metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11⋊C32, C22.C16, C88.2C4, C44.2C8, C176.2C2, C16.2D11, C8.3Dic11, C2.(C11⋊C16), C4.2(C11⋊C8), SmallGroup(352,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C11⋊C32 |
Generators and relations for C11⋊C32
G = < a,b | a11=b32=1, bab-1=a-1 >
Smallest permutation representation of C11⋊C32
►Regular action on 352 points
Generators in S352
(1 80 206 48 338 286 254 293 163 119 144)(2 145 120 164 294 255 287 339 49 207 81)(3 82 208 50 340 288 256 295 165 121 146)(4 147 122 166 296 225 257 341 51 209 83)(5 84 210 52 342 258 226 297 167 123 148)(6 149 124 168 298 227 259 343 53 211 85)(7 86 212 54 344 260 228 299 169 125 150)(8 151 126 170 300 229 261 345 55 213 87)(9 88 214 56 346 262 230 301 171 127 152)(10 153 128 172 302 231 263 347 57 215 89)(11 90 216 58 348 264 232 303 173 97 154)(12 155 98 174 304 233 265 349 59 217 91)(13 92 218 60 350 266 234 305 175 99 156)(14 157 100 176 306 235 267 351 61 219 93)(15 94 220 62 352 268 236 307 177 101 158)(16 159 102 178 308 237 269 321 63 221 95)(17 96 222 64 322 270 238 309 179 103 160)(18 129 104 180 310 239 271 323 33 223 65)(19 66 224 34 324 272 240 311 181 105 130)(20 131 106 182 312 241 273 325 35 193 67)(21 68 194 36 326 274 242 313 183 107 132)(22 133 108 184 314 243 275 327 37 195 69)(23 70 196 38 328 276 244 315 185 109 134)(24 135 110 186 316 245 277 329 39 197 71)(25 72 198 40 330 278 246 317 187 111 136)(26 137 112 188 318 247 279 331 41 199 73)(27 74 200 42 332 280 248 319 189 113 138)(28 139 114 190 320 249 281 333 43 201 75)(29 76 202 44 334 282 250 289 191 115 140)(30 141 116 192 290 251 283 335 45 203 77)(31 78 204 46 336 284 252 291 161 117 142)(32 143 118 162 292 253 285 337 47 205 79)
(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)
G:=sub<Sym(352)| (1,80,206,48,338,286,254,293,163,119,144)(2,145,120,164,294,255,287,339,49,207,81)(3,82,208,50,340,288,256,295,165,121,146)(4,147,122,166,296,225,257,341,51,209,83)(5,84,210,52,342,258,226,297,167,123,148)(6,149,124,168,298,227,259,343,53,211,85)(7,86,212,54,344,260,228,299,169,125,150)(8,151,126,170,300,229,261,345,55,213,87)(9,88,214,56,346,262,230,301,171,127,152)(10,153,128,172,302,231,263,347,57,215,89)(11,90,216,58,348,264,232,303,173,97,154)(12,155,98,174,304,233,265,349,59,217,91)(13,92,218,60,350,266,234,305,175,99,156)(14,157,100,176,306,235,267,351,61,219,93)(15,94,220,62,352,268,236,307,177,101,158)(16,159,102,178,308,237,269,321,63,221,95)(17,96,222,64,322,270,238,309,179,103,160)(18,129,104,180,310,239,271,323,33,223,65)(19,66,224,34,324,272,240,311,181,105,130)(20,131,106,182,312,241,273,325,35,193,67)(21,68,194,36,326,274,242,313,183,107,132)(22,133,108,184,314,243,275,327,37,195,69)(23,70,196,38,328,276,244,315,185,109,134)(24,135,110,186,316,245,277,329,39,197,71)(25,72,198,40,330,278,246,317,187,111,136)(26,137,112,188,318,247,279,331,41,199,73)(27,74,200,42,332,280,248,319,189,113,138)(28,139,114,190,320,249,281,333,43,201,75)(29,76,202,44,334,282,250,289,191,115,140)(30,141,116,192,290,251,283,335,45,203,77)(31,78,204,46,336,284,252,291,161,117,142)(32,143,118,162,292,253,285,337,47,205,79), (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)>;
G:=Group( (1,80,206,48,338,286,254,293,163,119,144)(2,145,120,164,294,255,287,339,49,207,81)(3,82,208,50,340,288,256,295,165,121,146)(4,147,122,166,296,225,257,341,51,209,83)(5,84,210,52,342,258,226,297,167,123,148)(6,149,124,168,298,227,259,343,53,211,85)(7,86,212,54,344,260,228,299,169,125,150)(8,151,126,170,300,229,261,345,55,213,87)(9,88,214,56,346,262,230,301,171,127,152)(10,153,128,172,302,231,263,347,57,215,89)(11,90,216,58,348,264,232,303,173,97,154)(12,155,98,174,304,233,265,349,59,217,91)(13,92,218,60,350,266,234,305,175,99,156)(14,157,100,176,306,235,267,351,61,219,93)(15,94,220,62,352,268,236,307,177,101,158)(16,159,102,178,308,237,269,321,63,221,95)(17,96,222,64,322,270,238,309,179,103,160)(18,129,104,180,310,239,271,323,33,223,65)(19,66,224,34,324,272,240,311,181,105,130)(20,131,106,182,312,241,273,325,35,193,67)(21,68,194,36,326,274,242,313,183,107,132)(22,133,108,184,314,243,275,327,37,195,69)(23,70,196,38,328,276,244,315,185,109,134)(24,135,110,186,316,245,277,329,39,197,71)(25,72,198,40,330,278,246,317,187,111,136)(26,137,112,188,318,247,279,331,41,199,73)(27,74,200,42,332,280,248,319,189,113,138)(28,139,114,190,320,249,281,333,43,201,75)(29,76,202,44,334,282,250,289,191,115,140)(30,141,116,192,290,251,283,335,45,203,77)(31,78,204,46,336,284,252,291,161,117,142)(32,143,118,162,292,253,285,337,47,205,79), (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) );
G=PermutationGroup([[(1,80,206,48,338,286,254,293,163,119,144),(2,145,120,164,294,255,287,339,49,207,81),(3,82,208,50,340,288,256,295,165,121,146),(4,147,122,166,296,225,257,341,51,209,83),(5,84,210,52,342,258,226,297,167,123,148),(6,149,124,168,298,227,259,343,53,211,85),(7,86,212,54,344,260,228,299,169,125,150),(8,151,126,170,300,229,261,345,55,213,87),(9,88,214,56,346,262,230,301,171,127,152),(10,153,128,172,302,231,263,347,57,215,89),(11,90,216,58,348,264,232,303,173,97,154),(12,155,98,174,304,233,265,349,59,217,91),(13,92,218,60,350,266,234,305,175,99,156),(14,157,100,176,306,235,267,351,61,219,93),(15,94,220,62,352,268,236,307,177,101,158),(16,159,102,178,308,237,269,321,63,221,95),(17,96,222,64,322,270,238,309,179,103,160),(18,129,104,180,310,239,271,323,33,223,65),(19,66,224,34,324,272,240,311,181,105,130),(20,131,106,182,312,241,273,325,35,193,67),(21,68,194,36,326,274,242,313,183,107,132),(22,133,108,184,314,243,275,327,37,195,69),(23,70,196,38,328,276,244,315,185,109,134),(24,135,110,186,316,245,277,329,39,197,71),(25,72,198,40,330,278,246,317,187,111,136),(26,137,112,188,318,247,279,331,41,199,73),(27,74,200,42,332,280,248,319,189,113,138),(28,139,114,190,320,249,281,333,43,201,75),(29,76,202,44,334,282,250,289,191,115,140),(30,141,116,192,290,251,283,335,45,203,77),(31,78,204,46,336,284,252,291,161,117,142),(32,143,118,162,292,253,285,337,47,205,79)], [(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)]])
112 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 16A | ··· | 16H | 22A | ··· | 22E | 32A | ··· | 32P | 44A | ··· | 44J | 88A | ··· | 88T | 176A | ··· | 176AN |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 16 | ··· | 16 | 22 | ··· | 22 | 32 | ··· | 32 | 44 | ··· | 44 | 88 | ··· | 88 | 176 | ··· | 176 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | D11 | Dic11 | C11⋊C8 | C11⋊C16 | C11⋊C32 |
| kernel | C11⋊C32 | C176 | C88 | C44 | C22 | C11 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 5 | 5 | 10 | 20 | 40 |
Matrix representation of C11⋊C32 ►in GL3(𝔽353) generated by
| 1 | 0 | 0 |
| 0 | 352 | 1 |
| 0 | 345 | 7 |
| 286 | 0 | 0 |
| 0 | 64 | 137 |
| 0 | 196 | 289 |
G:=sub<GL(3,GF(353))| [1,0,0,0,352,345,0,1,7],[286,0,0,0,64,196,0,137,289] >;
C11⋊C32 in GAP, Magma, Sage, TeX
C_{11}\rtimes C_{32} % in TeX
G:=Group("C11:C32"); // GroupNames label
G:=SmallGroup(352,1);
// by ID
G=gap.SmallGroup(352,1);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,12,31,50,69,11525]);
// Polycyclic
G:=Group<a,b|a^11=b^32=1,b*a*b^-1=a^-1>;
// generators/relations
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