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## G = C13×Q8⋊C4order 416 = 25·13

### Direct product of C13 and Q8⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

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

Generators and relations for C13×Q8⋊C4
G = < a,b,c,d | a13=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

Smallest permutation representation of C13×Q8⋊C4
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)
(1 362 222 167)(2 363 223 168)(3 364 224 169)(4 352 225 157)(5 353 226 158)(6 354 227 159)(7 355 228 160)(8 356 229 161)(9 357 230 162)(10 358 231 163)(11 359 232 164)(12 360 233 165)(13 361 234 166)(14 256 38 65)(15 257 39 53)(16 258 27 54)(17 259 28 55)(18 260 29 56)(19 248 30 57)(20 249 31 58)(21 250 32 59)(22 251 33 60)(23 252 34 61)(24 253 35 62)(25 254 36 63)(26 255 37 64)(40 214 380 341)(41 215 381 342)(42 216 382 343)(43 217 383 344)(44 218 384 345)(45 219 385 346)(46 220 386 347)(47 221 387 348)(48 209 388 349)(49 210 389 350)(50 211 390 351)(51 212 378 339)(52 213 379 340)(66 328 320 405)(67 329 321 406)(68 330 322 407)(69 331 323 408)(70 332 324 409)(71 333 325 410)(72 334 313 411)(73 335 314 412)(74 336 315 413)(75 337 316 414)(76 338 317 415)(77 326 318 416)(78 327 319 404)(79 135 205 179)(80 136 206 180)(81 137 207 181)(82 138 208 182)(83 139 196 170)(84 140 197 171)(85 141 198 172)(86 142 199 173)(87 143 200 174)(88 131 201 175)(89 132 202 176)(90 133 203 177)(91 134 204 178)(92 240 115 300)(93 241 116 301)(94 242 117 302)(95 243 105 303)(96 244 106 304)(97 245 107 305)(98 246 108 306)(99 247 109 307)(100 235 110 308)(101 236 111 309)(102 237 112 310)(103 238 113 311)(104 239 114 312)(118 149 393 276)(119 150 394 277)(120 151 395 278)(121 152 396 279)(122 153 397 280)(123 154 398 281)(124 155 399 282)(125 156 400 283)(126 144 401 284)(127 145 402 285)(128 146 403 286)(129 147 391 274)(130 148 392 275)(183 289 262 377)(184 290 263 365)(185 291 264 366)(186 292 265 367)(187 293 266 368)(188 294 267 369)(189 295 268 370)(190 296 269 371)(191 297 270 372)(192 298 271 373)(193 299 272 374)(194 287 273 375)(195 288 261 376)
(1 345 222 218)(2 346 223 219)(3 347 224 220)(4 348 225 221)(5 349 226 209)(6 350 227 210)(7 351 228 211)(8 339 229 212)(9 340 230 213)(10 341 231 214)(11 342 232 215)(12 343 233 216)(13 344 234 217)(14 410 38 333)(15 411 39 334)(16 412 27 335)(17 413 28 336)(18 414 29 337)(19 415 30 338)(20 416 31 326)(21 404 32 327)(22 405 33 328)(23 406 34 329)(24 407 35 330)(25 408 36 331)(26 409 37 332)(40 358 380 163)(41 359 381 164)(42 360 382 165)(43 361 383 166)(44 362 384 167)(45 363 385 168)(46 364 386 169)(47 352 387 157)(48 353 388 158)(49 354 389 159)(50 355 390 160)(51 356 378 161)(52 357 379 162)(53 72 257 313)(54 73 258 314)(55 74 259 315)(56 75 260 316)(57 76 248 317)(58 77 249 318)(59 78 250 319)(60 66 251 320)(61 67 252 321)(62 68 253 322)(63 69 254 323)(64 70 255 324)(65 71 256 325)(79 111 205 101)(80 112 206 102)(81 113 207 103)(82 114 208 104)(83 115 196 92)(84 116 197 93)(85 117 198 94)(86 105 199 95)(87 106 200 96)(88 107 201 97)(89 108 202 98)(90 109 203 99)(91 110 204 100)(118 191 393 270)(119 192 394 271)(120 193 395 272)(121 194 396 273)(122 195 397 261)(123 183 398 262)(124 184 399 263)(125 185 400 264)(126 186 401 265)(127 187 402 266)(128 188 403 267)(129 189 391 268)(130 190 392 269)(131 245 175 305)(132 246 176 306)(133 247 177 307)(134 235 178 308)(135 236 179 309)(136 237 180 310)(137 238 181 311)(138 239 182 312)(139 240 170 300)(140 241 171 301)(141 242 172 302)(142 243 173 303)(143 244 174 304)(144 367 284 292)(145 368 285 293)(146 369 286 294)(147 370 274 295)(148 371 275 296)(149 372 276 297)(150 373 277 298)(151 374 278 299)(152 375 279 287)(153 376 280 288)(154 377 281 289)(155 365 282 290)(156 366 283 291)
(1 80 406 149)(2 81 407 150)(3 82 408 151)(4 83 409 152)(5 84 410 153)(6 85 411 154)(7 86 412 155)(8 87 413 156)(9 88 414 144)(10 89 415 145)(11 90 416 146)(12 91 404 147)(13 79 405 148)(14 261 209 241)(15 262 210 242)(16 263 211 243)(17 264 212 244)(18 265 213 245)(19 266 214 246)(20 267 215 247)(21 268 216 235)(22 269 217 236)(23 270 218 237)(24 271 219 238)(25 272 220 239)(26 273 221 240)(27 184 351 303)(28 185 339 304)(29 186 340 305)(30 187 341 306)(31 188 342 307)(32 189 343 308)(33 190 344 309)(34 191 345 310)(35 192 346 311)(36 193 347 312)(37 194 348 300)(38 195 349 301)(39 183 350 302)(40 108 57 368)(41 109 58 369)(42 110 59 370)(43 111 60 371)(44 112 61 372)(45 113 62 373)(46 114 63 374)(47 115 64 375)(48 116 65 376)(49 117 53 377)(50 105 54 365)(51 106 55 366)(52 107 56 367)(66 130 361 179)(67 118 362 180)(68 119 363 181)(69 120 364 182)(70 121 352 170)(71 122 353 171)(72 123 354 172)(73 124 355 173)(74 125 356 174)(75 126 357 175)(76 127 358 176)(77 128 359 177)(78 129 360 178)(92 255 287 387)(93 256 288 388)(94 257 289 389)(95 258 290 390)(96 259 291 378)(97 260 292 379)(98 248 293 380)(99 249 294 381)(100 250 295 382)(101 251 296 383)(102 252 297 384)(103 253 298 385)(104 254 299 386)(131 316 401 162)(132 317 402 163)(133 318 403 164)(134 319 391 165)(135 320 392 166)(136 321 393 167)(137 322 394 168)(138 323 395 169)(139 324 396 157)(140 325 397 158)(141 313 398 159)(142 314 399 160)(143 315 400 161)(196 332 279 225)(197 333 280 226)(198 334 281 227)(199 335 282 228)(200 336 283 229)(201 337 284 230)(202 338 285 231)(203 326 286 232)(204 327 274 233)(205 328 275 234)(206 329 276 222)(207 330 277 223)(208 331 278 224)

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

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

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

182 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 13A ··· 13L 26A ··· 26AJ 52A ··· 52X 52Y ··· 52BT 104A ··· 104AV order 1 2 2 2 4 4 4 4 4 4 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

182 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C4 C13 C26 C26 C26 C52 D4 D4 SD16 Q16 D4×C13 D4×C13 C13×SD16 C13×Q16 kernel C13×Q8⋊C4 C13×C4⋊C4 C2×C104 Q8×C26 Q8×C13 Q8⋊C4 C4⋊C4 C2×C8 C2×Q8 Q8 C52 C2×C26 C26 C26 C4 C22 C2 C2 # reps 1 1 1 1 4 12 12 12 12 48 1 1 2 2 12 12 24 24

Matrix representation of C13×Q8⋊C4 in GL4(𝔽313) generated by

 103 0 0 0 0 103 0 0 0 0 27 0 0 0 0 27
,
 312 0 0 0 0 312 0 0 0 0 0 1 0 0 312 0
,
 159 2 0 0 193 154 0 0 0 0 230 263 0 0 263 83
,
 155 14 0 0 296 158 0 0 0 0 119 267 0 0 267 194
G:=sub<GL(4,GF(313))| [103,0,0,0,0,103,0,0,0,0,27,0,0,0,0,27],[312,0,0,0,0,312,0,0,0,0,0,312,0,0,1,0],[159,193,0,0,2,154,0,0,0,0,230,263,0,0,263,83],[155,296,0,0,14,158,0,0,0,0,119,267,0,0,267,194] >;

C13×Q8⋊C4 in GAP, Magma, Sage, TeX

C_{13}\times Q_8\rtimes C_4
% in TeX

G:=Group("C13xQ8:C4");
// GroupNames label

G:=SmallGroup(416,53);
// by ID

G=gap.SmallGroup(416,53);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,1255,6243,3129,117]);
// Polycyclic

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

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