,82,195),(69,180,83,194),(70,179,84,193),(85,319,99,333),(86,318,100,332),(87,317,101,331),(88,316,102,330),(89,315,103,329),(90,314,104,328),(91,313,105,327),(92,312,106,326),(93,311,107,325),(94,310,108,324),(95,309,109,323),(96,336,110,322),(97,335,111,321),(98,334,112,320),(113,361,127,347),(114,360,128,346),(115,359,129,345),(116,358,130,344),(117,357,131,343),(118,356,132,342),(119,355,133,341),(120,354,134,340),(121,353,135,339),(122,352,136,338),(123,351,137,337),(124,350,138,364),(125,349,139,363),(126,348,140,362),(197,300,211,286),(198,299,212,285),(199,298,213,284),(200,297,214,283),(201,296,215,282),(202,295,216,281),(203,294,217,308),(204,293,218,307),(205,292,219,306),(206,291,220,305),(207,290,221,304),(208,289,222,303),(209,288,223,302),(210,287,224,301),(225,367,239,381),(226,366,240,380),(227,365,241,379),(228,392,242,378),(229,391,243,377),(230,390,244,376),(231,389,245,375),(232,388,246,374),(233,387,247,373),(234,386,248,372),(235,385,249,371),(236,384,250,370),(237,383,251,369),(238,382,252,368),(253,443,267,429),(254,442,268,428),(255,441,269,427),(256,440,270,426),(257,439,271,425),(258,438,272,424),(259,437,273,423),(260,436,274,422),(261,435,275,421),(262,434,276,448),(263,433,277,447),(264,432,278,446),(265,431,279,445),(266,430,280,444)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | D7 | M4(2) | C4○D4 | D14 | D14 | C8○D4 | C4×D7 | C8⋊D7 | D4⋊2D7 | Q8×D7 | D28.C4 |
| kernel | C28.M4(2) | C4×C7⋊C8 | Dic7⋊C8 | C56⋊C4 | C7×C4⋊C8 | C4×Dic14 | Dic7⋊C4 | C4⋊Dic7 | C2×Dic14 | C7⋊C8 | C4⋊C8 | C28 | C28 | C42 | C2×C8 | C14 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 4 | 12 | 24 | 3 | 3 | 6 |
Matrix representation of C28.M4(2) ►in GL4(𝔽113) generated by
| 9 | 9 | 0 | 0 |
| 95 | 70 | 0 | 0 |
| 0 | 0 | 106 | 35 |
| 0 | 0 | 47 | 7 |
| 68 | 5 | 0 | 0 |
| 44 | 45 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 57 | 39 | 0 | 0 |
| 50 | 56 | 0 | 0 |
| 0 | 0 | 1 | 28 |
| 0 | 0 | 8 | 112 |
G:=sub<GL(4,GF(113))| [9,95,0,0,9,70,0,0,0,0,106,47,0,0,35,7],[68,44,0,0,5,45,0,0,0,0,15,0,0,0,0,15],[57,50,0,0,39,56,0,0,0,0,1,8,0,0,28,112] >;
C28.M4(2) in GAP, Magma, Sage, TeX
C_{28}.M_4(2) % in TeX
G:=Group("C28.M4(2)"); // GroupNames label
G:=SmallGroup(448,365);
// by ID
G=gap.SmallGroup(448,365);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,64,758,135,142,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^14,b*a*b^-1=a^13,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations