direct product, abelian, monomial, 3-elementary
Aliases: C3×C111, SmallGroup(333,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C111 |
| C1 — C3×C111 |
| C1 — C3×C111 |
Generators and relations for C3×C111
G = < a,b | a3=b111=1, ab=ba >
Smallest permutation representation of C3×C111
►Regular action on 333 points
Generators in S333
(1 283 127)(2 284 128)(3 285 129)(4 286 130)(5 287 131)(6 288 132)(7 289 133)(8 290 134)(9 291 135)(10 292 136)(11 293 137)(12 294 138)(13 295 139)(14 296 140)(15 297 141)(16 298 142)(17 299 143)(18 300 144)(19 301 145)(20 302 146)(21 303 147)(22 304 148)(23 305 149)(24 306 150)(25 307 151)(26 308 152)(27 309 153)(28 310 154)(29 311 155)(30 312 156)(31 313 157)(32 314 158)(33 315 159)(34 316 160)(35 317 161)(36 318 162)(37 319 163)(38 320 164)(39 321 165)(40 322 166)(41 323 167)(42 324 168)(43 325 169)(44 326 170)(45 327 171)(46 328 172)(47 329 173)(48 330 174)(49 331 175)(50 332 176)(51 333 177)(52 223 178)(53 224 179)(54 225 180)(55 226 181)(56 227 182)(57 228 183)(58 229 184)(59 230 185)(60 231 186)(61 232 187)(62 233 188)(63 234 189)(64 235 190)(65 236 191)(66 237 192)(67 238 193)(68 239 194)(69 240 195)(70 241 196)(71 242 197)(72 243 198)(73 244 199)(74 245 200)(75 246 201)(76 247 202)(77 248 203)(78 249 204)(79 250 205)(80 251 206)(81 252 207)(82 253 208)(83 254 209)(84 255 210)(85 256 211)(86 257 212)(87 258 213)(88 259 214)(89 260 215)(90 261 216)(91 262 217)(92 263 218)(93 264 219)(94 265 220)(95 266 221)(96 267 222)(97 268 112)(98 269 113)(99 270 114)(100 271 115)(101 272 116)(102 273 117)(103 274 118)(104 275 119)(105 276 120)(106 277 121)(107 278 122)(108 279 123)(109 280 124)(110 281 125)(111 282 126)
(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)
G:=sub<Sym(333)| (1,283,127)(2,284,128)(3,285,129)(4,286,130)(5,287,131)(6,288,132)(7,289,133)(8,290,134)(9,291,135)(10,292,136)(11,293,137)(12,294,138)(13,295,139)(14,296,140)(15,297,141)(16,298,142)(17,299,143)(18,300,144)(19,301,145)(20,302,146)(21,303,147)(22,304,148)(23,305,149)(24,306,150)(25,307,151)(26,308,152)(27,309,153)(28,310,154)(29,311,155)(30,312,156)(31,313,157)(32,314,158)(33,315,159)(34,316,160)(35,317,161)(36,318,162)(37,319,163)(38,320,164)(39,321,165)(40,322,166)(41,323,167)(42,324,168)(43,325,169)(44,326,170)(45,327,171)(46,328,172)(47,329,173)(48,330,174)(49,331,175)(50,332,176)(51,333,177)(52,223,178)(53,224,179)(54,225,180)(55,226,181)(56,227,182)(57,228,183)(58,229,184)(59,230,185)(60,231,186)(61,232,187)(62,233,188)(63,234,189)(64,235,190)(65,236,191)(66,237,192)(67,238,193)(68,239,194)(69,240,195)(70,241,196)(71,242,197)(72,243,198)(73,244,199)(74,245,200)(75,246,201)(76,247,202)(77,248,203)(78,249,204)(79,250,205)(80,251,206)(81,252,207)(82,253,208)(83,254,209)(84,255,210)(85,256,211)(86,257,212)(87,258,213)(88,259,214)(89,260,215)(90,261,216)(91,262,217)(92,263,218)(93,264,219)(94,265,220)(95,266,221)(96,267,222)(97,268,112)(98,269,113)(99,270,114)(100,271,115)(101,272,116)(102,273,117)(103,274,118)(104,275,119)(105,276,120)(106,277,121)(107,278,122)(108,279,123)(109,280,124)(110,281,125)(111,282,126), (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)>;
G:=Group( (1,283,127)(2,284,128)(3,285,129)(4,286,130)(5,287,131)(6,288,132)(7,289,133)(8,290,134)(9,291,135)(10,292,136)(11,293,137)(12,294,138)(13,295,139)(14,296,140)(15,297,141)(16,298,142)(17,299,143)(18,300,144)(19,301,145)(20,302,146)(21,303,147)(22,304,148)(23,305,149)(24,306,150)(25,307,151)(26,308,152)(27,309,153)(28,310,154)(29,311,155)(30,312,156)(31,313,157)(32,314,158)(33,315,159)(34,316,160)(35,317,161)(36,318,162)(37,319,163)(38,320,164)(39,321,165)(40,322,166)(41,323,167)(42,324,168)(43,325,169)(44,326,170)(45,327,171)(46,328,172)(47,329,173)(48,330,174)(49,331,175)(50,332,176)(51,333,177)(52,223,178)(53,224,179)(54,225,180)(55,226,181)(56,227,182)(57,228,183)(58,229,184)(59,230,185)(60,231,186)(61,232,187)(62,233,188)(63,234,189)(64,235,190)(65,236,191)(66,237,192)(67,238,193)(68,239,194)(69,240,195)(70,241,196)(71,242,197)(72,243,198)(73,244,199)(74,245,200)(75,246,201)(76,247,202)(77,248,203)(78,249,204)(79,250,205)(80,251,206)(81,252,207)(82,253,208)(83,254,209)(84,255,210)(85,256,211)(86,257,212)(87,258,213)(88,259,214)(89,260,215)(90,261,216)(91,262,217)(92,263,218)(93,264,219)(94,265,220)(95,266,221)(96,267,222)(97,268,112)(98,269,113)(99,270,114)(100,271,115)(101,272,116)(102,273,117)(103,274,118)(104,275,119)(105,276,120)(106,277,121)(107,278,122)(108,279,123)(109,280,124)(110,281,125)(111,282,126), (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) );
G=PermutationGroup([[(1,283,127),(2,284,128),(3,285,129),(4,286,130),(5,287,131),(6,288,132),(7,289,133),(8,290,134),(9,291,135),(10,292,136),(11,293,137),(12,294,138),(13,295,139),(14,296,140),(15,297,141),(16,298,142),(17,299,143),(18,300,144),(19,301,145),(20,302,146),(21,303,147),(22,304,148),(23,305,149),(24,306,150),(25,307,151),(26,308,152),(27,309,153),(28,310,154),(29,311,155),(30,312,156),(31,313,157),(32,314,158),(33,315,159),(34,316,160),(35,317,161),(36,318,162),(37,319,163),(38,320,164),(39,321,165),(40,322,166),(41,323,167),(42,324,168),(43,325,169),(44,326,170),(45,327,171),(46,328,172),(47,329,173),(48,330,174),(49,331,175),(50,332,176),(51,333,177),(52,223,178),(53,224,179),(54,225,180),(55,226,181),(56,227,182),(57,228,183),(58,229,184),(59,230,185),(60,231,186),(61,232,187),(62,233,188),(63,234,189),(64,235,190),(65,236,191),(66,237,192),(67,238,193),(68,239,194),(69,240,195),(70,241,196),(71,242,197),(72,243,198),(73,244,199),(74,245,200),(75,246,201),(76,247,202),(77,248,203),(78,249,204),(79,250,205),(80,251,206),(81,252,207),(82,253,208),(83,254,209),(84,255,210),(85,256,211),(86,257,212),(87,258,213),(88,259,214),(89,260,215),(90,261,216),(91,262,217),(92,263,218),(93,264,219),(94,265,220),(95,266,221),(96,267,222),(97,268,112),(98,269,113),(99,270,114),(100,271,115),(101,272,116),(102,273,117),(103,274,118),(104,275,119),(105,276,120),(106,277,121),(107,278,122),(108,279,123),(109,280,124),(110,281,125),(111,282,126)], [(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)]])
333 conjugacy classes
| class | 1 | 3A | ··· | 3H | 37A | ··· | 37AJ | 111A | ··· | 111KB |
| order | 1 | 3 | ··· | 3 | 37 | ··· | 37 | 111 | ··· | 111 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
333 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C37 | C111 |
| kernel | C3×C111 | C111 | C32 | C3 |
| # reps | 1 | 8 | 36 | 288 |
Matrix representation of C3×C111 ►in GL2(𝔽223) generated by
| 1 | 0 |
| 0 | 183 |
| 121 | 0 |
| 0 | 115 |
G:=sub<GL(2,GF(223))| [1,0,0,183],[121,0,0,115] >;
C3×C111 in GAP, Magma, Sage, TeX
C_3\times C_{111} % in TeX
G:=Group("C3xC111"); // GroupNames label
G:=SmallGroup(333,5);
// by ID
G=gap.SmallGroup(333,5);
# by ID
G:=PCGroup([3,-3,-3,-37]);
// Polycyclic
G:=Group<a,b|a^3=b^111=1,a*b=b*a>;
// generators/relations
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