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## G = C22×C13⋊2C8order 416 = 25·13

### Direct product of C22 and C13⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C22×C13⋊2C8
 Chief series C1 — C13 — C26 — C52 — C13⋊2C8 — C2×C13⋊2C8 — C22×C13⋊2C8
 Lower central C13 — C22×C13⋊2C8
 Upper central C1 — C22×C4

Generators and relations for C22×C132C8
G = < a,b,c,d | a2=b2=c13=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 208 in 76 conjugacy classes, 65 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, C23, C13, C2×C8, C22×C4, C26, C26, C22×C8, C52, C52, C2×C26, C132C8, C2×C52, C22×C26, C2×C132C8, C22×C52, C22×C132C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, D13, C22×C8, Dic13, D26, C132C8, C2×Dic13, C22×D13, C2×C132C8, C22×Dic13, C22×C132C8

Smallest permutation representation of C22×C132C8
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)(40 183)(41 184)(42 185)(43 186)(44 187)(45 188)(46 189)(47 190)(48 191)(49 192)(50 193)(51 194)(52 195)(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)(79 144)(80 145)(81 146)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 153)(89 154)(90 155)(91 156)(92 131)(93 132)(94 133)(95 134)(96 135)(97 136)(98 137)(99 138)(100 139)(101 140)(102 141)(103 142)(104 143)(209 385)(210 386)(211 387)(212 388)(213 389)(214 390)(215 378)(216 379)(217 380)(218 381)(219 382)(220 383)(221 384)(222 373)(223 374)(224 375)(225 376)(226 377)(227 365)(228 366)(229 367)(230 368)(231 369)(232 370)(233 371)(234 372)(235 407)(236 408)(237 409)(238 410)(239 411)(240 412)(241 413)(242 414)(243 415)(244 416)(245 404)(246 405)(247 406)(248 396)(249 397)(250 398)(251 399)(252 400)(253 401)(254 402)(255 403)(256 391)(257 392)(258 393)(259 394)(260 395)(261 328)(262 329)(263 330)(264 331)(265 332)(266 333)(267 334)(268 335)(269 336)(270 337)(271 338)(272 326)(273 327)(274 319)(275 320)(276 321)(277 322)(278 323)(279 324)(280 325)(281 313)(282 314)(283 315)(284 316)(285 317)(286 318)(287 355)(288 356)(289 357)(290 358)(291 359)(292 360)(293 361)(294 362)(295 363)(296 364)(297 352)(298 353)(299 354)(300 347)(301 348)(302 349)(303 350)(304 351)(305 339)(306 340)(307 341)(308 342)(309 343)(310 344)(311 345)(312 346)
(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 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(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)(144 183)(145 184)(146 185)(147 186)(148 187)(149 188)(150 189)(151 190)(152 191)(153 192)(154 193)(155 194)(156 195)(209 284)(210 285)(211 286)(212 274)(213 275)(214 276)(215 277)(216 278)(217 279)(218 280)(219 281)(220 282)(221 283)(222 268)(223 269)(224 270)(225 271)(226 272)(227 273)(228 261)(229 262)(230 263)(231 264)(232 265)(233 266)(234 267)(235 301)(236 302)(237 303)(238 304)(239 305)(240 306)(241 307)(242 308)(243 309)(244 310)(245 311)(246 312)(247 300)(248 293)(249 294)(250 295)(251 296)(252 297)(253 298)(254 299)(255 287)(256 288)(257 289)(258 290)(259 291)(260 292)(313 382)(314 383)(315 384)(316 385)(317 386)(318 387)(319 388)(320 389)(321 390)(322 378)(323 379)(324 380)(325 381)(326 377)(327 365)(328 366)(329 367)(330 368)(331 369)(332 370)(333 371)(334 372)(335 373)(336 374)(337 375)(338 376)(339 411)(340 412)(341 413)(342 414)(343 415)(344 416)(345 404)(346 405)(347 406)(348 407)(349 408)(350 409)(351 410)(352 400)(353 401)(354 402)(355 403)(356 391)(357 392)(358 393)(359 394)(360 395)(361 396)(362 397)(363 398)(364 399)
(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)
(1 404 40 387 14 397 27 375)(2 416 41 386 15 396 28 374)(3 415 42 385 16 395 29 373)(4 414 43 384 17 394 30 372)(5 413 44 383 18 393 31 371)(6 412 45 382 19 392 32 370)(7 411 46 381 20 391 33 369)(8 410 47 380 21 403 34 368)(9 409 48 379 22 402 35 367)(10 408 49 378 23 401 36 366)(11 407 50 390 24 400 37 365)(12 406 51 389 25 399 38 377)(13 405 52 388 26 398 39 376)(53 362 92 337 66 345 79 318)(54 361 93 336 67 344 80 317)(55 360 94 335 68 343 81 316)(56 359 95 334 69 342 82 315)(57 358 96 333 70 341 83 314)(58 357 97 332 71 340 84 313)(59 356 98 331 72 339 85 325)(60 355 99 330 73 351 86 324)(61 354 100 329 74 350 87 323)(62 353 101 328 75 349 88 322)(63 352 102 327 76 348 89 321)(64 364 103 326 77 347 90 320)(65 363 104 338 78 346 91 319)(105 311 144 286 118 294 131 270)(106 310 145 285 119 293 132 269)(107 309 146 284 120 292 133 268)(108 308 147 283 121 291 134 267)(109 307 148 282 122 290 135 266)(110 306 149 281 123 289 136 265)(111 305 150 280 124 288 137 264)(112 304 151 279 125 287 138 263)(113 303 152 278 126 299 139 262)(114 302 153 277 127 298 140 261)(115 301 154 276 128 297 141 273)(116 300 155 275 129 296 142 272)(117 312 156 274 130 295 143 271)(157 249 196 224 170 245 183 211)(158 248 197 223 171 244 184 210)(159 260 198 222 172 243 185 209)(160 259 199 234 173 242 186 221)(161 258 200 233 174 241 187 220)(162 257 201 232 175 240 188 219)(163 256 202 231 176 239 189 218)(164 255 203 230 177 238 190 217)(165 254 204 229 178 237 191 216)(166 253 205 228 179 236 192 215)(167 252 206 227 180 235 193 214)(168 251 207 226 181 247 194 213)(169 250 208 225 182 246 195 212)

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

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

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

128 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 8A ··· 8P 13A ··· 13F 26A ··· 26AP 52A ··· 52AV order 1 2 ··· 2 4 ··· 4 8 ··· 8 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 ··· 1 1 ··· 1 13 ··· 13 2 ··· 2 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 D13 Dic13 D26 Dic13 C13⋊2C8 kernel C22×C13⋊2C8 C2×C13⋊2C8 C22×C52 C2×C52 C22×C26 C2×C26 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 16 6 18 18 6 48

Matrix representation of C22×C132C8 in GL5(𝔽313)

 1 0 0 0 0 0 1 0 0 0 0 0 312 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 312 0 0 0 0 0 312 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 198 312 0 0 0 1 0
,
 188 0 0 0 0 0 312 0 0 0 0 0 312 0 0 0 0 0 36 244 0 0 0 173 277

G:=sub<GL(5,GF(313))| [1,0,0,0,0,0,1,0,0,0,0,0,312,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,312,0,0,0,0,0,312,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,198,1,0,0,0,312,0],[188,0,0,0,0,0,312,0,0,0,0,0,312,0,0,0,0,0,36,173,0,0,0,244,277] >;

C22×C132C8 in GAP, Magma, Sage, TeX

C_2^2\times C_{13}\rtimes_2C_8
% in TeX

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

G:=SmallGroup(416,141);
// by ID

G=gap.SmallGroup(416,141);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,69,13829]);
// 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^-1>;
// generators/relations

׿
×
𝔽