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## G = (C2×C28)⋊Q8order 448 = 26·7

### 6th semidirect product of C2×C28 and Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — (C2×C28)⋊Q8
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C22×Dic14 — (C2×C28)⋊Q8
 Lower central C7 — C2×C14 — (C2×C28)⋊Q8
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C28)⋊Q8
G = < a,b,c,d | a2=b28=c4=1, d2=c2, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b13, dcd-1=c-1 >

Subgroups: 796 in 186 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic7, Dic7, C28, C2×C14, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.67C23, C4×Dic7, Dic7⋊C4, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, C14.C42, C7×C2.C42, C2×C4×Dic7, C2×Dic7⋊C4, C22×Dic14, (C2×C28)⋊Q8
Quotients:

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

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

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

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

88 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 7A 7B 7C 14A ··· 14U 28A ··· 28AJ order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 2 2 2 4 4 4 4 14 ··· 14 28 28 28 28 2 2 2 2 ··· 2 4 ··· 4

88 irreducible representations

 dim 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 D4 Q8 Q8 D7 C4○D4 D14 Dic14 C4×D7 C4○D28 D4×D7 D4⋊2D7 Q8×D7 kernel (C2×C28)⋊Q8 C14.C42 C7×C2.C42 C2×C4×Dic7 C2×Dic7⋊C4 C22×Dic14 C2×Dic14 C2×Dic7 C2×Dic7 C2×C28 C2.C42 C2×C14 C22×C4 C2×C4 C2×C4 C22 C22 C22 C22 # reps 1 3 1 1 1 1 8 4 2 2 3 4 9 12 12 12 6 3 3

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 25 26 0 0 0 0 3 17 0 0 0 0 0 0 7 7 0 0 0 0 15 11 0 0 0 0 0 0 16 23 0 0 0 0 28 13
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 19 6 0 0 0 0 17 10 0 0 0 0 0 0 14 25 0 0 0 0 27 15
,
 4 8 0 0 0 0 9 25 0 0 0 0 0 0 21 4 0 0 0 0 20 8 0 0 0 0 0 0 28 0 0 0 0 0 0 28

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[25,3,0,0,0,0,26,17,0,0,0,0,0,0,7,15,0,0,0,0,7,11,0,0,0,0,0,0,16,28,0,0,0,0,23,13],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,19,17,0,0,0,0,6,10,0,0,0,0,0,0,14,27,0,0,0,0,25,15],[4,9,0,0,0,0,8,25,0,0,0,0,0,0,21,20,0,0,0,0,4,8,0,0,0,0,0,0,28,0,0,0,0,0,0,28] >;

(C2×C28)⋊Q8 in GAP, Magma, Sage, TeX

(C_2\times C_{28})\rtimes Q_8
% in TeX

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

G:=SmallGroup(448,180);
// by ID

G=gap.SmallGroup(448,180);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,387,58,18822]);
// Polycyclic

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

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