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G = C132C32order 416 = 25·13

The semidirect product of C13 and C32 acting via C32/C16=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C132C32, C52.5C8, C208.2C2, C104.7C4, C26.2C16, C16.2D13, C8.3Dic13, C2.(C132C16), C4.2(C132C8), SmallGroup(416,1)

Series: Derived Chief Lower central Upper central

C1C13 — C132C32
C1C13C26C52C104C208 — C132C32
C13 — C132C32
C1C16

Generators and relations for C132C32
 G = < a,b | a13=b32=1, bab-1=a-1 >

13C32

Smallest permutation representation of C132C32
Regular action on 416 points
Generators in S416
(1 216 174 340 360 135 276 307 54 232 392 96 124)(2 125 65 393 233 55 308 277 136 361 341 175 217)(3 218 176 342 362 137 278 309 56 234 394 66 126)(4 127 67 395 235 57 310 279 138 363 343 177 219)(5 220 178 344 364 139 280 311 58 236 396 68 128)(6 97 69 397 237 59 312 281 140 365 345 179 221)(7 222 180 346 366 141 282 313 60 238 398 70 98)(8 99 71 399 239 61 314 283 142 367 347 181 223)(9 224 182 348 368 143 284 315 62 240 400 72 100)(10 101 73 401 241 63 316 285 144 369 349 183 193)(11 194 184 350 370 145 286 317 64 242 402 74 102)(12 103 75 403 243 33 318 287 146 371 351 185 195)(13 196 186 352 372 147 288 319 34 244 404 76 104)(14 105 77 405 245 35 320 257 148 373 321 187 197)(15 198 188 322 374 149 258 289 36 246 406 78 106)(16 107 79 407 247 37 290 259 150 375 323 189 199)(17 200 190 324 376 151 260 291 38 248 408 80 108)(18 109 81 409 249 39 292 261 152 377 325 191 201)(19 202 192 326 378 153 262 293 40 250 410 82 110)(20 111 83 411 251 41 294 263 154 379 327 161 203)(21 204 162 328 380 155 264 295 42 252 412 84 112)(22 113 85 413 253 43 296 265 156 381 329 163 205)(23 206 164 330 382 157 266 297 44 254 414 86 114)(24 115 87 415 255 45 298 267 158 383 331 165 207)(25 208 166 332 384 159 268 299 46 256 416 88 116)(26 117 89 385 225 47 300 269 160 353 333 167 209)(27 210 168 334 354 129 270 301 48 226 386 90 118)(28 119 91 387 227 49 302 271 130 355 335 169 211)(29 212 170 336 356 131 272 303 50 228 388 92 120)(30 121 93 389 229 51 304 273 132 357 337 171 213)(31 214 172 338 358 133 274 305 52 230 390 94 122)(32 123 95 391 231 53 306 275 134 359 339 173 215)
(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)

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

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

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

128 conjugacy classes

class 1  2 4A4B8A8B8C8D13A···13F16A···16H26A···26F32A···32P52A···52L104A···104X208A···208AV
order1244888813···1316···1626···2632···3252···52104···104208···208
size111111112···21···12···213···132···22···22···2

128 irreducible representations

dim11111122222
type+++-
imageC1C2C4C8C16C32D13Dic13C132C8C132C16C132C32
kernelC132C32C208C104C52C26C13C16C8C4C2C1
# reps112481666122448

Matrix representation of C132C32 in GL2(𝔽1249) generated by

12481
1024224
,
725277
630524
G:=sub<GL(2,GF(1249))| [1248,1024,1,224],[725,630,277,524] >;

C132C32 in GAP, Magma, Sage, TeX

C_{13}\rtimes_2C_{32}
% in TeX

G:=Group("C13:2C32");
// GroupNames label

G:=SmallGroup(416,1);
// by ID

G=gap.SmallGroup(416,1);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,12,31,50,69,13829]);
// Polycyclic

G:=Group<a,b|a^13=b^32=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C132C32 in TeX

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