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G = C11⋊C32order 352 = 25·11

The semidirect product of C11 and C32 acting via C32/C16=C2

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

C1C11 — C11⋊C32
C1C11C22C44C88C176 — C11⋊C32
C11 — C11⋊C32
C1C16

Generators and relations for C11⋊C32
 G = < a,b | a11=b32=1, bab-1=a-1 >

11C32

Smallest permutation representation of C11⋊C32
Regular action on 352 points
Generators in S352
(1 33 257 312 205 345 143 66 102 171 252)(2 253 172 103 67 144 346 206 313 258 34)(3 35 259 314 207 347 145 68 104 173 254)(4 255 174 105 69 146 348 208 315 260 36)(5 37 261 316 209 349 147 70 106 175 256)(6 225 176 107 71 148 350 210 317 262 38)(7 39 263 318 211 351 149 72 108 177 226)(8 227 178 109 73 150 352 212 319 264 40)(9 41 265 320 213 321 151 74 110 179 228)(10 229 180 111 75 152 322 214 289 266 42)(11 43 267 290 215 323 153 76 112 181 230)(12 231 182 113 77 154 324 216 291 268 44)(13 45 269 292 217 325 155 78 114 183 232)(14 233 184 115 79 156 326 218 293 270 46)(15 47 271 294 219 327 157 80 116 185 234)(16 235 186 117 81 158 328 220 295 272 48)(17 49 273 296 221 329 159 82 118 187 236)(18 237 188 119 83 160 330 222 297 274 50)(19 51 275 298 223 331 129 84 120 189 238)(20 239 190 121 85 130 332 224 299 276 52)(21 53 277 300 193 333 131 86 122 191 240)(22 241 192 123 87 132 334 194 301 278 54)(23 55 279 302 195 335 133 88 124 161 242)(24 243 162 125 89 134 336 196 303 280 56)(25 57 281 304 197 337 135 90 126 163 244)(26 245 164 127 91 136 338 198 305 282 58)(27 59 283 306 199 339 137 92 128 165 246)(28 247 166 97 93 138 340 200 307 284 60)(29 61 285 308 201 341 139 94 98 167 248)(30 249 168 99 95 140 342 202 309 286 62)(31 63 287 310 203 343 141 96 100 169 250)(32 251 170 101 65 142 344 204 311 288 64)
(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)| (1,33,257,312,205,345,143,66,102,171,252)(2,253,172,103,67,144,346,206,313,258,34)(3,35,259,314,207,347,145,68,104,173,254)(4,255,174,105,69,146,348,208,315,260,36)(5,37,261,316,209,349,147,70,106,175,256)(6,225,176,107,71,148,350,210,317,262,38)(7,39,263,318,211,351,149,72,108,177,226)(8,227,178,109,73,150,352,212,319,264,40)(9,41,265,320,213,321,151,74,110,179,228)(10,229,180,111,75,152,322,214,289,266,42)(11,43,267,290,215,323,153,76,112,181,230)(12,231,182,113,77,154,324,216,291,268,44)(13,45,269,292,217,325,155,78,114,183,232)(14,233,184,115,79,156,326,218,293,270,46)(15,47,271,294,219,327,157,80,116,185,234)(16,235,186,117,81,158,328,220,295,272,48)(17,49,273,296,221,329,159,82,118,187,236)(18,237,188,119,83,160,330,222,297,274,50)(19,51,275,298,223,331,129,84,120,189,238)(20,239,190,121,85,130,332,224,299,276,52)(21,53,277,300,193,333,131,86,122,191,240)(22,241,192,123,87,132,334,194,301,278,54)(23,55,279,302,195,335,133,88,124,161,242)(24,243,162,125,89,134,336,196,303,280,56)(25,57,281,304,197,337,135,90,126,163,244)(26,245,164,127,91,136,338,198,305,282,58)(27,59,283,306,199,339,137,92,128,165,246)(28,247,166,97,93,138,340,200,307,284,60)(29,61,285,308,201,341,139,94,98,167,248)(30,249,168,99,95,140,342,202,309,286,62)(31,63,287,310,203,343,141,96,100,169,250)(32,251,170,101,65,142,344,204,311,288,64), (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( (1,33,257,312,205,345,143,66,102,171,252)(2,253,172,103,67,144,346,206,313,258,34)(3,35,259,314,207,347,145,68,104,173,254)(4,255,174,105,69,146,348,208,315,260,36)(5,37,261,316,209,349,147,70,106,175,256)(6,225,176,107,71,148,350,210,317,262,38)(7,39,263,318,211,351,149,72,108,177,226)(8,227,178,109,73,150,352,212,319,264,40)(9,41,265,320,213,321,151,74,110,179,228)(10,229,180,111,75,152,322,214,289,266,42)(11,43,267,290,215,323,153,76,112,181,230)(12,231,182,113,77,154,324,216,291,268,44)(13,45,269,292,217,325,155,78,114,183,232)(14,233,184,115,79,156,326,218,293,270,46)(15,47,271,294,219,327,157,80,116,185,234)(16,235,186,117,81,158,328,220,295,272,48)(17,49,273,296,221,329,159,82,118,187,236)(18,237,188,119,83,160,330,222,297,274,50)(19,51,275,298,223,331,129,84,120,189,238)(20,239,190,121,85,130,332,224,299,276,52)(21,53,277,300,193,333,131,86,122,191,240)(22,241,192,123,87,132,334,194,301,278,54)(23,55,279,302,195,335,133,88,124,161,242)(24,243,162,125,89,134,336,196,303,280,56)(25,57,281,304,197,337,135,90,126,163,244)(26,245,164,127,91,136,338,198,305,282,58)(27,59,283,306,199,339,137,92,128,165,246)(28,247,166,97,93,138,340,200,307,284,60)(29,61,285,308,201,341,139,94,98,167,248)(30,249,168,99,95,140,342,202,309,286,62)(31,63,287,310,203,343,141,96,100,169,250)(32,251,170,101,65,142,344,204,311,288,64), (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([(1,33,257,312,205,345,143,66,102,171,252),(2,253,172,103,67,144,346,206,313,258,34),(3,35,259,314,207,347,145,68,104,173,254),(4,255,174,105,69,146,348,208,315,260,36),(5,37,261,316,209,349,147,70,106,175,256),(6,225,176,107,71,148,350,210,317,262,38),(7,39,263,318,211,351,149,72,108,177,226),(8,227,178,109,73,150,352,212,319,264,40),(9,41,265,320,213,321,151,74,110,179,228),(10,229,180,111,75,152,322,214,289,266,42),(11,43,267,290,215,323,153,76,112,181,230),(12,231,182,113,77,154,324,216,291,268,44),(13,45,269,292,217,325,155,78,114,183,232),(14,233,184,115,79,156,326,218,293,270,46),(15,47,271,294,219,327,157,80,116,185,234),(16,235,186,117,81,158,328,220,295,272,48),(17,49,273,296,221,329,159,82,118,187,236),(18,237,188,119,83,160,330,222,297,274,50),(19,51,275,298,223,331,129,84,120,189,238),(20,239,190,121,85,130,332,224,299,276,52),(21,53,277,300,193,333,131,86,122,191,240),(22,241,192,123,87,132,334,194,301,278,54),(23,55,279,302,195,335,133,88,124,161,242),(24,243,162,125,89,134,336,196,303,280,56),(25,57,281,304,197,337,135,90,126,163,244),(26,245,164,127,91,136,338,198,305,282,58),(27,59,283,306,199,339,137,92,128,165,246),(28,247,166,97,93,138,340,200,307,284,60),(29,61,285,308,201,341,139,94,98,167,248),(30,249,168,99,95,140,342,202,309,286,62),(31,63,287,310,203,343,141,96,100,169,250),(32,251,170,101,65,142,344,204,311,288,64)], [(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 4A4B8A8B8C8D11A···11E16A···16H22A···22E32A···32P44A···44J88A···88T176A···176AN
order1244888811···1116···1622···2232···3244···4488···88176···176
size111111112···21···12···211···112···22···22···2

112 irreducible representations

dim11111122222
type+++-
imageC1C2C4C8C16C32D11Dic11C11⋊C8C11⋊C16C11⋊C32
kernelC11⋊C32C176C88C44C22C11C16C8C4C2C1
# reps112481655102040

Matrix representation of C11⋊C32 in GL3(𝔽353) generated by

100
03521
03457
,
28600
064137
0196289
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

Export

Subgroup lattice of C11⋊C32 in TeX

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