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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 [×6], C4 [×12], C22 [×7], C2×C4 [×18], Q8 [×16], C23, C13, C22×C4 [×3], C2×Q8 [×12], C26, C26 [×6], C22×Q8, C52 [×12], C2×C26 [×7], C2×C52 [×18], Q8×C13 [×16], C22×C26, C22×C52 [×3], Q8×C26 [×12], Q8×C2×C26
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C13, C2×Q8 [×6], C24, C26 [×15], C22×Q8, C2×C26 [×35], Q8×C13 [×4], C22×C26 [×15], Q8×C26 [×6], C23×C26, Q8×C2×C26

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

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

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

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

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