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G = C13⋊C32order 416 = 25·13

The semidirect product of C13 and C32 acting via C32/C8=C4

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

Aliases: C13⋊C32, C26.C16, C52.2C8, C104.3C4, C2.(C13⋊C16), C4.2(C13⋊C8), C8.4(C13⋊C4), C132C16.2C2, SmallGroup(416,3)

Series: Derived Chief Lower central Upper central

C1C13 — C13⋊C32
C1C13C26C52C104C132C16 — C13⋊C32
C13 — C13⋊C32
C1C8

Generators and relations for C13⋊C32
 G = < a,b | a13=b32=1, bab-1=a5 >

13C16
13C32

Smallest permutation representation of C13⋊C32
Regular action on 416 points
Generators in S416
(1 207 391 159 382 179 80 240 118 304 264 42 341)(2 180 265 392 241 342 383 305 208 81 43 160 119)(3 343 44 266 306 120 242 82 181 384 129 393 209)(4 121 130 45 83 210 307 353 344 243 394 267 182)(5 211 395 131 354 183 84 244 122 308 268 46 345)(6 184 269 396 245 346 355 309 212 85 47 132 123)(7 347 48 270 310 124 246 86 185 356 133 397 213)(8 125 134 49 87 214 311 357 348 247 398 271 186)(9 215 399 135 358 187 88 248 126 312 272 50 349)(10 188 273 400 249 350 359 313 216 89 51 136 127)(11 351 52 274 314 128 250 90 189 360 137 401 217)(12 97 138 53 91 218 315 361 352 251 402 275 190)(13 219 403 139 362 191 92 252 98 316 276 54 321)(14 192 277 404 253 322 363 317 220 93 55 140 99)(15 323 56 278 318 100 254 94 161 364 141 405 221)(16 101 142 57 95 222 319 365 324 255 406 279 162)(17 223 407 143 366 163 96 256 102 320 280 58 325)(18 164 281 408 225 326 367 289 224 65 59 144 103)(19 327 60 282 290 104 226 66 165 368 145 409 193)(20 105 146 61 67 194 291 369 328 227 410 283 166)(21 195 411 147 370 167 68 228 106 292 284 62 329)(22 168 285 412 229 330 371 293 196 69 63 148 107)(23 331 64 286 294 108 230 70 169 372 149 413 197)(24 109 150 33 71 198 295 373 332 231 414 287 170)(25 199 415 151 374 171 72 232 110 296 288 34 333)(26 172 257 416 233 334 375 297 200 73 35 152 111)(27 335 36 258 298 112 234 74 173 376 153 385 201)(28 113 154 37 75 202 299 377 336 235 386 259 174)(29 203 387 155 378 175 76 236 114 300 260 38 337)(30 176 261 388 237 338 379 301 204 77 39 156 115)(31 339 40 262 302 116 238 78 177 380 157 389 205)(32 117 158 41 79 206 303 381 340 239 390 263 178)
(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)| (1,207,391,159,382,179,80,240,118,304,264,42,341)(2,180,265,392,241,342,383,305,208,81,43,160,119)(3,343,44,266,306,120,242,82,181,384,129,393,209)(4,121,130,45,83,210,307,353,344,243,394,267,182)(5,211,395,131,354,183,84,244,122,308,268,46,345)(6,184,269,396,245,346,355,309,212,85,47,132,123)(7,347,48,270,310,124,246,86,185,356,133,397,213)(8,125,134,49,87,214,311,357,348,247,398,271,186)(9,215,399,135,358,187,88,248,126,312,272,50,349)(10,188,273,400,249,350,359,313,216,89,51,136,127)(11,351,52,274,314,128,250,90,189,360,137,401,217)(12,97,138,53,91,218,315,361,352,251,402,275,190)(13,219,403,139,362,191,92,252,98,316,276,54,321)(14,192,277,404,253,322,363,317,220,93,55,140,99)(15,323,56,278,318,100,254,94,161,364,141,405,221)(16,101,142,57,95,222,319,365,324,255,406,279,162)(17,223,407,143,366,163,96,256,102,320,280,58,325)(18,164,281,408,225,326,367,289,224,65,59,144,103)(19,327,60,282,290,104,226,66,165,368,145,409,193)(20,105,146,61,67,194,291,369,328,227,410,283,166)(21,195,411,147,370,167,68,228,106,292,284,62,329)(22,168,285,412,229,330,371,293,196,69,63,148,107)(23,331,64,286,294,108,230,70,169,372,149,413,197)(24,109,150,33,71,198,295,373,332,231,414,287,170)(25,199,415,151,374,171,72,232,110,296,288,34,333)(26,172,257,416,233,334,375,297,200,73,35,152,111)(27,335,36,258,298,112,234,74,173,376,153,385,201)(28,113,154,37,75,202,299,377,336,235,386,259,174)(29,203,387,155,378,175,76,236,114,300,260,38,337)(30,176,261,388,237,338,379,301,204,77,39,156,115)(31,339,40,262,302,116,238,78,177,380,157,389,205)(32,117,158,41,79,206,303,381,340,239,390,263,178), (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( (1,207,391,159,382,179,80,240,118,304,264,42,341)(2,180,265,392,241,342,383,305,208,81,43,160,119)(3,343,44,266,306,120,242,82,181,384,129,393,209)(4,121,130,45,83,210,307,353,344,243,394,267,182)(5,211,395,131,354,183,84,244,122,308,268,46,345)(6,184,269,396,245,346,355,309,212,85,47,132,123)(7,347,48,270,310,124,246,86,185,356,133,397,213)(8,125,134,49,87,214,311,357,348,247,398,271,186)(9,215,399,135,358,187,88,248,126,312,272,50,349)(10,188,273,400,249,350,359,313,216,89,51,136,127)(11,351,52,274,314,128,250,90,189,360,137,401,217)(12,97,138,53,91,218,315,361,352,251,402,275,190)(13,219,403,139,362,191,92,252,98,316,276,54,321)(14,192,277,404,253,322,363,317,220,93,55,140,99)(15,323,56,278,318,100,254,94,161,364,141,405,221)(16,101,142,57,95,222,319,365,324,255,406,279,162)(17,223,407,143,366,163,96,256,102,320,280,58,325)(18,164,281,408,225,326,367,289,224,65,59,144,103)(19,327,60,282,290,104,226,66,165,368,145,409,193)(20,105,146,61,67,194,291,369,328,227,410,283,166)(21,195,411,147,370,167,68,228,106,292,284,62,329)(22,168,285,412,229,330,371,293,196,69,63,148,107)(23,331,64,286,294,108,230,70,169,372,149,413,197)(24,109,150,33,71,198,295,373,332,231,414,287,170)(25,199,415,151,374,171,72,232,110,296,288,34,333)(26,172,257,416,233,334,375,297,200,73,35,152,111)(27,335,36,258,298,112,234,74,173,376,153,385,201)(28,113,154,37,75,202,299,377,336,235,386,259,174)(29,203,387,155,378,175,76,236,114,300,260,38,337)(30,176,261,388,237,338,379,301,204,77,39,156,115)(31,339,40,262,302,116,238,78,177,380,157,389,205)(32,117,158,41,79,206,303,381,340,239,390,263,178), (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([(1,207,391,159,382,179,80,240,118,304,264,42,341),(2,180,265,392,241,342,383,305,208,81,43,160,119),(3,343,44,266,306,120,242,82,181,384,129,393,209),(4,121,130,45,83,210,307,353,344,243,394,267,182),(5,211,395,131,354,183,84,244,122,308,268,46,345),(6,184,269,396,245,346,355,309,212,85,47,132,123),(7,347,48,270,310,124,246,86,185,356,133,397,213),(8,125,134,49,87,214,311,357,348,247,398,271,186),(9,215,399,135,358,187,88,248,126,312,272,50,349),(10,188,273,400,249,350,359,313,216,89,51,136,127),(11,351,52,274,314,128,250,90,189,360,137,401,217),(12,97,138,53,91,218,315,361,352,251,402,275,190),(13,219,403,139,362,191,92,252,98,316,276,54,321),(14,192,277,404,253,322,363,317,220,93,55,140,99),(15,323,56,278,318,100,254,94,161,364,141,405,221),(16,101,142,57,95,222,319,365,324,255,406,279,162),(17,223,407,143,366,163,96,256,102,320,280,58,325),(18,164,281,408,225,326,367,289,224,65,59,144,103),(19,327,60,282,290,104,226,66,165,368,145,409,193),(20,105,146,61,67,194,291,369,328,227,410,283,166),(21,195,411,147,370,167,68,228,106,292,284,62,329),(22,168,285,412,229,330,371,293,196,69,63,148,107),(23,331,64,286,294,108,230,70,169,372,149,413,197),(24,109,150,33,71,198,295,373,332,231,414,287,170),(25,199,415,151,374,171,72,232,110,296,288,34,333),(26,172,257,416,233,334,375,297,200,73,35,152,111),(27,335,36,258,298,112,234,74,173,376,153,385,201),(28,113,154,37,75,202,299,377,336,235,386,259,174),(29,203,387,155,378,175,76,236,114,300,260,38,337),(30,176,261,388,237,338,379,301,204,77,39,156,115),(31,339,40,262,302,116,238,78,177,380,157,389,205),(32,117,158,41,79,206,303,381,340,239,390,263,178)], [(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)])

56 conjugacy classes

class 1  2 4A4B8A8B8C8D13A13B13C16A···16H26A26B26C32A···32P52A···52F104A···104L
order1244888813131316···1626262632···3252···52104···104
size1111111144413···1344413···134···44···4

56 irreducible representations

dim1111114444
type+++-
imageC1C2C4C8C16C32C13⋊C4C13⋊C8C13⋊C16C13⋊C32
kernelC13⋊C32C132C16C104C52C26C13C8C4C2C1
# reps112481633612

Matrix representation of C13⋊C32 in GL5(𝔽1249)

10000
01248100
01248010
01248001
0682610639566
,
9080000
0319681209383
01242834107516
01101445799709
010121194641546

G:=sub<GL(5,GF(1249))| [1,0,0,0,0,0,1248,1248,1248,682,0,1,0,0,610,0,0,1,0,639,0,0,0,1,566],[908,0,0,0,0,0,319,1242,1101,1012,0,681,834,445,1194,0,209,1075,799,641,0,383,16,709,546] >;

C13⋊C32 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(416,3);
// by ID

G=gap.SmallGroup(416,3);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,12,31,50,69,9221,6923]);
// Polycyclic

G:=Group<a,b|a^13=b^32=1,b*a*b^-1=a^5>;
// generators/relations

Export

Subgroup lattice of C13⋊C32 in TeX

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