direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9×Dic11, C11⋊C36, C99⋊2C4, C33.C12, C22.C18, C66.2C6, C198.2C2, C18.2D11, C2.(C9×D11), C6.2(C3×D11), C3.(C3×Dic11), (C3×Dic11).C3, SmallGroup(396,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C9×Dic11 |
Generators and relations for C9×Dic11
G = < a,b,c | a9=b22=1, c2=b11, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C9×Dic11
►Regular action on 396 points
Generators in S396
(1 32 167 290 369 251 203 73 279)(2 33 168 291 370 252 204 74 280)(3 34 169 292 371 253 205 75 281)(4 35 170 293 372 254 206 76 282)(5 36 171 294 373 255 207 77 283)(6 37 172 295 374 256 208 78 284)(7 38 173 296 353 257 209 79 285)(8 39 174 297 354 258 210 80 286)(9 40 175 298 355 259 211 81 265)(10 41 176 299 356 260 212 82 266)(11 42 155 300 357 261 213 83 267)(12 43 156 301 358 262 214 84 268)(13 44 157 302 359 263 215 85 269)(14 23 158 303 360 264 216 86 270)(15 24 159 304 361 243 217 87 271)(16 25 160 305 362 244 218 88 272)(17 26 161 306 363 245 219 67 273)(18 27 162 307 364 246 220 68 274)(19 28 163 308 365 247 199 69 275)(20 29 164 287 366 248 200 70 276)(21 30 165 288 367 249 201 71 277)(22 31 166 289 368 250 202 72 278)(45 113 179 222 151 330 109 332 379)(46 114 180 223 152 309 110 333 380)(47 115 181 224 153 310 89 334 381)(48 116 182 225 154 311 90 335 382)(49 117 183 226 133 312 91 336 383)(50 118 184 227 134 313 92 337 384)(51 119 185 228 135 314 93 338 385)(52 120 186 229 136 315 94 339 386)(53 121 187 230 137 316 95 340 387)(54 122 188 231 138 317 96 341 388)(55 123 189 232 139 318 97 342 389)(56 124 190 233 140 319 98 343 390)(57 125 191 234 141 320 99 344 391)(58 126 192 235 142 321 100 345 392)(59 127 193 236 143 322 101 346 393)(60 128 194 237 144 323 102 347 394)(61 129 195 238 145 324 103 348 395)(62 130 196 239 146 325 104 349 396)(63 131 197 240 147 326 105 350 375)(64 132 198 241 148 327 106 351 376)(65 111 177 242 149 328 107 352 377)(66 112 178 221 150 329 108 331 378)
(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)
(1 136 12 147)(2 135 13 146)(3 134 14 145)(4 133 15 144)(5 154 16 143)(6 153 17 142)(7 152 18 141)(8 151 19 140)(9 150 20 139)(10 149 21 138)(11 148 22 137)(23 324 34 313)(24 323 35 312)(25 322 36 311)(26 321 37 310)(27 320 38 309)(28 319 39 330)(29 318 40 329)(30 317 41 328)(31 316 42 327)(32 315 43 326)(33 314 44 325)(45 247 56 258)(46 246 57 257)(47 245 58 256)(48 244 59 255)(49 243 60 254)(50 264 61 253)(51 263 62 252)(52 262 63 251)(53 261 64 250)(54 260 65 249)(55 259 66 248)(67 192 78 181)(68 191 79 180)(69 190 80 179)(70 189 81 178)(71 188 82 177)(72 187 83 198)(73 186 84 197)(74 185 85 196)(75 184 86 195)(76 183 87 194)(77 182 88 193)(89 161 100 172)(90 160 101 171)(91 159 102 170)(92 158 103 169)(93 157 104 168)(94 156 105 167)(95 155 106 166)(96 176 107 165)(97 175 108 164)(98 174 109 163)(99 173 110 162)(111 201 122 212)(112 200 123 211)(113 199 124 210)(114 220 125 209)(115 219 126 208)(116 218 127 207)(117 217 128 206)(118 216 129 205)(119 215 130 204)(120 214 131 203)(121 213 132 202)(221 276 232 265)(222 275 233 286)(223 274 234 285)(224 273 235 284)(225 272 236 283)(226 271 237 282)(227 270 238 281)(228 269 239 280)(229 268 240 279)(230 267 241 278)(231 266 242 277)(287 342 298 331)(288 341 299 352)(289 340 300 351)(290 339 301 350)(291 338 302 349)(292 337 303 348)(293 336 304 347)(294 335 305 346)(295 334 306 345)(296 333 307 344)(297 332 308 343)(353 380 364 391)(354 379 365 390)(355 378 366 389)(356 377 367 388)(357 376 368 387)(358 375 369 386)(359 396 370 385)(360 395 371 384)(361 394 372 383)(362 393 373 382)(363 392 374 381)
G:=sub<Sym(396)| (1,32,167,290,369,251,203,73,279)(2,33,168,291,370,252,204,74,280)(3,34,169,292,371,253,205,75,281)(4,35,170,293,372,254,206,76,282)(5,36,171,294,373,255,207,77,283)(6,37,172,295,374,256,208,78,284)(7,38,173,296,353,257,209,79,285)(8,39,174,297,354,258,210,80,286)(9,40,175,298,355,259,211,81,265)(10,41,176,299,356,260,212,82,266)(11,42,155,300,357,261,213,83,267)(12,43,156,301,358,262,214,84,268)(13,44,157,302,359,263,215,85,269)(14,23,158,303,360,264,216,86,270)(15,24,159,304,361,243,217,87,271)(16,25,160,305,362,244,218,88,272)(17,26,161,306,363,245,219,67,273)(18,27,162,307,364,246,220,68,274)(19,28,163,308,365,247,199,69,275)(20,29,164,287,366,248,200,70,276)(21,30,165,288,367,249,201,71,277)(22,31,166,289,368,250,202,72,278)(45,113,179,222,151,330,109,332,379)(46,114,180,223,152,309,110,333,380)(47,115,181,224,153,310,89,334,381)(48,116,182,225,154,311,90,335,382)(49,117,183,226,133,312,91,336,383)(50,118,184,227,134,313,92,337,384)(51,119,185,228,135,314,93,338,385)(52,120,186,229,136,315,94,339,386)(53,121,187,230,137,316,95,340,387)(54,122,188,231,138,317,96,341,388)(55,123,189,232,139,318,97,342,389)(56,124,190,233,140,319,98,343,390)(57,125,191,234,141,320,99,344,391)(58,126,192,235,142,321,100,345,392)(59,127,193,236,143,322,101,346,393)(60,128,194,237,144,323,102,347,394)(61,129,195,238,145,324,103,348,395)(62,130,196,239,146,325,104,349,396)(63,131,197,240,147,326,105,350,375)(64,132,198,241,148,327,106,351,376)(65,111,177,242,149,328,107,352,377)(66,112,178,221,150,329,108,331,378), (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), (1,136,12,147)(2,135,13,146)(3,134,14,145)(4,133,15,144)(5,154,16,143)(6,153,17,142)(7,152,18,141)(8,151,19,140)(9,150,20,139)(10,149,21,138)(11,148,22,137)(23,324,34,313)(24,323,35,312)(25,322,36,311)(26,321,37,310)(27,320,38,309)(28,319,39,330)(29,318,40,329)(30,317,41,328)(31,316,42,327)(32,315,43,326)(33,314,44,325)(45,247,56,258)(46,246,57,257)(47,245,58,256)(48,244,59,255)(49,243,60,254)(50,264,61,253)(51,263,62,252)(52,262,63,251)(53,261,64,250)(54,260,65,249)(55,259,66,248)(67,192,78,181)(68,191,79,180)(69,190,80,179)(70,189,81,178)(71,188,82,177)(72,187,83,198)(73,186,84,197)(74,185,85,196)(75,184,86,195)(76,183,87,194)(77,182,88,193)(89,161,100,172)(90,160,101,171)(91,159,102,170)(92,158,103,169)(93,157,104,168)(94,156,105,167)(95,155,106,166)(96,176,107,165)(97,175,108,164)(98,174,109,163)(99,173,110,162)(111,201,122,212)(112,200,123,211)(113,199,124,210)(114,220,125,209)(115,219,126,208)(116,218,127,207)(117,217,128,206)(118,216,129,205)(119,215,130,204)(120,214,131,203)(121,213,132,202)(221,276,232,265)(222,275,233,286)(223,274,234,285)(224,273,235,284)(225,272,236,283)(226,271,237,282)(227,270,238,281)(228,269,239,280)(229,268,240,279)(230,267,241,278)(231,266,242,277)(287,342,298,331)(288,341,299,352)(289,340,300,351)(290,339,301,350)(291,338,302,349)(292,337,303,348)(293,336,304,347)(294,335,305,346)(295,334,306,345)(296,333,307,344)(297,332,308,343)(353,380,364,391)(354,379,365,390)(355,378,366,389)(356,377,367,388)(357,376,368,387)(358,375,369,386)(359,396,370,385)(360,395,371,384)(361,394,372,383)(362,393,373,382)(363,392,374,381)>;
G:=Group( (1,32,167,290,369,251,203,73,279)(2,33,168,291,370,252,204,74,280)(3,34,169,292,371,253,205,75,281)(4,35,170,293,372,254,206,76,282)(5,36,171,294,373,255,207,77,283)(6,37,172,295,374,256,208,78,284)(7,38,173,296,353,257,209,79,285)(8,39,174,297,354,258,210,80,286)(9,40,175,298,355,259,211,81,265)(10,41,176,299,356,260,212,82,266)(11,42,155,300,357,261,213,83,267)(12,43,156,301,358,262,214,84,268)(13,44,157,302,359,263,215,85,269)(14,23,158,303,360,264,216,86,270)(15,24,159,304,361,243,217,87,271)(16,25,160,305,362,244,218,88,272)(17,26,161,306,363,245,219,67,273)(18,27,162,307,364,246,220,68,274)(19,28,163,308,365,247,199,69,275)(20,29,164,287,366,248,200,70,276)(21,30,165,288,367,249,201,71,277)(22,31,166,289,368,250,202,72,278)(45,113,179,222,151,330,109,332,379)(46,114,180,223,152,309,110,333,380)(47,115,181,224,153,310,89,334,381)(48,116,182,225,154,311,90,335,382)(49,117,183,226,133,312,91,336,383)(50,118,184,227,134,313,92,337,384)(51,119,185,228,135,314,93,338,385)(52,120,186,229,136,315,94,339,386)(53,121,187,230,137,316,95,340,387)(54,122,188,231,138,317,96,341,388)(55,123,189,232,139,318,97,342,389)(56,124,190,233,140,319,98,343,390)(57,125,191,234,141,320,99,344,391)(58,126,192,235,142,321,100,345,392)(59,127,193,236,143,322,101,346,393)(60,128,194,237,144,323,102,347,394)(61,129,195,238,145,324,103,348,395)(62,130,196,239,146,325,104,349,396)(63,131,197,240,147,326,105,350,375)(64,132,198,241,148,327,106,351,376)(65,111,177,242,149,328,107,352,377)(66,112,178,221,150,329,108,331,378), (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), (1,136,12,147)(2,135,13,146)(3,134,14,145)(4,133,15,144)(5,154,16,143)(6,153,17,142)(7,152,18,141)(8,151,19,140)(9,150,20,139)(10,149,21,138)(11,148,22,137)(23,324,34,313)(24,323,35,312)(25,322,36,311)(26,321,37,310)(27,320,38,309)(28,319,39,330)(29,318,40,329)(30,317,41,328)(31,316,42,327)(32,315,43,326)(33,314,44,325)(45,247,56,258)(46,246,57,257)(47,245,58,256)(48,244,59,255)(49,243,60,254)(50,264,61,253)(51,263,62,252)(52,262,63,251)(53,261,64,250)(54,260,65,249)(55,259,66,248)(67,192,78,181)(68,191,79,180)(69,190,80,179)(70,189,81,178)(71,188,82,177)(72,187,83,198)(73,186,84,197)(74,185,85,196)(75,184,86,195)(76,183,87,194)(77,182,88,193)(89,161,100,172)(90,160,101,171)(91,159,102,170)(92,158,103,169)(93,157,104,168)(94,156,105,167)(95,155,106,166)(96,176,107,165)(97,175,108,164)(98,174,109,163)(99,173,110,162)(111,201,122,212)(112,200,123,211)(113,199,124,210)(114,220,125,209)(115,219,126,208)(116,218,127,207)(117,217,128,206)(118,216,129,205)(119,215,130,204)(120,214,131,203)(121,213,132,202)(221,276,232,265)(222,275,233,286)(223,274,234,285)(224,273,235,284)(225,272,236,283)(226,271,237,282)(227,270,238,281)(228,269,239,280)(229,268,240,279)(230,267,241,278)(231,266,242,277)(287,342,298,331)(288,341,299,352)(289,340,300,351)(290,339,301,350)(291,338,302,349)(292,337,303,348)(293,336,304,347)(294,335,305,346)(295,334,306,345)(296,333,307,344)(297,332,308,343)(353,380,364,391)(354,379,365,390)(355,378,366,389)(356,377,367,388)(357,376,368,387)(358,375,369,386)(359,396,370,385)(360,395,371,384)(361,394,372,383)(362,393,373,382)(363,392,374,381) );
G=PermutationGroup([[(1,32,167,290,369,251,203,73,279),(2,33,168,291,370,252,204,74,280),(3,34,169,292,371,253,205,75,281),(4,35,170,293,372,254,206,76,282),(5,36,171,294,373,255,207,77,283),(6,37,172,295,374,256,208,78,284),(7,38,173,296,353,257,209,79,285),(8,39,174,297,354,258,210,80,286),(9,40,175,298,355,259,211,81,265),(10,41,176,299,356,260,212,82,266),(11,42,155,300,357,261,213,83,267),(12,43,156,301,358,262,214,84,268),(13,44,157,302,359,263,215,85,269),(14,23,158,303,360,264,216,86,270),(15,24,159,304,361,243,217,87,271),(16,25,160,305,362,244,218,88,272),(17,26,161,306,363,245,219,67,273),(18,27,162,307,364,246,220,68,274),(19,28,163,308,365,247,199,69,275),(20,29,164,287,366,248,200,70,276),(21,30,165,288,367,249,201,71,277),(22,31,166,289,368,250,202,72,278),(45,113,179,222,151,330,109,332,379),(46,114,180,223,152,309,110,333,380),(47,115,181,224,153,310,89,334,381),(48,116,182,225,154,311,90,335,382),(49,117,183,226,133,312,91,336,383),(50,118,184,227,134,313,92,337,384),(51,119,185,228,135,314,93,338,385),(52,120,186,229,136,315,94,339,386),(53,121,187,230,137,316,95,340,387),(54,122,188,231,138,317,96,341,388),(55,123,189,232,139,318,97,342,389),(56,124,190,233,140,319,98,343,390),(57,125,191,234,141,320,99,344,391),(58,126,192,235,142,321,100,345,392),(59,127,193,236,143,322,101,346,393),(60,128,194,237,144,323,102,347,394),(61,129,195,238,145,324,103,348,395),(62,130,196,239,146,325,104,349,396),(63,131,197,240,147,326,105,350,375),(64,132,198,241,148,327,106,351,376),(65,111,177,242,149,328,107,352,377),(66,112,178,221,150,329,108,331,378)], [(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)], [(1,136,12,147),(2,135,13,146),(3,134,14,145),(4,133,15,144),(5,154,16,143),(6,153,17,142),(7,152,18,141),(8,151,19,140),(9,150,20,139),(10,149,21,138),(11,148,22,137),(23,324,34,313),(24,323,35,312),(25,322,36,311),(26,321,37,310),(27,320,38,309),(28,319,39,330),(29,318,40,329),(30,317,41,328),(31,316,42,327),(32,315,43,326),(33,314,44,325),(45,247,56,258),(46,246,57,257),(47,245,58,256),(48,244,59,255),(49,243,60,254),(50,264,61,253),(51,263,62,252),(52,262,63,251),(53,261,64,250),(54,260,65,249),(55,259,66,248),(67,192,78,181),(68,191,79,180),(69,190,80,179),(70,189,81,178),(71,188,82,177),(72,187,83,198),(73,186,84,197),(74,185,85,196),(75,184,86,195),(76,183,87,194),(77,182,88,193),(89,161,100,172),(90,160,101,171),(91,159,102,170),(92,158,103,169),(93,157,104,168),(94,156,105,167),(95,155,106,166),(96,176,107,165),(97,175,108,164),(98,174,109,163),(99,173,110,162),(111,201,122,212),(112,200,123,211),(113,199,124,210),(114,220,125,209),(115,219,126,208),(116,218,127,207),(117,217,128,206),(118,216,129,205),(119,215,130,204),(120,214,131,203),(121,213,132,202),(221,276,232,265),(222,275,233,286),(223,274,234,285),(224,273,235,284),(225,272,236,283),(226,271,237,282),(227,270,238,281),(228,269,239,280),(229,268,240,279),(230,267,241,278),(231,266,242,277),(287,342,298,331),(288,341,299,352),(289,340,300,351),(290,339,301,350),(291,338,302,349),(292,337,303,348),(293,336,304,347),(294,335,305,346),(295,334,306,345),(296,333,307,344),(297,332,308,343),(353,380,364,391),(354,379,365,390),(355,378,366,389),(356,377,367,388),(357,376,368,387),(358,375,369,386),(359,396,370,385),(360,395,371,384),(361,394,372,383),(362,393,373,382),(363,392,374,381)]])
126 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 9A | ··· | 9F | 11A | ··· | 11E | 12A | 12B | 12C | 12D | 18A | ··· | 18F | 22A | ··· | 22E | 33A | ··· | 33J | 36A | ··· | 36L | 66A | ··· | 66J | 99A | ··· | 99AD | 198A | ··· | 198AD |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 9 | ··· | 9 | 11 | ··· | 11 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 22 | ··· | 22 | 33 | ··· | 33 | 36 | ··· | 36 | 66 | ··· | 66 | 99 | ··· | 99 | 198 | ··· | 198 |
| size | 1 | 1 | 1 | 1 | 11 | 11 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 11 | 11 | 11 | 11 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | D11 | Dic11 | C3×D11 | C3×Dic11 | C9×D11 | C9×Dic11 |
| kernel | C9×Dic11 | C198 | C3×Dic11 | C99 | C66 | Dic11 | C33 | C22 | C11 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 5 | 5 | 10 | 10 | 30 | 30 |
Matrix representation of C9×Dic11 ►in GL2(𝔽397) generated by
| 286 | 0 |
| 0 | 286 |
| 1 | 396 |
| 151 | 247 |
| 17 | 190 |
| 291 | 380 |
G:=sub<GL(2,GF(397))| [286,0,0,286],[1,151,396,247],[17,291,190,380] >;
C9×Dic11 in GAP, Magma, Sage, TeX
C_9\times {\rm Dic}_{11} % in TeX
G:=Group("C9xDic11"); // GroupNames label
G:=SmallGroup(396,2);
// by ID
G=gap.SmallGroup(396,2);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-11,30,66,9004]);
// Polycyclic
G:=Group<a,b,c|a^9=b^22=1,c^2=b^11,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export