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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 [×6], C4, C4 [×3], C22 [×7], C8 [×4], C2×C4 [×6], C23, C13, C2×C8 [×6], C22×C4, C26, C26 [×6], C22×C8, C52, C52 [×3], C2×C26 [×7], C132C8 [×4], C2×C52 [×6], C22×C26, C2×C132C8 [×6], C22×C52, C22×C132C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, D13, C22×C8, Dic13 [×4], D26 [×3], C132C8 [×4], C2×Dic13 [×6], C22×D13, C2×C132C8 [×6], 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)(209 387)(210 388)(211 389)(212 390)(213 378)(214 379)(215 380)(216 381)(217 382)(218 383)(219 384)(220 385)(221 386)(222 367)(223 368)(224 369)(225 370)(226 371)(227 372)(228 373)(229 374)(230 375)(231 376)(232 377)(233 365)(234 366)(235 404)(236 405)(237 406)(238 407)(239 408)(240 409)(241 410)(242 411)(243 412)(244 413)(245 414)(246 415)(247 416)(248 394)(249 395)(250 396)(251 397)(252 398)(253 399)(254 400)(255 401)(256 402)(257 403)(258 391)(259 392)(260 393)(261 333)(262 334)(263 335)(264 336)(265 337)(266 338)(267 326)(268 327)(269 328)(270 329)(271 330)(272 331)(273 332)(274 325)(275 313)(276 314)(277 315)(278 316)(279 317)(280 318)(281 319)(282 320)(283 321)(284 322)(285 323)(286 324)(287 358)(288 359)(289 360)(290 361)(291 362)(292 363)(293 364)(294 352)(295 353)(296 354)(297 355)(298 356)(299 357)(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)(209 282)(210 283)(211 284)(212 285)(213 286)(214 274)(215 275)(216 276)(217 277)(218 278)(219 279)(220 280)(221 281)(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)(235 307)(236 308)(237 309)(238 310)(239 311)(240 312)(241 300)(242 301)(243 302)(244 303)(245 304)(246 305)(247 306)(248 298)(249 299)(250 287)(251 288)(252 289)(253 290)(254 291)(255 292)(256 293)(257 294)(258 295)(259 296)(260 297)(313 380)(314 381)(315 382)(316 383)(317 384)(318 385)(319 386)(320 387)(321 388)(322 389)(323 390)(324 378)(325 379)(326 366)(327 367)(328 368)(329 369)(330 370)(331 371)(332 372)(333 373)(334 374)(335 375)(336 376)(337 377)(338 365)(339 415)(340 416)(341 404)(342 405)(343 406)(344 407)(345 408)(346 409)(347 410)(348 411)(349 412)(350 413)(351 414)(352 403)(353 391)(354 392)(355 393)(356 394)(357 395)(358 396)(359 397)(360 398)(361 399)(362 400)(363 401)(364 402)
(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)
(1 407 40 382 14 403 27 376)(2 406 41 381 15 402 28 375)(3 405 42 380 16 401 29 374)(4 404 43 379 17 400 30 373)(5 416 44 378 18 399 31 372)(6 415 45 390 19 398 32 371)(7 414 46 389 20 397 33 370)(8 413 47 388 21 396 34 369)(9 412 48 387 22 395 35 368)(10 411 49 386 23 394 36 367)(11 410 50 385 24 393 37 366)(12 409 51 384 25 392 38 365)(13 408 52 383 26 391 39 377)(53 352 92 336 66 344 79 315)(54 364 93 335 67 343 80 314)(55 363 94 334 68 342 81 313)(56 362 95 333 69 341 82 325)(57 361 96 332 70 340 83 324)(58 360 97 331 71 339 84 323)(59 359 98 330 72 351 85 322)(60 358 99 329 73 350 86 321)(61 357 100 328 74 349 87 320)(62 356 101 327 75 348 88 319)(63 355 102 326 76 347 89 318)(64 354 103 338 77 346 90 317)(65 353 104 337 78 345 91 316)(105 310 144 277 118 294 131 264)(106 309 145 276 119 293 132 263)(107 308 146 275 120 292 133 262)(108 307 147 274 121 291 134 261)(109 306 148 286 122 290 135 273)(110 305 149 285 123 289 136 272)(111 304 150 284 124 288 137 271)(112 303 151 283 125 287 138 270)(113 302 152 282 126 299 139 269)(114 301 153 281 127 298 140 268)(115 300 154 280 128 297 141 267)(116 312 155 279 129 296 142 266)(117 311 156 278 130 295 143 265)(157 257 196 231 170 238 183 217)(158 256 197 230 171 237 184 216)(159 255 198 229 172 236 185 215)(160 254 199 228 173 235 186 214)(161 253 200 227 174 247 187 213)(162 252 201 226 175 246 188 212)(163 251 202 225 176 245 189 211)(164 250 203 224 177 244 190 210)(165 249 204 223 178 243 191 209)(166 248 205 222 179 242 192 221)(167 260 206 234 180 241 193 220)(168 259 207 233 181 240 194 219)(169 258 208 232 182 239 195 218)

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

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

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

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