direct product, abelian, monomial, 3-elementary
Aliases: C3×C123, SmallGroup(369,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C123 |
| C1 — C3×C123 |
| C1 — C3×C123 |
Generators and relations for C3×C123
G = < a,b | a3=b123=1, ab=ba >
Smallest permutation representation of C3×C123
►Regular action on 369 points
Generators in S369
(1 302 203)(2 303 204)(3 304 205)(4 305 206)(5 306 207)(6 307 208)(7 308 209)(8 309 210)(9 310 211)(10 311 212)(11 312 213)(12 313 214)(13 314 215)(14 315 216)(15 316 217)(16 317 218)(17 318 219)(18 319 220)(19 320 221)(20 321 222)(21 322 223)(22 323 224)(23 324 225)(24 325 226)(25 326 227)(26 327 228)(27 328 229)(28 329 230)(29 330 231)(30 331 232)(31 332 233)(32 333 234)(33 334 235)(34 335 236)(35 336 237)(36 337 238)(37 338 239)(38 339 240)(39 340 241)(40 341 242)(41 342 243)(42 343 244)(43 344 245)(44 345 246)(45 346 124)(46 347 125)(47 348 126)(48 349 127)(49 350 128)(50 351 129)(51 352 130)(52 353 131)(53 354 132)(54 355 133)(55 356 134)(56 357 135)(57 358 136)(58 359 137)(59 360 138)(60 361 139)(61 362 140)(62 363 141)(63 364 142)(64 365 143)(65 366 144)(66 367 145)(67 368 146)(68 369 147)(69 247 148)(70 248 149)(71 249 150)(72 250 151)(73 251 152)(74 252 153)(75 253 154)(76 254 155)(77 255 156)(78 256 157)(79 257 158)(80 258 159)(81 259 160)(82 260 161)(83 261 162)(84 262 163)(85 263 164)(86 264 165)(87 265 166)(88 266 167)(89 267 168)(90 268 169)(91 269 170)(92 270 171)(93 271 172)(94 272 173)(95 273 174)(96 274 175)(97 275 176)(98 276 177)(99 277 178)(100 278 179)(101 279 180)(102 280 181)(103 281 182)(104 282 183)(105 283 184)(106 284 185)(107 285 186)(108 286 187)(109 287 188)(110 288 189)(111 289 190)(112 290 191)(113 291 192)(114 292 193)(115 293 194)(116 294 195)(117 295 196)(118 296 197)(119 297 198)(120 298 199)(121 299 200)(122 300 201)(123 301 202)
(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)
G:=sub<Sym(369)| (1,302,203)(2,303,204)(3,304,205)(4,305,206)(5,306,207)(6,307,208)(7,308,209)(8,309,210)(9,310,211)(10,311,212)(11,312,213)(12,313,214)(13,314,215)(14,315,216)(15,316,217)(16,317,218)(17,318,219)(18,319,220)(19,320,221)(20,321,222)(21,322,223)(22,323,224)(23,324,225)(24,325,226)(25,326,227)(26,327,228)(27,328,229)(28,329,230)(29,330,231)(30,331,232)(31,332,233)(32,333,234)(33,334,235)(34,335,236)(35,336,237)(36,337,238)(37,338,239)(38,339,240)(39,340,241)(40,341,242)(41,342,243)(42,343,244)(43,344,245)(44,345,246)(45,346,124)(46,347,125)(47,348,126)(48,349,127)(49,350,128)(50,351,129)(51,352,130)(52,353,131)(53,354,132)(54,355,133)(55,356,134)(56,357,135)(57,358,136)(58,359,137)(59,360,138)(60,361,139)(61,362,140)(62,363,141)(63,364,142)(64,365,143)(65,366,144)(66,367,145)(67,368,146)(68,369,147)(69,247,148)(70,248,149)(71,249,150)(72,250,151)(73,251,152)(74,252,153)(75,253,154)(76,254,155)(77,255,156)(78,256,157)(79,257,158)(80,258,159)(81,259,160)(82,260,161)(83,261,162)(84,262,163)(85,263,164)(86,264,165)(87,265,166)(88,266,167)(89,267,168)(90,268,169)(91,269,170)(92,270,171)(93,271,172)(94,272,173)(95,273,174)(96,274,175)(97,275,176)(98,276,177)(99,277,178)(100,278,179)(101,279,180)(102,280,181)(103,281,182)(104,282,183)(105,283,184)(106,284,185)(107,285,186)(108,286,187)(109,287,188)(110,288,189)(111,289,190)(112,290,191)(113,291,192)(114,292,193)(115,293,194)(116,294,195)(117,295,196)(118,296,197)(119,297,198)(120,298,199)(121,299,200)(122,300,201)(123,301,202), (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)>;
G:=Group( (1,302,203)(2,303,204)(3,304,205)(4,305,206)(5,306,207)(6,307,208)(7,308,209)(8,309,210)(9,310,211)(10,311,212)(11,312,213)(12,313,214)(13,314,215)(14,315,216)(15,316,217)(16,317,218)(17,318,219)(18,319,220)(19,320,221)(20,321,222)(21,322,223)(22,323,224)(23,324,225)(24,325,226)(25,326,227)(26,327,228)(27,328,229)(28,329,230)(29,330,231)(30,331,232)(31,332,233)(32,333,234)(33,334,235)(34,335,236)(35,336,237)(36,337,238)(37,338,239)(38,339,240)(39,340,241)(40,341,242)(41,342,243)(42,343,244)(43,344,245)(44,345,246)(45,346,124)(46,347,125)(47,348,126)(48,349,127)(49,350,128)(50,351,129)(51,352,130)(52,353,131)(53,354,132)(54,355,133)(55,356,134)(56,357,135)(57,358,136)(58,359,137)(59,360,138)(60,361,139)(61,362,140)(62,363,141)(63,364,142)(64,365,143)(65,366,144)(66,367,145)(67,368,146)(68,369,147)(69,247,148)(70,248,149)(71,249,150)(72,250,151)(73,251,152)(74,252,153)(75,253,154)(76,254,155)(77,255,156)(78,256,157)(79,257,158)(80,258,159)(81,259,160)(82,260,161)(83,261,162)(84,262,163)(85,263,164)(86,264,165)(87,265,166)(88,266,167)(89,267,168)(90,268,169)(91,269,170)(92,270,171)(93,271,172)(94,272,173)(95,273,174)(96,274,175)(97,275,176)(98,276,177)(99,277,178)(100,278,179)(101,279,180)(102,280,181)(103,281,182)(104,282,183)(105,283,184)(106,284,185)(107,285,186)(108,286,187)(109,287,188)(110,288,189)(111,289,190)(112,290,191)(113,291,192)(114,292,193)(115,293,194)(116,294,195)(117,295,196)(118,296,197)(119,297,198)(120,298,199)(121,299,200)(122,300,201)(123,301,202), (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) );
G=PermutationGroup([[(1,302,203),(2,303,204),(3,304,205),(4,305,206),(5,306,207),(6,307,208),(7,308,209),(8,309,210),(9,310,211),(10,311,212),(11,312,213),(12,313,214),(13,314,215),(14,315,216),(15,316,217),(16,317,218),(17,318,219),(18,319,220),(19,320,221),(20,321,222),(21,322,223),(22,323,224),(23,324,225),(24,325,226),(25,326,227),(26,327,228),(27,328,229),(28,329,230),(29,330,231),(30,331,232),(31,332,233),(32,333,234),(33,334,235),(34,335,236),(35,336,237),(36,337,238),(37,338,239),(38,339,240),(39,340,241),(40,341,242),(41,342,243),(42,343,244),(43,344,245),(44,345,246),(45,346,124),(46,347,125),(47,348,126),(48,349,127),(49,350,128),(50,351,129),(51,352,130),(52,353,131),(53,354,132),(54,355,133),(55,356,134),(56,357,135),(57,358,136),(58,359,137),(59,360,138),(60,361,139),(61,362,140),(62,363,141),(63,364,142),(64,365,143),(65,366,144),(66,367,145),(67,368,146),(68,369,147),(69,247,148),(70,248,149),(71,249,150),(72,250,151),(73,251,152),(74,252,153),(75,253,154),(76,254,155),(77,255,156),(78,256,157),(79,257,158),(80,258,159),(81,259,160),(82,260,161),(83,261,162),(84,262,163),(85,263,164),(86,264,165),(87,265,166),(88,266,167),(89,267,168),(90,268,169),(91,269,170),(92,270,171),(93,271,172),(94,272,173),(95,273,174),(96,274,175),(97,275,176),(98,276,177),(99,277,178),(100,278,179),(101,279,180),(102,280,181),(103,281,182),(104,282,183),(105,283,184),(106,284,185),(107,285,186),(108,286,187),(109,287,188),(110,288,189),(111,289,190),(112,290,191),(113,291,192),(114,292,193),(115,293,194),(116,294,195),(117,295,196),(118,296,197),(119,297,198),(120,298,199),(121,299,200),(122,300,201),(123,301,202)], [(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)]])
369 conjugacy classes
| class | 1 | 3A | ··· | 3H | 41A | ··· | 41AN | 123A | ··· | 123LH |
| order | 1 | 3 | ··· | 3 | 41 | ··· | 41 | 123 | ··· | 123 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
369 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C41 | C123 |
| kernel | C3×C123 | C123 | C32 | C3 |
| # reps | 1 | 8 | 40 | 320 |
Matrix representation of C3×C123 ►in GL2(𝔽739) generated by
| 1 | 0 |
| 0 | 320 |
| 433 | 0 |
| 0 | 433 |
G:=sub<GL(2,GF(739))| [1,0,0,320],[433,0,0,433] >;
C3×C123 in GAP, Magma, Sage, TeX
C_3\times C_{123} % in TeX
G:=Group("C3xC123"); // GroupNames label
G:=SmallGroup(369,2);
// by ID
G=gap.SmallGroup(369,2);
# by ID
G:=PCGroup([3,-3,-3,-41]);
// Polycyclic
G:=Group<a,b|a^3=b^123=1,a*b=b*a>;
// generators/relations
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