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G = C13×Q32order 416 = 25·13

Direct product of C13 and Q32

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C13×Q32, C16.C26, Q16.C26, C208.3C2, C26.17D8, C52.38D4, C104.26C22, C8.4(C2×C26), C4.3(D4×C13), C2.5(C13×D8), (C13×Q16).2C2, SmallGroup(416,63)

Series: Derived Chief Lower central Upper central

C1C8 — C13×Q32
C1C2C4C8C104C13×Q16 — C13×Q32
C1C2C4C8 — C13×Q32
C1C26C52C104 — C13×Q32

Generators and relations for C13×Q32
 G = < a,b,c | a13=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >

4C4
4C4
2Q8
2Q8
4C52
4C52
2Q8×C13
2Q8×C13

Smallest permutation representation of C13×Q32
Regular action on 416 points
Generators in S416
(1 175 267 40 100 81 215 246 116 352 23 307 381)(2 176 268 41 101 82 216 247 117 337 24 308 382)(3 161 269 42 102 83 217 248 118 338 25 309 383)(4 162 270 43 103 84 218 249 119 339 26 310 384)(5 163 271 44 104 85 219 250 120 340 27 311 369)(6 164 272 45 105 86 220 251 121 341 28 312 370)(7 165 257 46 106 87 221 252 122 342 29 313 371)(8 166 258 47 107 88 222 253 123 343 30 314 372)(9 167 259 48 108 89 223 254 124 344 31 315 373)(10 168 260 33 109 90 224 255 125 345 32 316 374)(11 169 261 34 110 91 209 256 126 346 17 317 375)(12 170 262 35 111 92 210 241 127 347 18 318 376)(13 171 263 36 112 93 211 242 128 348 19 319 377)(14 172 264 37 97 94 212 243 113 349 20 320 378)(15 173 265 38 98 95 213 244 114 350 21 305 379)(16 174 266 39 99 96 214 245 115 351 22 306 380)(49 201 329 297 181 396 138 284 407 149 361 233 79)(50 202 330 298 182 397 139 285 408 150 362 234 80)(51 203 331 299 183 398 140 286 409 151 363 235 65)(52 204 332 300 184 399 141 287 410 152 364 236 66)(53 205 333 301 185 400 142 288 411 153 365 237 67)(54 206 334 302 186 385 143 273 412 154 366 238 68)(55 207 335 303 187 386 144 274 413 155 367 239 69)(56 208 336 304 188 387 129 275 414 156 368 240 70)(57 193 321 289 189 388 130 276 415 157 353 225 71)(58 194 322 290 190 389 131 277 416 158 354 226 72)(59 195 323 291 191 390 132 278 401 159 355 227 73)(60 196 324 292 192 391 133 279 402 160 356 228 74)(61 197 325 293 177 392 134 280 403 145 357 229 75)(62 198 326 294 178 393 135 281 404 146 358 230 76)(63 199 327 295 179 394 136 282 405 147 359 231 77)(64 200 328 296 180 395 137 283 406 148 360 232 78)
(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)
(1 357 9 365)(2 356 10 364)(3 355 11 363)(4 354 12 362)(5 353 13 361)(6 368 14 360)(7 367 15 359)(8 366 16 358)(17 286 25 278)(18 285 26 277)(19 284 27 276)(20 283 28 275)(21 282 29 274)(22 281 30 273)(23 280 31 288)(24 279 32 287)(33 52 41 60)(34 51 42 59)(35 50 43 58)(36 49 44 57)(37 64 45 56)(38 63 46 55)(39 62 47 54)(40 61 48 53)(65 269 73 261)(66 268 74 260)(67 267 75 259)(68 266 76 258)(69 265 77 257)(70 264 78 272)(71 263 79 271)(72 262 80 270)(81 325 89 333)(82 324 90 332)(83 323 91 331)(84 322 92 330)(85 321 93 329)(86 336 94 328)(87 335 95 327)(88 334 96 326)(97 200 105 208)(98 199 106 207)(99 198 107 206)(100 197 108 205)(101 196 109 204)(102 195 110 203)(103 194 111 202)(104 193 112 201)(113 395 121 387)(114 394 122 386)(115 393 123 385)(116 392 124 400)(117 391 125 399)(118 390 126 398)(119 389 127 397)(120 388 128 396)(129 349 137 341)(130 348 138 340)(131 347 139 339)(132 346 140 338)(133 345 141 337)(134 344 142 352)(135 343 143 351)(136 342 144 350)(145 373 153 381)(146 372 154 380)(147 371 155 379)(148 370 156 378)(149 369 157 377)(150 384 158 376)(151 383 159 375)(152 382 160 374)(161 227 169 235)(162 226 170 234)(163 225 171 233)(164 240 172 232)(165 239 173 231)(166 238 174 230)(167 237 175 229)(168 236 176 228)(177 254 185 246)(178 253 186 245)(179 252 187 244)(180 251 188 243)(181 250 189 242)(182 249 190 241)(183 248 191 256)(184 247 192 255)(209 299 217 291)(210 298 218 290)(211 297 219 289)(212 296 220 304)(213 295 221 303)(214 294 222 302)(215 293 223 301)(216 292 224 300)(305 405 313 413)(306 404 314 412)(307 403 315 411)(308 402 316 410)(309 401 317 409)(310 416 318 408)(311 415 319 407)(312 414 320 406)

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

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

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

143 conjugacy classes

class 1  2 4A4B4C8A8B13A···13L16A16B16C16D26A···26L52A···52L52M···52AJ104A···104X208A···208AV
order124448813···131616161626···2652···5252···52104···104208···208
size11288221···122221···12···28···82···22···2

143 irreducible representations

dim111111222222
type+++++-
imageC1C2C2C13C26C26D4D8Q32D4×C13C13×D8C13×Q32
kernelC13×Q32C208C13×Q16Q32C16Q16C52C26C13C4C2C1
# reps112121224124122448

Matrix representation of C13×Q32 in GL2(𝔽1249) generated by

3490
0349
,
35678
910713
,
473743
220776
G:=sub<GL(2,GF(1249))| [349,0,0,349],[35,910,678,713],[473,220,743,776] >;

C13×Q32 in GAP, Magma, Sage, TeX

C_{13}\times Q_{32}
% in TeX

G:=Group("C13xQ32");
// GroupNames label

G:=SmallGroup(416,63);
// by ID

G=gap.SmallGroup(416,63);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,1248,649,1255,3747,1881,165,9364,4690,88]);
// Polycyclic

G:=Group<a,b,c|a^13=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C13×Q32 in TeX

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