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## G = C2×C28.6Q8order 448 = 26·7

### Direct product of C2 and C28.6Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C2×C28.6Q8
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C2×Dic7⋊C4 — C2×C28.6Q8
 Lower central C7 — C2×C14 — C2×C28.6Q8
 Upper central C1 — C23 — C2×C42

Generators and relations for C2×C28.6Q8
G = < a,b,c,d | a2=b28=c4=1, d2=b14c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b14c-1 >

Subgroups: 772 in 226 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C42, C2×C4⋊C4, C42.C2, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C2×C42.C2, Dic7⋊C4, C4⋊Dic7, C4×C28, C22×Dic7, C22×C28, C22×C28, C28.6Q8, C2×Dic7⋊C4, C2×C4⋊Dic7, C2×C4×C28, C2×C28.6Q8
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C4○D4, C24, D14, C42.C2, C22×Q8, C2×C4○D4, Dic14, C22×D7, C2×C42.C2, C2×Dic14, C4○D28, C23×D7, C28.6Q8, C22×Dic14, C2×C4○D28, C2×C28.6Q8

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

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

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

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

124 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 7A 7B 7C 14A ··· 14U 28A ··· 28BT order 1 2 ··· 2 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 ··· 2 28 ··· 28 2 2 2 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + - + + + - image C1 C2 C2 C2 C2 Q8 D7 C4○D4 D14 D14 Dic14 C4○D28 kernel C2×C28.6Q8 C28.6Q8 C2×Dic7⋊C4 C2×C4⋊Dic7 C2×C4×C28 C2×C28 C2×C42 C2×C14 C42 C22×C4 C2×C4 C22 # reps 1 8 4 2 1 4 3 8 12 9 24 48

Matrix representation of C2×C28.6Q8 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 10 7 0 0 0 0 22 1 0 0 0 0 0 0 9 4 0 0 0 0 25 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 23 22 0 0 0 0 26 6
,
 23 21 0 0 0 0 8 6 0 0 0 0 0 0 22 1 0 0 0 0 10 7 0 0 0 0 0 0 22 24 0 0 0 0 10 7

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[10,22,0,0,0,0,7,1,0,0,0,0,0,0,9,25,0,0,0,0,4,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,23,26,0,0,0,0,22,6],[23,8,0,0,0,0,21,6,0,0,0,0,0,0,22,10,0,0,0,0,1,7,0,0,0,0,0,0,22,10,0,0,0,0,24,7] >;

C2×C28.6Q8 in GAP, Magma, Sage, TeX

C_2\times C_{28}._6Q_8
% in TeX

G:=Group("C2xC28.6Q8");
// GroupNames label

G:=SmallGroup(448,922);
// by ID

G=gap.SmallGroup(448,922);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,100,675,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=c^4=1,d^2=b^14*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^14*c^-1>;
// generators/relations

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