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## G = Q8×C2×C26order 416 = 25·13

### Direct product of C2×C26 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C2×C26
 Chief series C1 — C2 — C26 — C52 — Q8×C13 — Q8×C26 — Q8×C2×C26
 Lower central C1 — C2 — Q8×C2×C26
 Upper central C1 — C22×C26 — Q8×C2×C26

Generators and relations for Q8×C2×C26
G = < a,b,c,d | a2=b26=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 156, all normal (8 characteristic)
C1, C2, C2, C4, C22, C2×C4, Q8, C23, C13, C22×C4, C2×Q8, C26, C26, C22×Q8, C52, C2×C26, C2×C52, Q8×C13, C22×C26, C22×C52, Q8×C26, Q8×C2×C26
Quotients: C1, C2, C22, Q8, C23, C13, C2×Q8, C24, C26, C22×Q8, C2×C26, Q8×C13, C22×C26, Q8×C26, C23×C26, Q8×C2×C26

Smallest permutation representation of Q8×C2×C26
Regular action on 416 points
Generators in S416
(1 385)(2 386)(3 387)(4 388)(5 389)(6 390)(7 365)(8 366)(9 367)(10 368)(11 369)(12 370)(13 371)(14 372)(15 373)(16 374)(17 375)(18 376)(19 377)(20 378)(21 379)(22 380)(23 381)(24 382)(25 383)(26 384)(27 176)(28 177)(29 178)(30 179)(31 180)(32 181)(33 182)(34 157)(35 158)(36 159)(37 160)(38 161)(39 162)(40 163)(41 164)(42 165)(43 166)(44 167)(45 168)(46 169)(47 170)(48 171)(49 172)(50 173)(51 174)(52 175)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)(62 123)(63 124)(64 125)(65 126)(66 127)(67 128)(68 129)(69 130)(70 105)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 113)(79 229)(80 230)(81 231)(82 232)(83 233)(84 234)(85 209)(86 210)(87 211)(88 212)(89 213)(90 214)(91 215)(92 216)(93 217)(94 218)(95 219)(96 220)(97 221)(98 222)(99 223)(100 224)(101 225)(102 226)(103 227)(104 228)(131 342)(132 343)(133 344)(134 345)(135 346)(136 347)(137 348)(138 349)(139 350)(140 351)(141 352)(142 353)(143 354)(144 355)(145 356)(146 357)(147 358)(148 359)(149 360)(150 361)(151 362)(152 363)(153 364)(154 339)(155 340)(156 341)(183 301)(184 302)(185 303)(186 304)(187 305)(188 306)(189 307)(190 308)(191 309)(192 310)(193 311)(194 312)(195 287)(196 288)(197 289)(198 290)(199 291)(200 292)(201 293)(202 294)(203 295)(204 296)(205 297)(206 298)(207 299)(208 300)(235 408)(236 409)(237 410)(238 411)(239 412)(240 413)(241 414)(242 415)(243 416)(244 391)(245 392)(246 393)(247 394)(248 395)(249 396)(250 397)(251 398)(252 399)(253 400)(254 401)(255 402)(256 403)(257 404)(258 405)(259 406)(260 407)(261 326)(262 327)(263 328)(264 329)(265 330)(266 331)(267 332)(268 333)(269 334)(270 335)(271 336)(272 337)(273 338)(274 313)(275 314)(276 315)(277 316)(278 317)(279 318)(280 319)(281 320)(282 321)(283 322)(284 323)(285 324)(286 325)
(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)
(1 57 294 334)(2 58 295 335)(3 59 296 336)(4 60 297 337)(5 61 298 338)(6 62 299 313)(7 63 300 314)(8 64 301 315)(9 65 302 316)(10 66 303 317)(11 67 304 318)(12 68 305 319)(13 69 306 320)(14 70 307 321)(15 71 308 322)(16 72 309 323)(17 73 310 324)(18 74 311 325)(19 75 312 326)(20 76 287 327)(21 77 288 328)(22 78 289 329)(23 53 290 330)(24 54 291 331)(25 55 292 332)(26 56 293 333)(27 98 412 143)(28 99 413 144)(29 100 414 145)(30 101 415 146)(31 102 416 147)(32 103 391 148)(33 104 392 149)(34 79 393 150)(35 80 394 151)(36 81 395 152)(37 82 396 153)(38 83 397 154)(39 84 398 155)(40 85 399 156)(41 86 400 131)(42 87 401 132)(43 88 402 133)(44 89 403 134)(45 90 404 135)(46 91 405 136)(47 92 406 137)(48 93 407 138)(49 94 408 139)(50 95 409 140)(51 96 410 141)(52 97 411 142)(105 189 282 372)(106 190 283 373)(107 191 284 374)(108 192 285 375)(109 193 286 376)(110 194 261 377)(111 195 262 378)(112 196 263 379)(113 197 264 380)(114 198 265 381)(115 199 266 382)(116 200 267 383)(117 201 268 384)(118 202 269 385)(119 203 270 386)(120 204 271 387)(121 205 272 388)(122 206 273 389)(123 207 274 390)(124 208 275 365)(125 183 276 366)(126 184 277 367)(127 185 278 368)(128 186 279 369)(129 187 280 370)(130 188 281 371)(157 229 246 361)(158 230 247 362)(159 231 248 363)(160 232 249 364)(161 233 250 339)(162 234 251 340)(163 209 252 341)(164 210 253 342)(165 211 254 343)(166 212 255 344)(167 213 256 345)(168 214 257 346)(169 215 258 347)(170 216 259 348)(171 217 260 349)(172 218 235 350)(173 219 236 351)(174 220 237 352)(175 221 238 353)(176 222 239 354)(177 223 240 355)(178 224 241 356)(179 225 242 357)(180 226 243 358)(181 227 244 359)(182 228 245 360)
(1 35 294 394)(2 36 295 395)(3 37 296 396)(4 38 297 397)(5 39 298 398)(6 40 299 399)(7 41 300 400)(8 42 301 401)(9 43 302 402)(10 44 303 403)(11 45 304 404)(12 46 305 405)(13 47 306 406)(14 48 307 407)(15 49 308 408)(16 50 309 409)(17 51 310 410)(18 52 311 411)(19 27 312 412)(20 28 287 413)(21 29 288 414)(22 30 289 415)(23 31 290 416)(24 32 291 391)(25 33 292 392)(26 34 293 393)(53 147 330 102)(54 148 331 103)(55 149 332 104)(56 150 333 79)(57 151 334 80)(58 152 335 81)(59 153 336 82)(60 154 337 83)(61 155 338 84)(62 156 313 85)(63 131 314 86)(64 132 315 87)(65 133 316 88)(66 134 317 89)(67 135 318 90)(68 136 319 91)(69 137 320 92)(70 138 321 93)(71 139 322 94)(72 140 323 95)(73 141 324 96)(74 142 325 97)(75 143 326 98)(76 144 327 99)(77 145 328 100)(78 146 329 101)(105 349 282 217)(106 350 283 218)(107 351 284 219)(108 352 285 220)(109 353 286 221)(110 354 261 222)(111 355 262 223)(112 356 263 224)(113 357 264 225)(114 358 265 226)(115 359 266 227)(116 360 267 228)(117 361 268 229)(118 362 269 230)(119 363 270 231)(120 364 271 232)(121 339 272 233)(122 340 273 234)(123 341 274 209)(124 342 275 210)(125 343 276 211)(126 344 277 212)(127 345 278 213)(128 346 279 214)(129 347 280 215)(130 348 281 216)(157 201 246 384)(158 202 247 385)(159 203 248 386)(160 204 249 387)(161 205 250 388)(162 206 251 389)(163 207 252 390)(164 208 253 365)(165 183 254 366)(166 184 255 367)(167 185 256 368)(168 186 257 369)(169 187 258 370)(170 188 259 371)(171 189 260 372)(172 190 235 373)(173 191 236 374)(174 192 237 375)(175 193 238 376)(176 194 239 377)(177 195 240 378)(178 196 241 379)(179 197 242 380)(180 198 243 381)(181 199 244 382)(182 200 245 383)

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

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

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

260 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 13A ··· 13L 26A ··· 26CF 52A ··· 52EN order 1 2 ··· 2 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

260 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + - image C1 C2 C2 C13 C26 C26 Q8 Q8×C13 kernel Q8×C2×C26 C22×C52 Q8×C26 C22×Q8 C22×C4 C2×Q8 C2×C26 C22 # reps 1 3 12 12 36 144 4 48

Matrix representation of Q8×C2×C26 in GL4(𝔽53) generated by

 52 0 0 0 0 1 0 0 0 0 52 0 0 0 0 52
,
 1 0 0 0 0 52 0 0 0 0 11 0 0 0 0 11
,
 1 0 0 0 0 52 0 0 0 0 1 2 0 0 52 52
,
 52 0 0 0 0 1 0 0 0 0 6 48 0 0 18 47
G:=sub<GL(4,GF(53))| [52,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[1,0,0,0,0,52,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,52,0,0,0,0,1,52,0,0,2,52],[52,0,0,0,0,1,0,0,0,0,6,18,0,0,48,47] >;

Q8×C2×C26 in GAP, Magma, Sage, TeX

Q_8\times C_2\times C_{26}
% in TeX

G:=Group("Q8xC2xC26");
// GroupNames label

G:=SmallGroup(416,229);
// by ID

G=gap.SmallGroup(416,229);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-13,-2,1248,2521,1255]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^26=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽