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## G = C2×C28.Q8order 448 = 26·7

### Direct product of C2 and C28.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×C28.Q8
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×C7⋊C8 — C22×C7⋊C8 — C2×C28.Q8
 Lower central C7 — C14 — C28 — C2×C28.Q8
 Upper central C1 — C23 — C22×C4 — C2×C4⋊C4

Generators and relations for C2×C28.Q8
G = < a,b,c,d | a2=b28=c4=1, d2=b21c2, ab=ba, ac=ca, ad=da, cbc-1=b15, dbd-1=b13, dcd-1=b21c-1 >

Subgroups: 452 in 130 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, Dic7, C28, C28, C28, C2×C14, C2×C14, C2.D8, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C2×C28, C22×C14, C2×C2.D8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C7×C4⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, C22×C28, C28.Q8, C22×C7⋊C8, C2×C4⋊Dic7, C14×C4⋊C4, C2×C28.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, D14, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, Dic14, C4×D7, C7⋊D4, C22×D7, C2×C2.D8, Dic7⋊C4, D4⋊D7, C7⋊Q16, C2×Dic14, C2×C4×D7, C2×C7⋊D4, C28.Q8, C2×Dic7⋊C4, C2×D4⋊D7, C2×C7⋊Q16, C2×C28.Q8

Smallest permutation representation of C2×C28.Q8
Regular action on 448 points
Generators in S448
(1 243)(2 244)(3 245)(4 246)(5 247)(6 248)(7 249)(8 250)(9 251)(10 252)(11 225)(12 226)(13 227)(14 228)(15 229)(16 230)(17 231)(18 232)(19 233)(20 234)(21 235)(22 236)(23 237)(24 238)(25 239)(26 240)(27 241)(28 242)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 99)(57 148)(58 149)(59 150)(60 151)(61 152)(62 153)(63 154)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 161)(71 162)(72 163)(73 164)(74 165)(75 166)(76 167)(77 168)(78 141)(79 142)(80 143)(81 144)(82 145)(83 146)(84 147)(113 366)(114 367)(115 368)(116 369)(117 370)(118 371)(119 372)(120 373)(121 374)(122 375)(123 376)(124 377)(125 378)(126 379)(127 380)(128 381)(129 382)(130 383)(131 384)(132 385)(133 386)(134 387)(135 388)(136 389)(137 390)(138 391)(139 392)(140 365)(169 402)(170 403)(171 404)(172 405)(173 406)(174 407)(175 408)(176 409)(177 410)(178 411)(179 412)(180 413)(181 414)(182 415)(183 416)(184 417)(185 418)(186 419)(187 420)(188 393)(189 394)(190 395)(191 396)(192 397)(193 398)(194 399)(195 400)(196 401)(197 321)(198 322)(199 323)(200 324)(201 325)(202 326)(203 327)(204 328)(205 329)(206 330)(207 331)(208 332)(209 333)(210 334)(211 335)(212 336)(213 309)(214 310)(215 311)(216 312)(217 313)(218 314)(219 315)(220 316)(221 317)(222 318)(223 319)(224 320)(253 281)(254 282)(255 283)(256 284)(257 285)(258 286)(259 287)(260 288)(261 289)(262 290)(263 291)(264 292)(265 293)(266 294)(267 295)(268 296)(269 297)(270 298)(271 299)(272 300)(273 301)(274 302)(275 303)(276 304)(277 305)(278 306)(279 307)(280 308)(337 434)(338 435)(339 436)(340 437)(341 438)(342 439)(343 440)(344 441)(345 442)(346 443)(347 444)(348 445)(349 446)(350 447)(351 448)(352 421)(353 422)(354 423)(355 424)(356 425)(357 426)(358 427)(359 428)(360 429)(361 430)(362 431)(363 432)(364 433)
(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)
(1 366 286 159)(2 381 287 146)(3 368 288 161)(4 383 289 148)(5 370 290 163)(6 385 291 150)(7 372 292 165)(8 387 293 152)(9 374 294 167)(10 389 295 154)(11 376 296 141)(12 391 297 156)(13 378 298 143)(14 365 299 158)(15 380 300 145)(16 367 301 160)(17 382 302 147)(18 369 303 162)(19 384 304 149)(20 371 305 164)(21 386 306 151)(22 373 307 166)(23 388 308 153)(24 375 281 168)(25 390 282 155)(26 377 283 142)(27 392 284 157)(28 379 285 144)(29 317 433 183)(30 332 434 170)(31 319 435 185)(32 334 436 172)(33 321 437 187)(34 336 438 174)(35 323 439 189)(36 310 440 176)(37 325 441 191)(38 312 442 178)(39 327 443 193)(40 314 444 180)(41 329 445 195)(42 316 446 182)(43 331 447 169)(44 318 448 184)(45 333 421 171)(46 320 422 186)(47 335 423 173)(48 322 424 188)(49 309 425 175)(50 324 426 190)(51 311 427 177)(52 326 428 192)(53 313 429 179)(54 328 430 194)(55 315 431 181)(56 330 432 196)(57 246 130 261)(58 233 131 276)(59 248 132 263)(60 235 133 278)(61 250 134 265)(62 237 135 280)(63 252 136 267)(64 239 137 254)(65 226 138 269)(66 241 139 256)(67 228 140 271)(68 243 113 258)(69 230 114 273)(70 245 115 260)(71 232 116 275)(72 247 117 262)(73 234 118 277)(74 249 119 264)(75 236 120 279)(76 251 121 266)(77 238 122 253)(78 225 123 268)(79 240 124 255)(80 227 125 270)(81 242 126 257)(82 229 127 272)(83 244 128 259)(84 231 129 274)(85 220 349 415)(86 207 350 402)(87 222 351 417)(88 209 352 404)(89 224 353 419)(90 211 354 406)(91 198 355 393)(92 213 356 408)(93 200 357 395)(94 215 358 410)(95 202 359 397)(96 217 360 412)(97 204 361 399)(98 219 362 414)(99 206 363 401)(100 221 364 416)(101 208 337 403)(102 223 338 418)(103 210 339 405)(104 197 340 420)(105 212 341 407)(106 199 342 394)(107 214 343 409)(108 201 344 396)(109 216 345 411)(110 203 346 398)(111 218 347 413)(112 205 348 400)
(1 319 307 178 15 333 293 192)(2 332 308 191 16 318 294 177)(3 317 281 176 17 331 295 190)(4 330 282 189 18 316 296 175)(5 315 283 174 19 329 297 188)(6 328 284 187 20 314 298 173)(7 313 285 172 21 327 299 186)(8 326 286 185 22 312 300 171)(9 311 287 170 23 325 301 184)(10 324 288 183 24 310 302 169)(11 309 289 196 25 323 303 182)(12 322 290 181 26 336 304 195)(13 335 291 194 27 321 305 180)(14 320 292 179 28 334 306 193)(29 161 426 389 43 147 440 375)(30 146 427 374 44 160 441 388)(31 159 428 387 45 145 442 373)(32 144 429 372 46 158 443 386)(33 157 430 385 47 143 444 371)(34 142 431 370 48 156 445 384)(35 155 432 383 49 141 446 369)(36 168 433 368 50 154 447 382)(37 153 434 381 51 167 448 367)(38 166 435 366 52 152 421 380)(39 151 436 379 53 165 422 365)(40 164 437 392 54 150 423 378)(41 149 438 377 55 163 424 391)(42 162 439 390 56 148 425 376)(57 356 123 85 71 342 137 99)(58 341 124 98 72 355 138 112)(59 354 125 111 73 340 139 97)(60 339 126 96 74 353 140 110)(61 352 127 109 75 338 113 95)(62 337 128 94 76 351 114 108)(63 350 129 107 77 364 115 93)(64 363 130 92 78 349 116 106)(65 348 131 105 79 362 117 91)(66 361 132 90 80 347 118 104)(67 346 133 103 81 360 119 89)(68 359 134 88 82 345 120 102)(69 344 135 101 83 358 121 87)(70 357 136 86 84 343 122 100)(197 277 413 227 211 263 399 241)(198 262 414 240 212 276 400 226)(199 275 415 225 213 261 401 239)(200 260 416 238 214 274 402 252)(201 273 417 251 215 259 403 237)(202 258 418 236 216 272 404 250)(203 271 419 249 217 257 405 235)(204 256 420 234 218 270 406 248)(205 269 393 247 219 255 407 233)(206 254 394 232 220 268 408 246)(207 267 395 245 221 253 409 231)(208 280 396 230 222 266 410 244)(209 265 397 243 223 279 411 229)(210 278 398 228 224 264 412 242)

G:=sub<Sym(448)| (1,243)(2,244)(3,245)(4,246)(5,247)(6,248)(7,249)(8,250)(9,251)(10,252)(11,225)(12,226)(13,227)(14,228)(15,229)(16,230)(17,231)(18,232)(19,233)(20,234)(21,235)(22,236)(23,237)(24,238)(25,239)(26,240)(27,241)(28,242)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,99)(57,148)(58,149)(59,150)(60,151)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,161)(71,162)(72,163)(73,164)(74,165)(75,166)(76,167)(77,168)(78,141)(79,142)(80,143)(81,144)(82,145)(83,146)(84,147)(113,366)(114,367)(115,368)(116,369)(117,370)(118,371)(119,372)(120,373)(121,374)(122,375)(123,376)(124,377)(125,378)(126,379)(127,380)(128,381)(129,382)(130,383)(131,384)(132,385)(133,386)(134,387)(135,388)(136,389)(137,390)(138,391)(139,392)(140,365)(169,402)(170,403)(171,404)(172,405)(173,406)(174,407)(175,408)(176,409)(177,410)(178,411)(179,412)(180,413)(181,414)(182,415)(183,416)(184,417)(185,418)(186,419)(187,420)(188,393)(189,394)(190,395)(191,396)(192,397)(193,398)(194,399)(195,400)(196,401)(197,321)(198,322)(199,323)(200,324)(201,325)(202,326)(203,327)(204,328)(205,329)(206,330)(207,331)(208,332)(209,333)(210,334)(211,335)(212,336)(213,309)(214,310)(215,311)(216,312)(217,313)(218,314)(219,315)(220,316)(221,317)(222,318)(223,319)(224,320)(253,281)(254,282)(255,283)(256,284)(257,285)(258,286)(259,287)(260,288)(261,289)(262,290)(263,291)(264,292)(265,293)(266,294)(267,295)(268,296)(269,297)(270,298)(271,299)(272,300)(273,301)(274,302)(275,303)(276,304)(277,305)(278,306)(279,307)(280,308)(337,434)(338,435)(339,436)(340,437)(341,438)(342,439)(343,440)(344,441)(345,442)(346,443)(347,444)(348,445)(349,446)(350,447)(351,448)(352,421)(353,422)(354,423)(355,424)(356,425)(357,426)(358,427)(359,428)(360,429)(361,430)(362,431)(363,432)(364,433), (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), (1,366,286,159)(2,381,287,146)(3,368,288,161)(4,383,289,148)(5,370,290,163)(6,385,291,150)(7,372,292,165)(8,387,293,152)(9,374,294,167)(10,389,295,154)(11,376,296,141)(12,391,297,156)(13,378,298,143)(14,365,299,158)(15,380,300,145)(16,367,301,160)(17,382,302,147)(18,369,303,162)(19,384,304,149)(20,371,305,164)(21,386,306,151)(22,373,307,166)(23,388,308,153)(24,375,281,168)(25,390,282,155)(26,377,283,142)(27,392,284,157)(28,379,285,144)(29,317,433,183)(30,332,434,170)(31,319,435,185)(32,334,436,172)(33,321,437,187)(34,336,438,174)(35,323,439,189)(36,310,440,176)(37,325,441,191)(38,312,442,178)(39,327,443,193)(40,314,444,180)(41,329,445,195)(42,316,446,182)(43,331,447,169)(44,318,448,184)(45,333,421,171)(46,320,422,186)(47,335,423,173)(48,322,424,188)(49,309,425,175)(50,324,426,190)(51,311,427,177)(52,326,428,192)(53,313,429,179)(54,328,430,194)(55,315,431,181)(56,330,432,196)(57,246,130,261)(58,233,131,276)(59,248,132,263)(60,235,133,278)(61,250,134,265)(62,237,135,280)(63,252,136,267)(64,239,137,254)(65,226,138,269)(66,241,139,256)(67,228,140,271)(68,243,113,258)(69,230,114,273)(70,245,115,260)(71,232,116,275)(72,247,117,262)(73,234,118,277)(74,249,119,264)(75,236,120,279)(76,251,121,266)(77,238,122,253)(78,225,123,268)(79,240,124,255)(80,227,125,270)(81,242,126,257)(82,229,127,272)(83,244,128,259)(84,231,129,274)(85,220,349,415)(86,207,350,402)(87,222,351,417)(88,209,352,404)(89,224,353,419)(90,211,354,406)(91,198,355,393)(92,213,356,408)(93,200,357,395)(94,215,358,410)(95,202,359,397)(96,217,360,412)(97,204,361,399)(98,219,362,414)(99,206,363,401)(100,221,364,416)(101,208,337,403)(102,223,338,418)(103,210,339,405)(104,197,340,420)(105,212,341,407)(106,199,342,394)(107,214,343,409)(108,201,344,396)(109,216,345,411)(110,203,346,398)(111,218,347,413)(112,205,348,400), (1,319,307,178,15,333,293,192)(2,332,308,191,16,318,294,177)(3,317,281,176,17,331,295,190)(4,330,282,189,18,316,296,175)(5,315,283,174,19,329,297,188)(6,328,284,187,20,314,298,173)(7,313,285,172,21,327,299,186)(8,326,286,185,22,312,300,171)(9,311,287,170,23,325,301,184)(10,324,288,183,24,310,302,169)(11,309,289,196,25,323,303,182)(12,322,290,181,26,336,304,195)(13,335,291,194,27,321,305,180)(14,320,292,179,28,334,306,193)(29,161,426,389,43,147,440,375)(30,146,427,374,44,160,441,388)(31,159,428,387,45,145,442,373)(32,144,429,372,46,158,443,386)(33,157,430,385,47,143,444,371)(34,142,431,370,48,156,445,384)(35,155,432,383,49,141,446,369)(36,168,433,368,50,154,447,382)(37,153,434,381,51,167,448,367)(38,166,435,366,52,152,421,380)(39,151,436,379,53,165,422,365)(40,164,437,392,54,150,423,378)(41,149,438,377,55,163,424,391)(42,162,439,390,56,148,425,376)(57,356,123,85,71,342,137,99)(58,341,124,98,72,355,138,112)(59,354,125,111,73,340,139,97)(60,339,126,96,74,353,140,110)(61,352,127,109,75,338,113,95)(62,337,128,94,76,351,114,108)(63,350,129,107,77,364,115,93)(64,363,130,92,78,349,116,106)(65,348,131,105,79,362,117,91)(66,361,132,90,80,347,118,104)(67,346,133,103,81,360,119,89)(68,359,134,88,82,345,120,102)(69,344,135,101,83,358,121,87)(70,357,136,86,84,343,122,100)(197,277,413,227,211,263,399,241)(198,262,414,240,212,276,400,226)(199,275,415,225,213,261,401,239)(200,260,416,238,214,274,402,252)(201,273,417,251,215,259,403,237)(202,258,418,236,216,272,404,250)(203,271,419,249,217,257,405,235)(204,256,420,234,218,270,406,248)(205,269,393,247,219,255,407,233)(206,254,394,232,220,268,408,246)(207,267,395,245,221,253,409,231)(208,280,396,230,222,266,410,244)(209,265,397,243,223,279,411,229)(210,278,398,228,224,264,412,242)>;

G:=Group( (1,243)(2,244)(3,245)(4,246)(5,247)(6,248)(7,249)(8,250)(9,251)(10,252)(11,225)(12,226)(13,227)(14,228)(15,229)(16,230)(17,231)(18,232)(19,233)(20,234)(21,235)(22,236)(23,237)(24,238)(25,239)(26,240)(27,241)(28,242)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,99)(57,148)(58,149)(59,150)(60,151)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,161)(71,162)(72,163)(73,164)(74,165)(75,166)(76,167)(77,168)(78,141)(79,142)(80,143)(81,144)(82,145)(83,146)(84,147)(113,366)(114,367)(115,368)(116,369)(117,370)(118,371)(119,372)(120,373)(121,374)(122,375)(123,376)(124,377)(125,378)(126,379)(127,380)(128,381)(129,382)(130,383)(131,384)(132,385)(133,386)(134,387)(135,388)(136,389)(137,390)(138,391)(139,392)(140,365)(169,402)(170,403)(171,404)(172,405)(173,406)(174,407)(175,408)(176,409)(177,410)(178,411)(179,412)(180,413)(181,414)(182,415)(183,416)(184,417)(185,418)(186,419)(187,420)(188,393)(189,394)(190,395)(191,396)(192,397)(193,398)(194,399)(195,400)(196,401)(197,321)(198,322)(199,323)(200,324)(201,325)(202,326)(203,327)(204,328)(205,329)(206,330)(207,331)(208,332)(209,333)(210,334)(211,335)(212,336)(213,309)(214,310)(215,311)(216,312)(217,313)(218,314)(219,315)(220,316)(221,317)(222,318)(223,319)(224,320)(253,281)(254,282)(255,283)(256,284)(257,285)(258,286)(259,287)(260,288)(261,289)(262,290)(263,291)(264,292)(265,293)(266,294)(267,295)(268,296)(269,297)(270,298)(271,299)(272,300)(273,301)(274,302)(275,303)(276,304)(277,305)(278,306)(279,307)(280,308)(337,434)(338,435)(339,436)(340,437)(341,438)(342,439)(343,440)(344,441)(345,442)(346,443)(347,444)(348,445)(349,446)(350,447)(351,448)(352,421)(353,422)(354,423)(355,424)(356,425)(357,426)(358,427)(359,428)(360,429)(361,430)(362,431)(363,432)(364,433), (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), (1,366,286,159)(2,381,287,146)(3,368,288,161)(4,383,289,148)(5,370,290,163)(6,385,291,150)(7,372,292,165)(8,387,293,152)(9,374,294,167)(10,389,295,154)(11,376,296,141)(12,391,297,156)(13,378,298,143)(14,365,299,158)(15,380,300,145)(16,367,301,160)(17,382,302,147)(18,369,303,162)(19,384,304,149)(20,371,305,164)(21,386,306,151)(22,373,307,166)(23,388,308,153)(24,375,281,168)(25,390,282,155)(26,377,283,142)(27,392,284,157)(28,379,285,144)(29,317,433,183)(30,332,434,170)(31,319,435,185)(32,334,436,172)(33,321,437,187)(34,336,438,174)(35,323,439,189)(36,310,440,176)(37,325,441,191)(38,312,442,178)(39,327,443,193)(40,314,444,180)(41,329,445,195)(42,316,446,182)(43,331,447,169)(44,318,448,184)(45,333,421,171)(46,320,422,186)(47,335,423,173)(48,322,424,188)(49,309,425,175)(50,324,426,190)(51,311,427,177)(52,326,428,192)(53,313,429,179)(54,328,430,194)(55,315,431,181)(56,330,432,196)(57,246,130,261)(58,233,131,276)(59,248,132,263)(60,235,133,278)(61,250,134,265)(62,237,135,280)(63,252,136,267)(64,239,137,254)(65,226,138,269)(66,241,139,256)(67,228,140,271)(68,243,113,258)(69,230,114,273)(70,245,115,260)(71,232,116,275)(72,247,117,262)(73,234,118,277)(74,249,119,264)(75,236,120,279)(76,251,121,266)(77,238,122,253)(78,225,123,268)(79,240,124,255)(80,227,125,270)(81,242,126,257)(82,229,127,272)(83,244,128,259)(84,231,129,274)(85,220,349,415)(86,207,350,402)(87,222,351,417)(88,209,352,404)(89,224,353,419)(90,211,354,406)(91,198,355,393)(92,213,356,408)(93,200,357,395)(94,215,358,410)(95,202,359,397)(96,217,360,412)(97,204,361,399)(98,219,362,414)(99,206,363,401)(100,221,364,416)(101,208,337,403)(102,223,338,418)(103,210,339,405)(104,197,340,420)(105,212,341,407)(106,199,342,394)(107,214,343,409)(108,201,344,396)(109,216,345,411)(110,203,346,398)(111,218,347,413)(112,205,348,400), (1,319,307,178,15,333,293,192)(2,332,308,191,16,318,294,177)(3,317,281,176,17,331,295,190)(4,330,282,189,18,316,296,175)(5,315,283,174,19,329,297,188)(6,328,284,187,20,314,298,173)(7,313,285,172,21,327,299,186)(8,326,286,185,22,312,300,171)(9,311,287,170,23,325,301,184)(10,324,288,183,24,310,302,169)(11,309,289,196,25,323,303,182)(12,322,290,181,26,336,304,195)(13,335,291,194,27,321,305,180)(14,320,292,179,28,334,306,193)(29,161,426,389,43,147,440,375)(30,146,427,374,44,160,441,388)(31,159,428,387,45,145,442,373)(32,144,429,372,46,158,443,386)(33,157,430,385,47,143,444,371)(34,142,431,370,48,156,445,384)(35,155,432,383,49,141,446,369)(36,168,433,368,50,154,447,382)(37,153,434,381,51,167,448,367)(38,166,435,366,52,152,421,380)(39,151,436,379,53,165,422,365)(40,164,437,392,54,150,423,378)(41,149,438,377,55,163,424,391)(42,162,439,390,56,148,425,376)(57,356,123,85,71,342,137,99)(58,341,124,98,72,355,138,112)(59,354,125,111,73,340,139,97)(60,339,126,96,74,353,140,110)(61,352,127,109,75,338,113,95)(62,337,128,94,76,351,114,108)(63,350,129,107,77,364,115,93)(64,363,130,92,78,349,116,106)(65,348,131,105,79,362,117,91)(66,361,132,90,80,347,118,104)(67,346,133,103,81,360,119,89)(68,359,134,88,82,345,120,102)(69,344,135,101,83,358,121,87)(70,357,136,86,84,343,122,100)(197,277,413,227,211,263,399,241)(198,262,414,240,212,276,400,226)(199,275,415,225,213,261,401,239)(200,260,416,238,214,274,402,252)(201,273,417,251,215,259,403,237)(202,258,418,236,216,272,404,250)(203,271,419,249,217,257,405,235)(204,256,420,234,218,270,406,248)(205,269,393,247,219,255,407,233)(206,254,394,232,220,268,408,246)(207,267,395,245,221,253,409,231)(208,280,396,230,222,266,410,244)(209,265,397,243,223,279,411,229)(210,278,398,228,224,264,412,242) );

G=PermutationGroup([[(1,243),(2,244),(3,245),(4,246),(5,247),(6,248),(7,249),(8,250),(9,251),(10,252),(11,225),(12,226),(13,227),(14,228),(15,229),(16,230),(17,231),(18,232),(19,233),(20,234),(21,235),(22,236),(23,237),(24,238),(25,239),(26,240),(27,241),(28,242),(29,100),(30,101),(31,102),(32,103),(33,104),(34,105),(35,106),(36,107),(37,108),(38,109),(39,110),(40,111),(41,112),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,92),(50,93),(51,94),(52,95),(53,96),(54,97),(55,98),(56,99),(57,148),(58,149),(59,150),(60,151),(61,152),(62,153),(63,154),(64,155),(65,156),(66,157),(67,158),(68,159),(69,160),(70,161),(71,162),(72,163),(73,164),(74,165),(75,166),(76,167),(77,168),(78,141),(79,142),(80,143),(81,144),(82,145),(83,146),(84,147),(113,366),(114,367),(115,368),(116,369),(117,370),(118,371),(119,372),(120,373),(121,374),(122,375),(123,376),(124,377),(125,378),(126,379),(127,380),(128,381),(129,382),(130,383),(131,384),(132,385),(133,386),(134,387),(135,388),(136,389),(137,390),(138,391),(139,392),(140,365),(169,402),(170,403),(171,404),(172,405),(173,406),(174,407),(175,408),(176,409),(177,410),(178,411),(179,412),(180,413),(181,414),(182,415),(183,416),(184,417),(185,418),(186,419),(187,420),(188,393),(189,394),(190,395),(191,396),(192,397),(193,398),(194,399),(195,400),(196,401),(197,321),(198,322),(199,323),(200,324),(201,325),(202,326),(203,327),(204,328),(205,329),(206,330),(207,331),(208,332),(209,333),(210,334),(211,335),(212,336),(213,309),(214,310),(215,311),(216,312),(217,313),(218,314),(219,315),(220,316),(221,317),(222,318),(223,319),(224,320),(253,281),(254,282),(255,283),(256,284),(257,285),(258,286),(259,287),(260,288),(261,289),(262,290),(263,291),(264,292),(265,293),(266,294),(267,295),(268,296),(269,297),(270,298),(271,299),(272,300),(273,301),(274,302),(275,303),(276,304),(277,305),(278,306),(279,307),(280,308),(337,434),(338,435),(339,436),(340,437),(341,438),(342,439),(343,440),(344,441),(345,442),(346,443),(347,444),(348,445),(349,446),(350,447),(351,448),(352,421),(353,422),(354,423),(355,424),(356,425),(357,426),(358,427),(359,428),(360,429),(361,430),(362,431),(363,432),(364,433)], [(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)], [(1,366,286,159),(2,381,287,146),(3,368,288,161),(4,383,289,148),(5,370,290,163),(6,385,291,150),(7,372,292,165),(8,387,293,152),(9,374,294,167),(10,389,295,154),(11,376,296,141),(12,391,297,156),(13,378,298,143),(14,365,299,158),(15,380,300,145),(16,367,301,160),(17,382,302,147),(18,369,303,162),(19,384,304,149),(20,371,305,164),(21,386,306,151),(22,373,307,166),(23,388,308,153),(24,375,281,168),(25,390,282,155),(26,377,283,142),(27,392,284,157),(28,379,285,144),(29,317,433,183),(30,332,434,170),(31,319,435,185),(32,334,436,172),(33,321,437,187),(34,336,438,174),(35,323,439,189),(36,310,440,176),(37,325,441,191),(38,312,442,178),(39,327,443,193),(40,314,444,180),(41,329,445,195),(42,316,446,182),(43,331,447,169),(44,318,448,184),(45,333,421,171),(46,320,422,186),(47,335,423,173),(48,322,424,188),(49,309,425,175),(50,324,426,190),(51,311,427,177),(52,326,428,192),(53,313,429,179),(54,328,430,194),(55,315,431,181),(56,330,432,196),(57,246,130,261),(58,233,131,276),(59,248,132,263),(60,235,133,278),(61,250,134,265),(62,237,135,280),(63,252,136,267),(64,239,137,254),(65,226,138,269),(66,241,139,256),(67,228,140,271),(68,243,113,258),(69,230,114,273),(70,245,115,260),(71,232,116,275),(72,247,117,262),(73,234,118,277),(74,249,119,264),(75,236,120,279),(76,251,121,266),(77,238,122,253),(78,225,123,268),(79,240,124,255),(80,227,125,270),(81,242,126,257),(82,229,127,272),(83,244,128,259),(84,231,129,274),(85,220,349,415),(86,207,350,402),(87,222,351,417),(88,209,352,404),(89,224,353,419),(90,211,354,406),(91,198,355,393),(92,213,356,408),(93,200,357,395),(94,215,358,410),(95,202,359,397),(96,217,360,412),(97,204,361,399),(98,219,362,414),(99,206,363,401),(100,221,364,416),(101,208,337,403),(102,223,338,418),(103,210,339,405),(104,197,340,420),(105,212,341,407),(106,199,342,394),(107,214,343,409),(108,201,344,396),(109,216,345,411),(110,203,346,398),(111,218,347,413),(112,205,348,400)], [(1,319,307,178,15,333,293,192),(2,332,308,191,16,318,294,177),(3,317,281,176,17,331,295,190),(4,330,282,189,18,316,296,175),(5,315,283,174,19,329,297,188),(6,328,284,187,20,314,298,173),(7,313,285,172,21,327,299,186),(8,326,286,185,22,312,300,171),(9,311,287,170,23,325,301,184),(10,324,288,183,24,310,302,169),(11,309,289,196,25,323,303,182),(12,322,290,181,26,336,304,195),(13,335,291,194,27,321,305,180),(14,320,292,179,28,334,306,193),(29,161,426,389,43,147,440,375),(30,146,427,374,44,160,441,388),(31,159,428,387,45,145,442,373),(32,144,429,372,46,158,443,386),(33,157,430,385,47,143,444,371),(34,142,431,370,48,156,445,384),(35,155,432,383,49,141,446,369),(36,168,433,368,50,154,447,382),(37,153,434,381,51,167,448,367),(38,166,435,366,52,152,421,380),(39,151,436,379,53,165,422,365),(40,164,437,392,54,150,423,378),(41,149,438,377,55,163,424,391),(42,162,439,390,56,148,425,376),(57,356,123,85,71,342,137,99),(58,341,124,98,72,355,138,112),(59,354,125,111,73,340,139,97),(60,339,126,96,74,353,140,110),(61,352,127,109,75,338,113,95),(62,337,128,94,76,351,114,108),(63,350,129,107,77,364,115,93),(64,363,130,92,78,349,116,106),(65,348,131,105,79,362,117,91),(66,361,132,90,80,347,118,104),(67,346,133,103,81,360,119,89),(68,359,134,88,82,345,120,102),(69,344,135,101,83,358,121,87),(70,357,136,86,84,343,122,100),(197,277,413,227,211,263,399,241),(198,262,414,240,212,276,400,226),(199,275,415,225,213,261,401,239),(200,260,416,238,214,274,402,252),(201,273,417,251,215,259,403,237),(202,258,418,236,216,272,404,250),(203,271,419,249,217,257,405,235),(204,256,420,234,218,270,406,248),(205,269,393,247,219,255,407,233),(206,254,394,232,220,268,408,246),(207,267,395,245,221,253,409,231),(208,280,396,230,222,266,410,244),(209,265,397,243,223,279,411,229),(210,278,398,228,224,264,412,242)]])

88 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 7A 7B 7C 8A ··· 8H 14A ··· 14U 28A ··· 28AJ order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 4 4 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 2 2 2 4 4 4 4 28 28 28 28 2 2 2 14 ··· 14 2 ··· 2 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + + + - + + - + - image C1 C2 C2 C2 C2 C4 D4 Q8 D4 D7 D8 Q16 D14 D14 Dic14 C4×D7 C7⋊D4 C7⋊D4 D4⋊D7 C7⋊Q16 kernel C2×C28.Q8 C28.Q8 C22×C7⋊C8 C2×C4⋊Dic7 C14×C4⋊C4 C2×C7⋊C8 C2×C28 C2×C28 C22×C14 C2×C4⋊C4 C2×C14 C2×C14 C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 1 2 1 3 4 4 6 3 12 12 6 6 6 6

Matrix representation of C2×C28.Q8 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 34 89 0 0 0 0 59 88 0 0 0 0 0 0 103 24 0 0 0 0 89 1 0 0 0 0 0 0 1 36 0 0 0 0 69 112
,
 9 8 0 0 0 0 18 104 0 0 0 0 0 0 98 0 0 0 0 0 0 98 0 0 0 0 0 0 110 111 0 0 0 0 4 3
,
 76 10 0 0 0 0 89 37 0 0 0 0 0 0 112 24 0 0 0 0 0 1 0 0 0 0 0 0 62 99 0 0 0 0 105 0

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[34,59,0,0,0,0,89,88,0,0,0,0,0,0,103,89,0,0,0,0,24,1,0,0,0,0,0,0,1,69,0,0,0,0,36,112],[9,18,0,0,0,0,8,104,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,110,4,0,0,0,0,111,3],[76,89,0,0,0,0,10,37,0,0,0,0,0,0,112,0,0,0,0,0,24,1,0,0,0,0,0,0,62,105,0,0,0,0,99,0] >;

C2×C28.Q8 in GAP, Magma, Sage, TeX

C_2\times C_{28}.Q_8
% in TeX

G:=Group("C2xC28.Q8");
// GroupNames label

G:=SmallGroup(448,496);
// by ID

G=gap.SmallGroup(448,496);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,422,58,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=c^4=1,d^2=b^21*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^15,d*b*d^-1=b^13,d*c*d^-1=b^21*c^-1>;
// generators/relations

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