direct product, abelian, monomial, 11-elementary
Aliases: C11×C33, SmallGroup(363,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C11×C33 |
| C1 — C11×C33 |
| C1 — C11×C33 |
Generators and relations for C11×C33
G = < a,b | a11=b33=1, ab=ba >
Smallest permutation representation of C11×C33
►Regular action on 363 points
Generators in S363
(1 329 211 152 99 49 354 236 178 286 130)(2 330 212 153 67 50 355 237 179 287 131)(3 298 213 154 68 51 356 238 180 288 132)(4 299 214 155 69 52 357 239 181 289 100)(5 300 215 156 70 53 358 240 182 290 101)(6 301 216 157 71 54 359 241 183 291 102)(7 302 217 158 72 55 360 242 184 292 103)(8 303 218 159 73 56 361 243 185 293 104)(9 304 219 160 74 57 362 244 186 294 105)(10 305 220 161 75 58 363 245 187 295 106)(11 306 221 162 76 59 331 246 188 296 107)(12 307 222 163 77 60 332 247 189 297 108)(13 308 223 164 78 61 333 248 190 265 109)(14 309 224 165 79 62 334 249 191 266 110)(15 310 225 133 80 63 335 250 192 267 111)(16 311 226 134 81 64 336 251 193 268 112)(17 312 227 135 82 65 337 252 194 269 113)(18 313 228 136 83 66 338 253 195 270 114)(19 314 229 137 84 34 339 254 196 271 115)(20 315 230 138 85 35 340 255 197 272 116)(21 316 231 139 86 36 341 256 198 273 117)(22 317 199 140 87 37 342 257 166 274 118)(23 318 200 141 88 38 343 258 167 275 119)(24 319 201 142 89 39 344 259 168 276 120)(25 320 202 143 90 40 345 260 169 277 121)(26 321 203 144 91 41 346 261 170 278 122)(27 322 204 145 92 42 347 262 171 279 123)(28 323 205 146 93 43 348 263 172 280 124)(29 324 206 147 94 44 349 264 173 281 125)(30 325 207 148 95 45 350 232 174 282 126)(31 326 208 149 96 46 351 233 175 283 127)(32 327 209 150 97 47 352 234 176 284 128)(33 328 210 151 98 48 353 235 177 285 129)
(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)
G:=sub<Sym(363)| (1,329,211,152,99,49,354,236,178,286,130)(2,330,212,153,67,50,355,237,179,287,131)(3,298,213,154,68,51,356,238,180,288,132)(4,299,214,155,69,52,357,239,181,289,100)(5,300,215,156,70,53,358,240,182,290,101)(6,301,216,157,71,54,359,241,183,291,102)(7,302,217,158,72,55,360,242,184,292,103)(8,303,218,159,73,56,361,243,185,293,104)(9,304,219,160,74,57,362,244,186,294,105)(10,305,220,161,75,58,363,245,187,295,106)(11,306,221,162,76,59,331,246,188,296,107)(12,307,222,163,77,60,332,247,189,297,108)(13,308,223,164,78,61,333,248,190,265,109)(14,309,224,165,79,62,334,249,191,266,110)(15,310,225,133,80,63,335,250,192,267,111)(16,311,226,134,81,64,336,251,193,268,112)(17,312,227,135,82,65,337,252,194,269,113)(18,313,228,136,83,66,338,253,195,270,114)(19,314,229,137,84,34,339,254,196,271,115)(20,315,230,138,85,35,340,255,197,272,116)(21,316,231,139,86,36,341,256,198,273,117)(22,317,199,140,87,37,342,257,166,274,118)(23,318,200,141,88,38,343,258,167,275,119)(24,319,201,142,89,39,344,259,168,276,120)(25,320,202,143,90,40,345,260,169,277,121)(26,321,203,144,91,41,346,261,170,278,122)(27,322,204,145,92,42,347,262,171,279,123)(28,323,205,146,93,43,348,263,172,280,124)(29,324,206,147,94,44,349,264,173,281,125)(30,325,207,148,95,45,350,232,174,282,126)(31,326,208,149,96,46,351,233,175,283,127)(32,327,209,150,97,47,352,234,176,284,128)(33,328,210,151,98,48,353,235,177,285,129), (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)>;
G:=Group( (1,329,211,152,99,49,354,236,178,286,130)(2,330,212,153,67,50,355,237,179,287,131)(3,298,213,154,68,51,356,238,180,288,132)(4,299,214,155,69,52,357,239,181,289,100)(5,300,215,156,70,53,358,240,182,290,101)(6,301,216,157,71,54,359,241,183,291,102)(7,302,217,158,72,55,360,242,184,292,103)(8,303,218,159,73,56,361,243,185,293,104)(9,304,219,160,74,57,362,244,186,294,105)(10,305,220,161,75,58,363,245,187,295,106)(11,306,221,162,76,59,331,246,188,296,107)(12,307,222,163,77,60,332,247,189,297,108)(13,308,223,164,78,61,333,248,190,265,109)(14,309,224,165,79,62,334,249,191,266,110)(15,310,225,133,80,63,335,250,192,267,111)(16,311,226,134,81,64,336,251,193,268,112)(17,312,227,135,82,65,337,252,194,269,113)(18,313,228,136,83,66,338,253,195,270,114)(19,314,229,137,84,34,339,254,196,271,115)(20,315,230,138,85,35,340,255,197,272,116)(21,316,231,139,86,36,341,256,198,273,117)(22,317,199,140,87,37,342,257,166,274,118)(23,318,200,141,88,38,343,258,167,275,119)(24,319,201,142,89,39,344,259,168,276,120)(25,320,202,143,90,40,345,260,169,277,121)(26,321,203,144,91,41,346,261,170,278,122)(27,322,204,145,92,42,347,262,171,279,123)(28,323,205,146,93,43,348,263,172,280,124)(29,324,206,147,94,44,349,264,173,281,125)(30,325,207,148,95,45,350,232,174,282,126)(31,326,208,149,96,46,351,233,175,283,127)(32,327,209,150,97,47,352,234,176,284,128)(33,328,210,151,98,48,353,235,177,285,129), (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) );
G=PermutationGroup([[(1,329,211,152,99,49,354,236,178,286,130),(2,330,212,153,67,50,355,237,179,287,131),(3,298,213,154,68,51,356,238,180,288,132),(4,299,214,155,69,52,357,239,181,289,100),(5,300,215,156,70,53,358,240,182,290,101),(6,301,216,157,71,54,359,241,183,291,102),(7,302,217,158,72,55,360,242,184,292,103),(8,303,218,159,73,56,361,243,185,293,104),(9,304,219,160,74,57,362,244,186,294,105),(10,305,220,161,75,58,363,245,187,295,106),(11,306,221,162,76,59,331,246,188,296,107),(12,307,222,163,77,60,332,247,189,297,108),(13,308,223,164,78,61,333,248,190,265,109),(14,309,224,165,79,62,334,249,191,266,110),(15,310,225,133,80,63,335,250,192,267,111),(16,311,226,134,81,64,336,251,193,268,112),(17,312,227,135,82,65,337,252,194,269,113),(18,313,228,136,83,66,338,253,195,270,114),(19,314,229,137,84,34,339,254,196,271,115),(20,315,230,138,85,35,340,255,197,272,116),(21,316,231,139,86,36,341,256,198,273,117),(22,317,199,140,87,37,342,257,166,274,118),(23,318,200,141,88,38,343,258,167,275,119),(24,319,201,142,89,39,344,259,168,276,120),(25,320,202,143,90,40,345,260,169,277,121),(26,321,203,144,91,41,346,261,170,278,122),(27,322,204,145,92,42,347,262,171,279,123),(28,323,205,146,93,43,348,263,172,280,124),(29,324,206,147,94,44,349,264,173,281,125),(30,325,207,148,95,45,350,232,174,282,126),(31,326,208,149,96,46,351,233,175,283,127),(32,327,209,150,97,47,352,234,176,284,128),(33,328,210,151,98,48,353,235,177,285,129)], [(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)]])
363 conjugacy classes
| class | 1 | 3A | 3B | 11A | ··· | 11DP | 33A | ··· | 33IF |
| order | 1 | 3 | 3 | 11 | ··· | 11 | 33 | ··· | 33 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
363 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C11 | C33 |
| kernel | C11×C33 | C112 | C33 | C11 |
| # reps | 1 | 2 | 120 | 240 |
Matrix representation of C11×C33 ►in GL2(𝔽67) generated by
| 9 | 0 |
| 0 | 22 |
| 22 | 0 |
| 0 | 35 |
G:=sub<GL(2,GF(67))| [9,0,0,22],[22,0,0,35] >;
C11×C33 in GAP, Magma, Sage, TeX
C_{11}\times C_{33} % in TeX
G:=Group("C11xC33"); // GroupNames label
G:=SmallGroup(363,3);
// by ID
G=gap.SmallGroup(363,3);
# by ID
G:=PCGroup([3,-3,-11,-11]);
// Polycyclic
G:=Group<a,b|a^11=b^33=1,a*b=b*a>;
// generators/relations
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