Copied to
clipboard

G = C13×C2.C42order 416 = 25·13

Direct product of C13 and C2.C42

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

Aliases: C13×C2.C42, C26.11C42, (C2×C52)⋊9C4, (C2×C4)⋊2C52, C2.1(C4×C52), (C2×C26).7Q8, (C2×C26).45D4, C26.17(C4⋊C4), (C22×C52).2C2, C22.7(C2×C52), (C22×C4).1C26, C22.7(D4×C13), C22.2(Q8×C13), C23.12(C2×C26), C26.30(C22⋊C4), (C22×C26).48C22, C2.1(C13×C4⋊C4), (C2×C26).56(C2×C4), C2.1(C13×C22⋊C4), SmallGroup(416,45)

Series: Derived Chief Lower central Upper central

C1C2 — C13×C2.C42
C1C2C22C23C22×C26C22×C52 — C13×C2.C42
C1C2 — C13×C2.C42
C1C22×C26 — C13×C2.C42

Generators and relations for C13×C2.C42
 G = < a,b,c,d | a13=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 100 in 76 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2 [×6], C4 [×6], C22, C22 [×6], C2×C4 [×6], C2×C4 [×6], C23, C13, C22×C4 [×3], C26, C26 [×6], C2.C42, C52 [×6], C2×C26, C2×C26 [×6], C2×C52 [×6], C2×C52 [×6], C22×C26, C22×C52 [×3], C13×C2.C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C13, C42, C22⋊C4 [×3], C4⋊C4 [×3], C26 [×3], C2.C42, C52 [×6], C2×C26, C2×C52 [×3], D4×C13 [×3], Q8×C13, C4×C52, C13×C22⋊C4 [×3], C13×C4⋊C4 [×3], C13×C2.C42

Smallest permutation representation of C13×C2.C42
Regular action on 416 points
Generators in S416
(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)
(1 127)(2 128)(3 129)(4 130)(5 118)(6 119)(7 120)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 132)(15 133)(16 134)(17 135)(18 136)(19 137)(20 138)(21 139)(22 140)(23 141)(24 142)(25 143)(26 131)(27 336)(28 337)(29 338)(30 326)(31 327)(32 328)(33 329)(34 330)(35 331)(36 332)(37 333)(38 334)(39 335)(40 379)(41 380)(42 381)(43 382)(44 383)(45 384)(46 385)(47 386)(48 387)(49 388)(50 389)(51 390)(52 378)(53 113)(54 114)(55 115)(56 116)(57 117)(58 105)(59 106)(60 107)(61 108)(62 109)(63 110)(64 111)(65 112)(66 209)(67 210)(68 211)(69 212)(70 213)(71 214)(72 215)(73 216)(74 217)(75 218)(76 219)(77 220)(78 221)(79 166)(80 167)(81 168)(82 169)(83 157)(84 158)(85 159)(86 160)(87 161)(88 162)(89 163)(90 164)(91 165)(92 402)(93 403)(94 391)(95 392)(96 393)(97 394)(98 395)(99 396)(100 397)(101 398)(102 399)(103 400)(104 401)(144 350)(145 351)(146 339)(147 340)(148 341)(149 342)(150 343)(151 344)(152 345)(153 346)(154 347)(155 348)(156 349)(170 192)(171 193)(172 194)(173 195)(174 183)(175 184)(176 185)(177 186)(178 187)(179 188)(180 189)(181 190)(182 191)(196 296)(197 297)(198 298)(199 299)(200 287)(201 288)(202 289)(203 290)(204 291)(205 292)(206 293)(207 294)(208 295)(222 259)(223 260)(224 248)(225 249)(226 250)(227 251)(228 252)(229 253)(230 254)(231 255)(232 256)(233 257)(234 258)(235 279)(236 280)(237 281)(238 282)(239 283)(240 284)(241 285)(242 286)(243 274)(244 275)(245 276)(246 277)(247 278)(261 366)(262 367)(263 368)(264 369)(265 370)(266 371)(267 372)(268 373)(269 374)(270 375)(271 376)(272 377)(273 365)(300 322)(301 323)(302 324)(303 325)(304 313)(305 314)(306 315)(307 316)(308 317)(309 318)(310 319)(311 320)(312 321)(352 409)(353 410)(354 411)(355 412)(356 413)(357 414)(358 415)(359 416)(360 404)(361 405)(362 406)(363 407)(364 408)
(1 224 98 355)(2 225 99 356)(3 226 100 357)(4 227 101 358)(5 228 102 359)(6 229 103 360)(7 230 104 361)(8 231 92 362)(9 232 93 363)(10 233 94 364)(11 234 95 352)(12 222 96 353)(13 223 97 354)(14 334 61 220)(15 335 62 221)(16 336 63 209)(17 337 64 210)(18 338 65 211)(19 326 53 212)(20 327 54 213)(21 328 55 214)(22 329 56 215)(23 330 57 216)(24 331 58 217)(25 332 59 218)(26 333 60 219)(27 110 66 134)(28 111 67 135)(29 112 68 136)(30 113 69 137)(31 114 70 138)(32 115 71 139)(33 116 72 140)(34 117 73 141)(35 105 74 142)(36 106 75 143)(37 107 76 131)(38 108 77 132)(39 109 78 133)(40 305 206 189)(41 306 207 190)(42 307 208 191)(43 308 196 192)(44 309 197 193)(45 310 198 194)(46 311 199 195)(47 312 200 183)(48 300 201 184)(49 301 202 185)(50 302 203 186)(51 303 204 187)(52 304 205 188)(79 285 265 147)(80 286 266 148)(81 274 267 149)(82 275 268 150)(83 276 269 151)(84 277 270 152)(85 278 271 153)(86 279 272 154)(87 280 273 155)(88 281 261 156)(89 282 262 144)(90 283 263 145)(91 284 264 146)(118 252 399 416)(119 253 400 404)(120 254 401 405)(121 255 402 406)(122 256 403 407)(123 257 391 408)(124 258 392 409)(125 259 393 410)(126 260 394 411)(127 248 395 412)(128 249 396 413)(129 250 397 414)(130 251 398 415)(157 245 374 344)(158 246 375 345)(159 247 376 346)(160 235 377 347)(161 236 365 348)(162 237 366 349)(163 238 367 350)(164 239 368 351)(165 240 369 339)(166 241 370 340)(167 242 371 341)(168 243 372 342)(169 244 373 343)(170 382 317 296)(171 383 318 297)(172 384 319 298)(173 385 320 299)(174 386 321 287)(175 387 322 288)(176 388 323 289)(177 389 324 290)(178 390 325 291)(179 378 313 292)(180 379 314 293)(181 380 315 294)(182 381 316 295)
(1 271 183 56)(2 272 184 57)(3 273 185 58)(4 261 186 59)(5 262 187 60)(6 263 188 61)(7 264 189 62)(8 265 190 63)(9 266 191 64)(10 267 192 65)(11 268 193 53)(12 269 194 54)(13 270 195 55)(14 103 90 304)(15 104 91 305)(16 92 79 306)(17 93 80 307)(18 94 81 308)(19 95 82 309)(20 96 83 310)(21 97 84 311)(22 98 85 312)(23 99 86 300)(24 100 87 301)(25 101 88 302)(26 102 89 303)(27 362 241 207)(28 363 242 208)(29 364 243 196)(30 352 244 197)(31 353 245 198)(32 354 246 199)(33 355 247 200)(34 356 235 201)(35 357 236 202)(36 358 237 203)(37 359 238 204)(38 360 239 205)(39 361 240 206)(40 78 230 339)(41 66 231 340)(42 67 232 341)(43 68 233 342)(44 69 234 343)(45 70 222 344)(46 71 223 345)(47 72 224 346)(48 73 225 347)(49 74 226 348)(50 75 227 349)(51 76 228 350)(52 77 229 351)(105 129 365 176)(106 130 366 177)(107 118 367 178)(108 119 368 179)(109 120 369 180)(110 121 370 181)(111 122 371 182)(112 123 372 170)(113 124 373 171)(114 125 374 172)(115 126 375 173)(116 127 376 174)(117 128 377 175)(131 399 163 325)(132 400 164 313)(133 401 165 314)(134 402 166 315)(135 403 167 316)(136 391 168 317)(137 392 169 318)(138 393 157 319)(139 394 158 320)(140 395 159 321)(141 396 160 322)(142 397 161 323)(143 398 162 324)(144 390 219 252)(145 378 220 253)(146 379 221 254)(147 380 209 255)(148 381 210 256)(149 382 211 257)(150 383 212 258)(151 384 213 259)(152 385 214 260)(153 386 215 248)(154 387 216 249)(155 388 217 250)(156 389 218 251)(274 296 338 408)(275 297 326 409)(276 298 327 410)(277 299 328 411)(278 287 329 412)(279 288 330 413)(280 289 331 414)(281 290 332 415)(282 291 333 416)(283 292 334 404)(284 293 335 405)(285 294 336 406)(286 295 337 407)

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

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

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

260 conjugacy classes

class 1 2A···2G4A···4L13A···13L26A···26CF52A···52EN
order12···24···413···1326···2652···52
size11···12···21···11···12···2

260 irreducible representations

dim1111112222
type+++-
imageC1C2C4C13C26C52D4Q8D4×C13Q8×C13
kernelC13×C2.C42C22×C52C2×C52C2.C42C22×C4C2×C4C2×C26C2×C26C22C22
# reps13121236144313612

Matrix representation of C13×C2.C42 in GL4(𝔽53) generated by

1000
0100
00240
00024
,
1000
0100
00520
00052
,
52000
02300
00421
003249
,
23000
03000
0001
0010
G:=sub<GL(4,GF(53))| [1,0,0,0,0,1,0,0,0,0,24,0,0,0,0,24],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,23,0,0,0,0,4,32,0,0,21,49],[23,0,0,0,0,30,0,0,0,0,0,1,0,0,1,0] >;

C13×C2.C42 in GAP, Magma, Sage, TeX

C_{13}\times C_2.C_4^2
% in TeX

G:=Group("C13xC2.C4^2");
// GroupNames label

G:=SmallGroup(416,45);
// by ID

G=gap.SmallGroup(416,45);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,1255]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^2=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations

׿
×
𝔽