metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.10D8, C28.6Q16, C28.10SD16, C42.11D14, C4⋊Q8.2D7, C4.9(Q8⋊D7), C4⋊C4.2Dic7, C4.13(D4⋊D7), (C2×C28).108D4, C7⋊3(C4.10D8), C4.7(D4.D7), C28⋊C8.13C2, C4.7(C7⋊Q16), (C4×C28).49C22, C2.5(D4⋊Dic7), C2.5(Q8⋊Dic7), C14.25(D4⋊C4), C14.13(Q8⋊C4), C2.4(C28.10D4), C14.9(C4.10D4), C22.43(C23.D7), (C7×C4⋊C4).2C4, (C7×C4⋊Q8).2C2, (C2×C28).173(C2×C4), (C2×C4).13(C2×Dic7), (C2×C4).178(C7⋊D4), (C2×C14).105(C22⋊C4), SmallGroup(448,104)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.10D8
G = < a,b,c | a28=b8=1, c2=a21, bab-1=a-1, cac-1=a13, cbc-1=a7b-1 >
Subgroups: 204 in 64 conjugacy classes, 35 normal (31 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C28, C28, C2×C14, C4⋊C8, C4⋊Q8, C7⋊C8, C2×C28, C2×C28, C7×Q8, C4.10D8, C2×C7⋊C8, C4×C28, C7×C4⋊C4, C7×C4⋊C4, Q8×C14, C28⋊C8, C7×C4⋊Q8, C28.10D8
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, Q16, Dic7, D14, C4.10D4, D4⋊C4, Q8⋊C4, C2×Dic7, C7⋊D4, C4.10D8, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C23.D7, D4⋊Dic7, Q8⋊Dic7, C28.10D4, C28.10D8
►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)
(1 423 408 262 104 236 34 192)(2 422 409 261 105 235 35 191)(3 421 410 260 106 234 36 190)(4 448 411 259 107 233 37 189)(5 447 412 258 108 232 38 188)(6 446 413 257 109 231 39 187)(7 445 414 256 110 230 40 186)(8 444 415 255 111 229 41 185)(9 443 416 254 112 228 42 184)(10 442 417 253 85 227 43 183)(11 441 418 280 86 226 44 182)(12 440 419 279 87 225 45 181)(13 439 420 278 88 252 46 180)(14 438 393 277 89 251 47 179)(15 437 394 276 90 250 48 178)(16 436 395 275 91 249 49 177)(17 435 396 274 92 248 50 176)(18 434 397 273 93 247 51 175)(19 433 398 272 94 246 52 174)(20 432 399 271 95 245 53 173)(21 431 400 270 96 244 54 172)(22 430 401 269 97 243 55 171)(23 429 402 268 98 242 56 170)(24 428 403 267 99 241 29 169)(25 427 404 266 100 240 30 196)(26 426 405 265 101 239 31 195)(27 425 406 264 102 238 32 194)(28 424 407 263 103 237 33 193)(57 299 325 344 202 137 152 379)(58 298 326 343 203 136 153 378)(59 297 327 342 204 135 154 377)(60 296 328 341 205 134 155 376)(61 295 329 340 206 133 156 375)(62 294 330 339 207 132 157 374)(63 293 331 338 208 131 158 373)(64 292 332 337 209 130 159 372)(65 291 333 364 210 129 160 371)(66 290 334 363 211 128 161 370)(67 289 335 362 212 127 162 369)(68 288 336 361 213 126 163 368)(69 287 309 360 214 125 164 367)(70 286 310 359 215 124 165 366)(71 285 311 358 216 123 166 365)(72 284 312 357 217 122 167 392)(73 283 313 356 218 121 168 391)(74 282 314 355 219 120 141 390)(75 281 315 354 220 119 142 389)(76 308 316 353 221 118 143 388)(77 307 317 352 222 117 144 387)(78 306 318 351 223 116 145 386)(79 305 319 350 224 115 146 385)(80 304 320 349 197 114 147 384)(81 303 321 348 198 113 148 383)(82 302 322 347 199 140 149 382)(83 301 323 346 200 139 150 381)(84 300 324 345 201 138 151 380)
(1 296 22 289 15 282 8 303)(2 281 23 302 16 295 9 288)(3 294 24 287 17 308 10 301)(4 307 25 300 18 293 11 286)(5 292 26 285 19 306 12 299)(6 305 27 298 20 291 13 284)(7 290 28 283 21 304 14 297)(29 360 50 353 43 346 36 339)(30 345 51 338 44 359 37 352)(31 358 52 351 45 344 38 337)(32 343 53 364 46 357 39 350)(33 356 54 349 47 342 40 363)(34 341 55 362 48 355 41 348)(35 354 56 347 49 340 42 361)(57 426 78 447 71 440 64 433)(58 439 79 432 72 425 65 446)(59 424 80 445 73 438 66 431)(60 437 81 430 74 423 67 444)(61 422 82 443 75 436 68 429)(62 435 83 428 76 421 69 442)(63 448 84 441 77 434 70 427)(85 139 106 132 99 125 92 118)(86 124 107 117 100 138 93 131)(87 137 108 130 101 123 94 116)(88 122 109 115 102 136 95 129)(89 135 110 128 103 121 96 114)(90 120 111 113 104 134 97 127)(91 133 112 126 105 119 98 140)(141 262 162 255 155 276 148 269)(142 275 163 268 156 261 149 254)(143 260 164 253 157 274 150 267)(144 273 165 266 158 259 151 280)(145 258 166 279 159 272 152 265)(146 271 167 264 160 257 153 278)(147 256 168 277 161 270 154 263)(169 316 190 309 183 330 176 323)(170 329 191 322 184 315 177 336)(171 314 192 335 185 328 178 321)(172 327 193 320 186 313 179 334)(173 312 194 333 187 326 180 319)(174 325 195 318 188 311 181 332)(175 310 196 331 189 324 182 317)(197 230 218 251 211 244 204 237)(198 243 219 236 212 229 205 250)(199 228 220 249 213 242 206 235)(200 241 221 234 214 227 207 248)(201 226 222 247 215 240 208 233)(202 239 223 232 216 225 209 246)(203 252 224 245 217 238 210 231)(365 398 386 419 379 412 372 405)(366 411 387 404 380 397 373 418)(367 396 388 417 381 410 374 403)(368 409 389 402 382 395 375 416)(369 394 390 415 383 408 376 401)(370 407 391 400 384 393 377 414)(371 420 392 413 385 406 378 399)
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), (1,423,408,262,104,236,34,192)(2,422,409,261,105,235,35,191)(3,421,410,260,106,234,36,190)(4,448,411,259,107,233,37,189)(5,447,412,258,108,232,38,188)(6,446,413,257,109,231,39,187)(7,445,414,256,110,230,40,186)(8,444,415,255,111,229,41,185)(9,443,416,254,112,228,42,184)(10,442,417,253,85,227,43,183)(11,441,418,280,86,226,44,182)(12,440,419,279,87,225,45,181)(13,439,420,278,88,252,46,180)(14,438,393,277,89,251,47,179)(15,437,394,276,90,250,48,178)(16,436,395,275,91,249,49,177)(17,435,396,274,92,248,50,176)(18,434,397,273,93,247,51,175)(19,433,398,272,94,246,52,174)(20,432,399,271,95,245,53,173)(21,431,400,270,96,244,54,172)(22,430,401,269,97,243,55,171)(23,429,402,268,98,242,56,170)(24,428,403,267,99,241,29,169)(25,427,404,266,100,240,30,196)(26,426,405,265,101,239,31,195)(27,425,406,264,102,238,32,194)(28,424,407,263,103,237,33,193)(57,299,325,344,202,137,152,379)(58,298,326,343,203,136,153,378)(59,297,327,342,204,135,154,377)(60,296,328,341,205,134,155,376)(61,295,329,340,206,133,156,375)(62,294,330,339,207,132,157,374)(63,293,331,338,208,131,158,373)(64,292,332,337,209,130,159,372)(65,291,333,364,210,129,160,371)(66,290,334,363,211,128,161,370)(67,289,335,362,212,127,162,369)(68,288,336,361,213,126,163,368)(69,287,309,360,214,125,164,367)(70,286,310,359,215,124,165,366)(71,285,311,358,216,123,166,365)(72,284,312,357,217,122,167,392)(73,283,313,356,218,121,168,391)(74,282,314,355,219,120,141,390)(75,281,315,354,220,119,142,389)(76,308,316,353,221,118,143,388)(77,307,317,352,222,117,144,387)(78,306,318,351,223,116,145,386)(79,305,319,350,224,115,146,385)(80,304,320,349,197,114,147,384)(81,303,321,348,198,113,148,383)(82,302,322,347,199,140,149,382)(83,301,323,346,200,139,150,381)(84,300,324,345,201,138,151,380), (1,296,22,289,15,282,8,303)(2,281,23,302,16,295,9,288)(3,294,24,287,17,308,10,301)(4,307,25,300,18,293,11,286)(5,292,26,285,19,306,12,299)(6,305,27,298,20,291,13,284)(7,290,28,283,21,304,14,297)(29,360,50,353,43,346,36,339)(30,345,51,338,44,359,37,352)(31,358,52,351,45,344,38,337)(32,343,53,364,46,357,39,350)(33,356,54,349,47,342,40,363)(34,341,55,362,48,355,41,348)(35,354,56,347,49,340,42,361)(57,426,78,447,71,440,64,433)(58,439,79,432,72,425,65,446)(59,424,80,445,73,438,66,431)(60,437,81,430,74,423,67,444)(61,422,82,443,75,436,68,429)(62,435,83,428,76,421,69,442)(63,448,84,441,77,434,70,427)(85,139,106,132,99,125,92,118)(86,124,107,117,100,138,93,131)(87,137,108,130,101,123,94,116)(88,122,109,115,102,136,95,129)(89,135,110,128,103,121,96,114)(90,120,111,113,104,134,97,127)(91,133,112,126,105,119,98,140)(141,262,162,255,155,276,148,269)(142,275,163,268,156,261,149,254)(143,260,164,253,157,274,150,267)(144,273,165,266,158,259,151,280)(145,258,166,279,159,272,152,265)(146,271,167,264,160,257,153,278)(147,256,168,277,161,270,154,263)(169,316,190,309,183,330,176,323)(170,329,191,322,184,315,177,336)(171,314,192,335,185,328,178,321)(172,327,193,320,186,313,179,334)(173,312,194,333,187,326,180,319)(174,325,195,318,188,311,181,332)(175,310,196,331,189,324,182,317)(197,230,218,251,211,244,204,237)(198,243,219,236,212,229,205,250)(199,228,220,249,213,242,206,235)(200,241,221,234,214,227,207,248)(201,226,222,247,215,240,208,233)(202,239,223,232,216,225,209,246)(203,252,224,245,217,238,210,231)(365,398,386,419,379,412,372,405)(366,411,387,404,380,397,373,418)(367,396,388,417,381,410,374,403)(368,409,389,402,382,395,375,416)(369,394,390,415,383,408,376,401)(370,407,391,400,384,393,377,414)(371,420,392,413,385,406,378,399)>;
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), (1,423,408,262,104,236,34,192)(2,422,409,261,105,235,35,191)(3,421,410,260,106,234,36,190)(4,448,411,259,107,233,37,189)(5,447,412,258,108,232,38,188)(6,446,413,257,109,231,39,187)(7,445,414,256,110,230,40,186)(8,444,415,255,111,229,41,185)(9,443,416,254,112,228,42,184)(10,442,417,253,85,227,43,183)(11,441,418,280,86,226,44,182)(12,440,419,279,87,225,45,181)(13,439,420,278,88,252,46,180)(14,438,393,277,89,251,47,179)(15,437,394,276,90,250,48,178)(16,436,395,275,91,249,49,177)(17,435,396,274,92,248,50,176)(18,434,397,273,93,247,51,175)(19,433,398,272,94,246,52,174)(20,432,399,271,95,245,53,173)(21,431,400,270,96,244,54,172)(22,430,401,269,97,243,55,171)(23,429,402,268,98,242,56,170)(24,428,403,267,99,241,29,169)(25,427,404,266,100,240,30,196)(26,426,405,265,101,239,31,195)(27,425,406,264,102,238,32,194)(28,424,407,263,103,237,33,193)(57,299,325,344,202,137,152,379)(58,298,326,343,203,136,153,378)(59,297,327,342,204,135,154,377)(60,296,328,341,205,134,155,376)(61,295,329,340,206,133,156,375)(62,294,330,339,207,132,157,374)(63,293,331,338,208,131,158,373)(64,292,332,337,209,130,159,372)(65,291,333,364,210,129,160,371)(66,290,334,363,211,128,161,370)(67,289,335,362,212,127,162,369)(68,288,336,361,213,126,163,368)(69,287,309,360,214,125,164,367)(70,286,310,359,215,124,165,366)(71,285,311,358,216,123,166,365)(72,284,312,357,217,122,167,392)(73,283,313,356,218,121,168,391)(74,282,314,355,219,120,141,390)(75,281,315,354,220,119,142,389)(76,308,316,353,221,118,143,388)(77,307,317,352,222,117,144,387)(78,306,318,351,223,116,145,386)(79,305,319,350,224,115,146,385)(80,304,320,349,197,114,147,384)(81,303,321,348,198,113,148,383)(82,302,322,347,199,140,149,382)(83,301,323,346,200,139,150,381)(84,300,324,345,201,138,151,380), (1,296,22,289,15,282,8,303)(2,281,23,302,16,295,9,288)(3,294,24,287,17,308,10,301)(4,307,25,300,18,293,11,286)(5,292,26,285,19,306,12,299)(6,305,27,298,20,291,13,284)(7,290,28,283,21,304,14,297)(29,360,50,353,43,346,36,339)(30,345,51,338,44,359,37,352)(31,358,52,351,45,344,38,337)(32,343,53,364,46,357,39,350)(33,356,54,349,47,342,40,363)(34,341,55,362,48,355,41,348)(35,354,56,347,49,340,42,361)(57,426,78,447,71,440,64,433)(58,439,79,432,72,425,65,446)(59,424,80,445,73,438,66,431)(60,437,81,430,74,423,67,444)(61,422,82,443,75,436,68,429)(62,435,83,428,76,421,69,442)(63,448,84,441,77,434,70,427)(85,139,106,132,99,125,92,118)(86,124,107,117,100,138,93,131)(87,137,108,130,101,123,94,116)(88,122,109,115,102,136,95,129)(89,135,110,128,103,121,96,114)(90,120,111,113,104,134,97,127)(91,133,112,126,105,119,98,140)(141,262,162,255,155,276,148,269)(142,275,163,268,156,261,149,254)(143,260,164,253,157,274,150,267)(144,273,165,266,158,259,151,280)(145,258,166,279,159,272,152,265)(146,271,167,264,160,257,153,278)(147,256,168,277,161,270,154,263)(169,316,190,309,183,330,176,323)(170,329,191,322,184,315,177,336)(171,314,192,335,185,328,178,321)(172,327,193,320,186,313,179,334)(173,312,194,333,187,326,180,319)(174,325,195,318,188,311,181,332)(175,310,196,331,189,324,182,317)(197,230,218,251,211,244,204,237)(198,243,219,236,212,229,205,250)(199,228,220,249,213,242,206,235)(200,241,221,234,214,227,207,248)(201,226,222,247,215,240,208,233)(202,239,223,232,216,225,209,246)(203,252,224,245,217,238,210,231)(365,398,386,419,379,412,372,405)(366,411,387,404,380,397,373,418)(367,396,388,417,381,410,374,403)(368,409,389,402,382,395,375,416)(369,394,390,415,383,408,376,401)(370,407,391,400,384,393,377,414)(371,420,392,413,385,406,378,399) );
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)], [(1,423,408,262,104,236,34,192),(2,422,409,261,105,235,35,191),(3,421,410,260,106,234,36,190),(4,448,411,259,107,233,37,189),(5,447,412,258,108,232,38,188),(6,446,413,257,109,231,39,187),(7,445,414,256,110,230,40,186),(8,444,415,255,111,229,41,185),(9,443,416,254,112,228,42,184),(10,442,417,253,85,227,43,183),(11,441,418,280,86,226,44,182),(12,440,419,279,87,225,45,181),(13,439,420,278,88,252,46,180),(14,438,393,277,89,251,47,179),(15,437,394,276,90,250,48,178),(16,436,395,275,91,249,49,177),(17,435,396,274,92,248,50,176),(18,434,397,273,93,247,51,175),(19,433,398,272,94,246,52,174),(20,432,399,271,95,245,53,173),(21,431,400,270,96,244,54,172),(22,430,401,269,97,243,55,171),(23,429,402,268,98,242,56,170),(24,428,403,267,99,241,29,169),(25,427,404,266,100,240,30,196),(26,426,405,265,101,239,31,195),(27,425,406,264,102,238,32,194),(28,424,407,263,103,237,33,193),(57,299,325,344,202,137,152,379),(58,298,326,343,203,136,153,378),(59,297,327,342,204,135,154,377),(60,296,328,341,205,134,155,376),(61,295,329,340,206,133,156,375),(62,294,330,339,207,132,157,374),(63,293,331,338,208,131,158,373),(64,292,332,337,209,130,159,372),(65,291,333,364,210,129,160,371),(66,290,334,363,211,128,161,370),(67,289,335,362,212,127,162,369),(68,288,336,361,213,126,163,368),(69,287,309,360,214,125,164,367),(70,286,310,359,215,124,165,366),(71,285,311,358,216,123,166,365),(72,284,312,357,217,122,167,392),(73,283,313,356,218,121,168,391),(74,282,314,355,219,120,141,390),(75,281,315,354,220,119,142,389),(76,308,316,353,221,118,143,388),(77,307,317,352,222,117,144,387),(78,306,318,351,223,116,145,386),(79,305,319,350,224,115,146,385),(80,304,320,349,197,114,147,384),(81,303,321,348,198,113,148,383),(82,302,322,347,199,140,149,382),(83,301,323,346,200,139,150,381),(84,300,324,345,201,138,151,380)], [(1,296,22,289,15,282,8,303),(2,281,23,302,16,295,9,288),(3,294,24,287,17,308,10,301),(4,307,25,300,18,293,11,286),(5,292,26,285,19,306,12,299),(6,305,27,298,20,291,13,284),(7,290,28,283,21,304,14,297),(29,360,50,353,43,346,36,339),(30,345,51,338,44,359,37,352),(31,358,52,351,45,344,38,337),(32,343,53,364,46,357,39,350),(33,356,54,349,47,342,40,363),(34,341,55,362,48,355,41,348),(35,354,56,347,49,340,42,361),(57,426,78,447,71,440,64,433),(58,439,79,432,72,425,65,446),(59,424,80,445,73,438,66,431),(60,437,81,430,74,423,67,444),(61,422,82,443,75,436,68,429),(62,435,83,428,76,421,69,442),(63,448,84,441,77,434,70,427),(85,139,106,132,99,125,92,118),(86,124,107,117,100,138,93,131),(87,137,108,130,101,123,94,116),(88,122,109,115,102,136,95,129),(89,135,110,128,103,121,96,114),(90,120,111,113,104,134,97,127),(91,133,112,126,105,119,98,140),(141,262,162,255,155,276,148,269),(142,275,163,268,156,261,149,254),(143,260,164,253,157,274,150,267),(144,273,165,266,158,259,151,280),(145,258,166,279,159,272,152,265),(146,271,167,264,160,257,153,278),(147,256,168,277,161,270,154,263),(169,316,190,309,183,330,176,323),(170,329,191,322,184,315,177,336),(171,314,192,335,185,328,178,321),(172,327,193,320,186,313,179,334),(173,312,194,333,187,326,180,319),(174,325,195,318,188,311,181,332),(175,310,196,331,189,324,182,317),(197,230,218,251,211,244,204,237),(198,243,219,236,212,229,205,250),(199,228,220,249,213,242,206,235),(200,241,221,234,214,227,207,248),(201,226,222,247,215,240,208,233),(202,239,223,232,216,225,209,246),(203,252,224,245,217,238,210,231),(365,398,386,419,379,412,372,405),(366,411,387,404,380,397,373,418),(367,396,388,417,381,410,374,403),(368,409,389,402,382,395,375,416),(369,394,390,415,383,408,376,401),(370,407,391,400,384,393,377,414),(371,420,392,413,385,406,378,399)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | D4 | D7 | D8 | SD16 | Q16 | D14 | Dic7 | C7⋊D4 | C4.10D4 | D4⋊D7 | D4.D7 | Q8⋊D7 | C7⋊Q16 | C28.10D4 |
| kernel | C28.10D8 | C28⋊C8 | C7×C4⋊Q8 | C7×C4⋊C4 | C2×C28 | C4⋊Q8 | C28 | C28 | C28 | C42 | C4⋊C4 | C2×C4 | C14 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 3 | 2 | 4 | 2 | 3 | 6 | 12 | 1 | 3 | 3 | 3 | 3 | 6 |
Matrix representation of C28.10D8 ►in GL6(𝔽113)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 79 |
| 0 | 0 | 0 | 0 | 42 | 104 |
| 12 | 9 | 0 | 0 | 0 | 0 |
| 9 | 101 | 0 | 0 | 0 | 0 |
| 0 | 0 | 100 | 13 | 0 | 0 |
| 0 | 0 | 100 | 100 | 0 | 0 |
| 0 | 0 | 0 | 0 | 65 | 8 |
| 0 | 0 | 0 | 0 | 37 | 48 |
| 31 | 31 | 0 | 0 | 0 | 0 |
| 82 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 26 | 0 | 0 |
| 0 | 0 | 26 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 65 | 8 |
| 0 | 0 | 0 | 0 | 37 | 48 |
G:=sub<GL(6,GF(113))| [0,112,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,33,42,0,0,0,0,79,104],[12,9,0,0,0,0,9,101,0,0,0,0,0,0,100,100,0,0,0,0,13,100,0,0,0,0,0,0,65,37,0,0,0,0,8,48],[31,82,0,0,0,0,31,31,0,0,0,0,0,0,112,26,0,0,0,0,26,1,0,0,0,0,0,0,65,37,0,0,0,0,8,48] >;
C28.10D8 in GAP, Magma, Sage, TeX
C_{28}._{10}D_8 % in TeX
G:=Group("C28.10D8"); // GroupNames label
G:=SmallGroup(448,104);
// by ID
G=gap.SmallGroup(448,104);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,120,219,100,1571,570,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=1,c^2=a^21,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^7*b^-1>;
// generators/relations