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## G = C22×C13⋊C8order 416 = 25·13

### Direct product of C22 and C13⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C22×C13⋊C8
 Chief series C1 — C13 — C26 — Dic13 — C13⋊C8 — C2×C13⋊C8 — C22×C13⋊C8
 Lower central C13 — C22×C13⋊C8
 Upper central C1 — C23

Generators and relations for C22×C13⋊C8
G = < a,b,c,d | a2=b2=c13=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 340 in 76 conjugacy classes, 54 normal (9 characteristic)
C1, C2, C2, C4, C22, C8, C2×C4, C23, C13, C2×C8, C22×C4, C26, C26, C22×C8, Dic13, Dic13, C2×C26, C13⋊C8, C2×Dic13, C22×C26, C2×C13⋊C8, C22×Dic13, C22×C13⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C22×C8, C13⋊C4, C13⋊C8, C2×C13⋊C4, C2×C13⋊C8, C22×C13⋊C4, C22×C13⋊C8

Smallest permutation representation of C22×C13⋊C8
Regular action on 416 points
Generators in S416
(1 170)(2 171)(3 172)(4 173)(5 174)(6 175)(7 176)(8 177)(9 178)(10 179)(11 180)(12 181)(13 182)(14 157)(15 158)(16 159)(17 160)(18 161)(19 162)(20 163)(21 164)(22 165)(23 166)(24 167)(25 168)(26 169)(27 196)(28 197)(29 198)(30 199)(31 200)(32 201)(33 202)(34 203)(35 204)(36 205)(37 206)(38 207)(39 208)(40 187)(41 188)(42 189)(43 190)(44 191)(45 192)(46 193)(47 194)(48 195)(49 183)(50 184)(51 185)(52 186)(53 118)(54 119)(55 120)(56 121)(57 122)(58 123)(59 124)(60 125)(61 126)(62 127)(63 128)(64 129)(65 130)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 156)(80 144)(81 145)(82 146)(83 147)(84 148)(85 149)(86 150)(87 151)(88 152)(89 153)(90 154)(91 155)(92 136)(93 137)(94 138)(95 139)(96 140)(97 141)(98 142)(99 143)(100 131)(101 132)(102 133)(103 134)(104 135)(209 390)(210 378)(211 379)(212 380)(213 381)(214 382)(215 383)(216 384)(217 385)(218 386)(219 387)(220 388)(221 389)(222 369)(223 370)(224 371)(225 372)(226 373)(227 374)(228 375)(229 376)(230 377)(231 365)(232 366)(233 367)(234 368)(235 406)(236 407)(237 408)(238 409)(239 410)(240 411)(241 412)(242 413)(243 414)(244 415)(245 416)(246 404)(247 405)(248 403)(249 391)(250 392)(251 393)(252 394)(253 395)(254 396)(255 397)(256 398)(257 399)(258 400)(259 401)(260 402)(261 331)(262 332)(263 333)(264 334)(265 335)(266 336)(267 337)(268 338)(269 326)(270 327)(271 328)(272 329)(273 330)(274 324)(275 325)(276 313)(277 314)(278 315)(279 316)(280 317)(281 318)(282 319)(283 320)(284 321)(285 322)(286 323)(287 363)(288 364)(289 352)(290 353)(291 354)(292 355)(293 356)(294 357)(295 358)(296 359)(297 360)(298 361)(299 362)(300 342)(301 343)(302 344)(303 345)(304 346)(305 347)(306 348)(307 349)(308 350)(309 351)(310 339)(311 340)(312 341)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 100)(28 101)(29 102)(30 103)(31 104)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 87)(41 88)(42 89)(43 90)(44 91)(45 79)(46 80)(47 81)(48 82)(49 83)(50 84)(51 85)(52 86)(105 170)(106 171)(107 172)(108 173)(109 174)(110 175)(111 176)(112 177)(113 178)(114 179)(115 180)(116 181)(117 182)(118 157)(119 158)(120 159)(121 160)(122 161)(123 162)(124 163)(125 164)(126 165)(127 166)(128 167)(129 168)(130 169)(131 196)(132 197)(133 198)(134 199)(135 200)(136 201)(137 202)(138 203)(139 204)(140 205)(141 206)(142 207)(143 208)(144 193)(145 194)(146 195)(147 183)(148 184)(149 185)(150 186)(151 187)(152 188)(153 189)(154 190)(155 191)(156 192)(209 278)(210 279)(211 280)(212 281)(213 282)(214 283)(215 284)(216 285)(217 286)(218 274)(219 275)(220 276)(221 277)(222 269)(223 270)(224 271)(225 272)(226 273)(227 261)(228 262)(229 263)(230 264)(231 265)(232 266)(233 267)(234 268)(235 303)(236 304)(237 305)(238 306)(239 307)(240 308)(241 309)(242 310)(243 311)(244 312)(245 300)(246 301)(247 302)(248 290)(249 291)(250 292)(251 293)(252 294)(253 295)(254 296)(255 297)(256 298)(257 299)(258 287)(259 288)(260 289)(313 388)(314 389)(315 390)(316 378)(317 379)(318 380)(319 381)(320 382)(321 383)(322 384)(323 385)(324 386)(325 387)(326 369)(327 370)(328 371)(329 372)(330 373)(331 374)(332 375)(333 376)(334 377)(335 365)(336 366)(337 367)(338 368)(339 413)(340 414)(341 415)(342 416)(343 404)(344 405)(345 406)(346 407)(347 408)(348 409)(349 410)(350 411)(351 412)(352 402)(353 403)(354 391)(355 392)(356 393)(357 394)(358 395)(359 396)(360 397)(361 398)(362 399)(363 400)(364 401)
(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)
(1 304 42 282 14 294 34 271)(2 312 41 274 15 289 33 263)(3 307 40 279 16 297 32 268)(4 302 52 284 17 292 31 273)(5 310 51 276 18 287 30 265)(6 305 50 281 19 295 29 270)(7 300 49 286 20 290 28 262)(8 308 48 278 21 298 27 267)(9 303 47 283 22 293 39 272)(10 311 46 275 23 288 38 264)(11 306 45 280 24 296 37 269)(12 301 44 285 25 291 36 261)(13 309 43 277 26 299 35 266)(53 252 94 224 66 236 89 213)(54 260 93 229 67 244 88 218)(55 255 92 234 68 239 87 210)(56 250 104 226 69 247 86 215)(57 258 103 231 70 242 85 220)(58 253 102 223 71 237 84 212)(59 248 101 228 72 245 83 217)(60 256 100 233 73 240 82 209)(61 251 99 225 74 235 81 214)(62 259 98 230 75 243 80 219)(63 254 97 222 76 238 79 211)(64 249 96 227 77 246 91 216)(65 257 95 232 78 241 90 221)(105 407 153 381 118 394 138 371)(106 415 152 386 119 402 137 376)(107 410 151 378 120 397 136 368)(108 405 150 383 121 392 135 373)(109 413 149 388 122 400 134 365)(110 408 148 380 123 395 133 370)(111 416 147 385 124 403 132 375)(112 411 146 390 125 398 131 367)(113 406 145 382 126 393 143 372)(114 414 144 387 127 401 142 377)(115 409 156 379 128 396 141 369)(116 404 155 384 129 391 140 374)(117 412 154 389 130 399 139 366)(157 357 203 328 170 346 189 319)(158 352 202 333 171 341 188 324)(159 360 201 338 172 349 187 316)(160 355 200 330 173 344 186 321)(161 363 199 335 174 339 185 313)(162 358 198 327 175 347 184 318)(163 353 197 332 176 342 183 323)(164 361 196 337 177 350 195 315)(165 356 208 329 178 345 194 320)(166 364 207 334 179 340 193 325)(167 359 206 326 180 348 192 317)(168 354 205 331 181 343 191 322)(169 362 204 336 182 351 190 314)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 8A ··· 8P 13A 13B 13C 26A ··· 26U order 1 2 ··· 2 4 ··· 4 8 ··· 8 13 13 13 26 ··· 26 size 1 1 ··· 1 13 ··· 13 13 ··· 13 4 4 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 4 4 4 type + + + + - + image C1 C2 C2 C4 C4 C8 C13⋊C4 C13⋊C8 C2×C13⋊C4 kernel C22×C13⋊C8 C2×C13⋊C8 C22×Dic13 C2×Dic13 C22×C26 C2×C26 C23 C22 C22 # reps 1 6 1 6 2 16 3 12 9

Matrix representation of C22×C13⋊C8 in GL7(𝔽313)

 1 0 0 0 0 0 0 0 312 0 0 0 0 0 0 0 312 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 312 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 312 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 213 100 284 312 0 0 0 214 100 284 312 0 0 0 213 101 284 312 0 0 0 213 100 285 312
,
 188 0 0 0 0 0 0 0 312 0 0 0 0 0 0 0 312 0 0 0 0 0 0 0 141 142 38 260 0 0 0 14 115 178 131 0 0 0 207 292 154 30 0 0 0 131 304 22 216

G:=sub<GL(7,GF(313))| [1,0,0,0,0,0,0,0,312,0,0,0,0,0,0,0,312,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[312,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,312,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,213,214,213,213,0,0,0,100,100,101,100,0,0,0,284,284,284,285,0,0,0,312,312,312,312],[188,0,0,0,0,0,0,0,312,0,0,0,0,0,0,0,312,0,0,0,0,0,0,0,141,14,207,131,0,0,0,142,115,292,304,0,0,0,38,178,154,22,0,0,0,260,131,30,216] >;

C22×C13⋊C8 in GAP, Magma, Sage, TeX

C_2^2\times C_{13}\rtimes C_8
% in TeX

G:=Group("C2^2xC13:C8");
// GroupNames label

G:=SmallGroup(416,209);
// by ID

G=gap.SmallGroup(416,209);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,69,9221,1751]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^13=d^8=1,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^5>;
// generators/relations

׿
×
𝔽