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## G = C13×C4⋊Q8order 416 = 25·13

### Direct product of C13 and C4⋊Q8

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C13×C4⋊Q8
G = < a,b,c,d | a13=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 84 in 68 conjugacy classes, 52 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C13, C42, C4⋊C4, C2×Q8, C26, C26, C4⋊Q8, C52, C52, C2×C26, C2×C52, C2×C52, Q8×C13, C4×C52, C13×C4⋊C4, Q8×C26, C13×C4⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C13, C2×D4, C2×Q8, C26, C4⋊Q8, C2×C26, D4×C13, Q8×C13, C22×C26, D4×C26, Q8×C26, C13×C4⋊Q8

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

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

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

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

182 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 13A ··· 13L 26A ··· 26AJ 52A ··· 52BT 52BU ··· 52DP order 1 2 2 2 4 ··· 4 4 4 4 4 13 ··· 13 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4

182 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C13 C26 C26 C26 D4 Q8 D4×C13 Q8×C13 kernel C13×C4⋊Q8 C4×C52 C13×C4⋊C4 Q8×C26 C4⋊Q8 C42 C4⋊C4 C2×Q8 C52 C52 C4 C4 # reps 1 1 4 2 12 12 48 24 2 4 24 48

Matrix representation of C13×C4⋊Q8 in GL4(𝔽53) generated by

 42 0 0 0 0 42 0 0 0 0 16 0 0 0 0 16
,
 30 0 0 0 0 23 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 23 51 0 0 0 30
,
 0 1 0 0 1 0 0 0 0 0 10 15 0 0 18 43
G:=sub<GL(4,GF(53))| [42,0,0,0,0,42,0,0,0,0,16,0,0,0,0,16],[30,0,0,0,0,23,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,23,0,0,0,51,30],[0,1,0,0,1,0,0,0,0,0,10,18,0,0,15,43] >;

C13×C4⋊Q8 in GAP, Magma, Sage, TeX

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

G:=Group("C13xC4:Q8");
// GroupNames label

G:=SmallGroup(416,189);
// by ID

G=gap.SmallGroup(416,189);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,624,1273,631,3818,950]);
// Polycyclic

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

׿
×
𝔽