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## G = C2×C13⋊Q16order 416 = 25·13

### Direct product of C2 and C13⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C13⋊Q16
 Chief series C1 — C13 — C26 — C52 — Dic26 — C2×Dic26 — C2×C13⋊Q16
 Lower central C13 — C26 — C52 — C2×C13⋊Q16
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for C2×C13⋊Q16
G = < a,b,c,d | a2=b13=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 304 in 60 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×4], C22, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], Q8 [×4], C13, C2×C8, Q16 [×4], C2×Q8, C2×Q8, C26, C26 [×2], C2×Q16, Dic13 [×2], C52 [×2], C52 [×2], C2×C26, C132C8 [×2], Dic26 [×2], Dic26, C2×Dic13, C2×C52, C2×C52, Q8×C13 [×2], Q8×C13, C2×C132C8, C13⋊Q16 [×4], C2×Dic26, Q8×C26, C2×C13⋊Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, Q16 [×2], C2×D4, D13, C2×Q16, D26 [×3], C13⋊D4 [×2], C22×D13, C13⋊Q16 [×2], C2×C13⋊D4, C2×C13⋊Q16

Smallest permutation representation of C2×C13⋊Q16
Regular action on 416 points
Generators in S416
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 141)(38 142)(39 143)(40 144)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 161)(58 162)(59 163)(60 164)(61 165)(62 166)(63 167)(64 168)(65 169)(66 170)(67 171)(68 172)(69 173)(70 174)(71 175)(72 176)(73 177)(74 178)(75 179)(76 180)(77 181)(78 182)(79 183)(80 184)(81 185)(82 186)(83 187)(84 188)(85 189)(86 190)(87 191)(88 192)(89 193)(90 194)(91 195)(92 196)(93 197)(94 198)(95 199)(96 200)(97 201)(98 202)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)(209 325)(210 313)(211 314)(212 315)(213 316)(214 317)(215 318)(216 319)(217 320)(218 321)(219 322)(220 323)(221 324)(222 328)(223 329)(224 330)(225 331)(226 332)(227 333)(228 334)(229 335)(230 336)(231 337)(232 338)(233 326)(234 327)(235 349)(236 350)(237 351)(238 339)(239 340)(240 341)(241 342)(242 343)(243 344)(244 345)(245 346)(246 347)(247 348)(248 353)(249 354)(250 355)(251 356)(252 357)(253 358)(254 359)(255 360)(256 361)(257 362)(258 363)(259 364)(260 352)(261 375)(262 376)(263 377)(264 365)(265 366)(266 367)(267 368)(268 369)(269 370)(270 371)(271 372)(272 373)(273 374)(274 378)(275 379)(276 380)(277 381)(278 382)(279 383)(280 384)(281 385)(282 386)(283 387)(284 388)(285 389)(286 390)(287 402)(288 403)(289 391)(290 392)(291 393)(292 394)(293 395)(294 396)(295 397)(296 398)(297 399)(298 400)(299 401)(300 410)(301 411)(302 412)(303 413)(304 414)(305 415)(306 416)(307 404)(308 405)(309 406)(310 407)(311 408)(312 409)
(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)
(1 293 27 285 14 311 40 268)(2 292 28 284 15 310 41 267)(3 291 29 283 16 309 42 266)(4 290 30 282 17 308 43 265)(5 289 31 281 18 307 44 264)(6 288 32 280 19 306 45 263)(7 287 33 279 20 305 46 262)(8 299 34 278 21 304 47 261)(9 298 35 277 22 303 48 273)(10 297 36 276 23 302 49 272)(11 296 37 275 24 301 50 271)(12 295 38 274 25 300 51 270)(13 294 39 286 26 312 52 269)(53 236 92 219 66 251 79 230)(54 235 93 218 67 250 80 229)(55 247 94 217 68 249 81 228)(56 246 95 216 69 248 82 227)(57 245 96 215 70 260 83 226)(58 244 97 214 71 259 84 225)(59 243 98 213 72 258 85 224)(60 242 99 212 73 257 86 223)(61 241 100 211 74 256 87 222)(62 240 101 210 75 255 88 234)(63 239 102 209 76 254 89 233)(64 238 103 221 77 253 90 232)(65 237 104 220 78 252 91 231)(105 395 131 389 118 408 144 369)(106 394 132 388 119 407 145 368)(107 393 133 387 120 406 146 367)(108 392 134 386 121 405 147 366)(109 391 135 385 122 404 148 365)(110 403 136 384 123 416 149 377)(111 402 137 383 124 415 150 376)(112 401 138 382 125 414 151 375)(113 400 139 381 126 413 152 374)(114 399 140 380 127 412 153 373)(115 398 141 379 128 411 154 372)(116 397 142 378 129 410 155 371)(117 396 143 390 130 409 156 370)(157 350 196 322 170 356 183 336)(158 349 197 321 171 355 184 335)(159 348 198 320 172 354 185 334)(160 347 199 319 173 353 186 333)(161 346 200 318 174 352 187 332)(162 345 201 317 175 364 188 331)(163 344 202 316 176 363 189 330)(164 343 203 315 177 362 190 329)(165 342 204 314 178 361 191 328)(166 341 205 313 179 360 192 327)(167 340 206 325 180 359 193 326)(168 339 207 324 181 358 194 338)(169 351 208 323 182 357 195 337)
(1 170 14 157)(2 171 15 158)(3 172 16 159)(4 173 17 160)(5 174 18 161)(6 175 19 162)(7 176 20 163)(8 177 21 164)(9 178 22 165)(10 179 23 166)(11 180 24 167)(12 181 25 168)(13 182 26 169)(27 196 40 183)(28 197 41 184)(29 198 42 185)(30 199 43 186)(31 200 44 187)(32 201 45 188)(33 202 46 189)(34 203 47 190)(35 204 48 191)(36 205 49 192)(37 206 50 193)(38 207 51 194)(39 208 52 195)(53 105 66 118)(54 106 67 119)(55 107 68 120)(56 108 69 121)(57 109 70 122)(58 110 71 123)(59 111 72 124)(60 112 73 125)(61 113 74 126)(62 114 75 127)(63 115 76 128)(64 116 77 129)(65 117 78 130)(79 131 92 144)(80 132 93 145)(81 133 94 146)(82 134 95 147)(83 135 96 148)(84 136 97 149)(85 137 98 150)(86 138 99 151)(87 139 100 152)(88 140 101 153)(89 141 102 154)(90 142 103 155)(91 143 104 156)(209 411 233 398)(210 412 234 399)(211 413 222 400)(212 414 223 401)(213 415 224 402)(214 416 225 403)(215 404 226 391)(216 405 227 392)(217 406 228 393)(218 407 229 394)(219 408 230 395)(220 409 231 396)(221 410 232 397)(235 368 250 388)(236 369 251 389)(237 370 252 390)(238 371 253 378)(239 372 254 379)(240 373 255 380)(241 374 256 381)(242 375 257 382)(243 376 258 383)(244 377 259 384)(245 365 260 385)(246 366 248 386)(247 367 249 387)(261 362 278 343)(262 363 279 344)(263 364 280 345)(264 352 281 346)(265 353 282 347)(266 354 283 348)(267 355 284 349)(268 356 285 350)(269 357 286 351)(270 358 274 339)(271 359 275 340)(272 360 276 341)(273 361 277 342)(287 316 305 330)(288 317 306 331)(289 318 307 332)(290 319 308 333)(291 320 309 334)(292 321 310 335)(293 322 311 336)(294 323 312 337)(295 324 300 338)(296 325 301 326)(297 313 302 327)(298 314 303 328)(299 315 304 329)

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 2 4 4 4 4 4 4 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 2 2 4 4 52 52 26 26 26 26 2 ··· 2 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + - + + + - image C1 C2 C2 C2 C2 D4 D4 Q16 D13 D26 D26 C13⋊D4 C13⋊D4 C13⋊Q16 kernel C2×C13⋊Q16 C2×C13⋊2C8 C13⋊Q16 C2×Dic26 Q8×C26 C52 C2×C26 C26 C2×Q8 C2×C4 Q8 C4 C22 C2 # reps 1 1 4 1 1 1 1 4 6 6 12 12 12 12

Matrix representation of C2×C13⋊Q16 in GL4(𝔽313) generated by

 312 0 0 0 0 312 0 0 0 0 1 0 0 0 0 1
,
 85 312 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 70 84 0 0 87 243 0 0 0 0 120 203 0 0 37 0
,
 312 0 0 0 0 312 0 0 0 0 120 105 0 0 152 193
G:=sub<GL(4,GF(313))| [312,0,0,0,0,312,0,0,0,0,1,0,0,0,0,1],[85,1,0,0,312,0,0,0,0,0,1,0,0,0,0,1],[70,87,0,0,84,243,0,0,0,0,120,37,0,0,203,0],[312,0,0,0,0,312,0,0,0,0,120,152,0,0,105,193] >;

C2×C13⋊Q16 in GAP, Magma, Sage, TeX

C_2\times C_{13}\rtimes Q_{16}
% in TeX

G:=Group("C2xC13:Q16");
// GroupNames label

G:=SmallGroup(416,164);
// by ID

G=gap.SmallGroup(416,164);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,218,86,579,159,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^13=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽