direct product, abelian, monomial, 3-elementary
Aliases: C3×C99, SmallGroup(297,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C99 |
| C1 — C3×C99 |
| C1 — C3×C99 |
Generators and relations for C3×C99
G = < a,b | a3=b99=1, ab=ba >
Smallest permutation representation of C3×C99
►Regular action on 297 points
Generators in S297
(1 155 247)(2 156 248)(3 157 249)(4 158 250)(5 159 251)(6 160 252)(7 161 253)(8 162 254)(9 163 255)(10 164 256)(11 165 257)(12 166 258)(13 167 259)(14 168 260)(15 169 261)(16 170 262)(17 171 263)(18 172 264)(19 173 265)(20 174 266)(21 175 267)(22 176 268)(23 177 269)(24 178 270)(25 179 271)(26 180 272)(27 181 273)(28 182 274)(29 183 275)(30 184 276)(31 185 277)(32 186 278)(33 187 279)(34 188 280)(35 189 281)(36 190 282)(37 191 283)(38 192 284)(39 193 285)(40 194 286)(41 195 287)(42 196 288)(43 197 289)(44 198 290)(45 100 291)(46 101 292)(47 102 293)(48 103 294)(49 104 295)(50 105 296)(51 106 297)(52 107 199)(53 108 200)(54 109 201)(55 110 202)(56 111 203)(57 112 204)(58 113 205)(59 114 206)(60 115 207)(61 116 208)(62 117 209)(63 118 210)(64 119 211)(65 120 212)(66 121 213)(67 122 214)(68 123 215)(69 124 216)(70 125 217)(71 126 218)(72 127 219)(73 128 220)(74 129 221)(75 130 222)(76 131 223)(77 132 224)(78 133 225)(79 134 226)(80 135 227)(81 136 228)(82 137 229)(83 138 230)(84 139 231)(85 140 232)(86 141 233)(87 142 234)(88 143 235)(89 144 236)(90 145 237)(91 146 238)(92 147 239)(93 148 240)(94 149 241)(95 150 242)(96 151 243)(97 152 244)(98 153 245)(99 154 246)
(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)
G:=sub<Sym(297)| (1,155,247)(2,156,248)(3,157,249)(4,158,250)(5,159,251)(6,160,252)(7,161,253)(8,162,254)(9,163,255)(10,164,256)(11,165,257)(12,166,258)(13,167,259)(14,168,260)(15,169,261)(16,170,262)(17,171,263)(18,172,264)(19,173,265)(20,174,266)(21,175,267)(22,176,268)(23,177,269)(24,178,270)(25,179,271)(26,180,272)(27,181,273)(28,182,274)(29,183,275)(30,184,276)(31,185,277)(32,186,278)(33,187,279)(34,188,280)(35,189,281)(36,190,282)(37,191,283)(38,192,284)(39,193,285)(40,194,286)(41,195,287)(42,196,288)(43,197,289)(44,198,290)(45,100,291)(46,101,292)(47,102,293)(48,103,294)(49,104,295)(50,105,296)(51,106,297)(52,107,199)(53,108,200)(54,109,201)(55,110,202)(56,111,203)(57,112,204)(58,113,205)(59,114,206)(60,115,207)(61,116,208)(62,117,209)(63,118,210)(64,119,211)(65,120,212)(66,121,213)(67,122,214)(68,123,215)(69,124,216)(70,125,217)(71,126,218)(72,127,219)(73,128,220)(74,129,221)(75,130,222)(76,131,223)(77,132,224)(78,133,225)(79,134,226)(80,135,227)(81,136,228)(82,137,229)(83,138,230)(84,139,231)(85,140,232)(86,141,233)(87,142,234)(88,143,235)(89,144,236)(90,145,237)(91,146,238)(92,147,239)(93,148,240)(94,149,241)(95,150,242)(96,151,243)(97,152,244)(98,153,245)(99,154,246), (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)>;
G:=Group( (1,155,247)(2,156,248)(3,157,249)(4,158,250)(5,159,251)(6,160,252)(7,161,253)(8,162,254)(9,163,255)(10,164,256)(11,165,257)(12,166,258)(13,167,259)(14,168,260)(15,169,261)(16,170,262)(17,171,263)(18,172,264)(19,173,265)(20,174,266)(21,175,267)(22,176,268)(23,177,269)(24,178,270)(25,179,271)(26,180,272)(27,181,273)(28,182,274)(29,183,275)(30,184,276)(31,185,277)(32,186,278)(33,187,279)(34,188,280)(35,189,281)(36,190,282)(37,191,283)(38,192,284)(39,193,285)(40,194,286)(41,195,287)(42,196,288)(43,197,289)(44,198,290)(45,100,291)(46,101,292)(47,102,293)(48,103,294)(49,104,295)(50,105,296)(51,106,297)(52,107,199)(53,108,200)(54,109,201)(55,110,202)(56,111,203)(57,112,204)(58,113,205)(59,114,206)(60,115,207)(61,116,208)(62,117,209)(63,118,210)(64,119,211)(65,120,212)(66,121,213)(67,122,214)(68,123,215)(69,124,216)(70,125,217)(71,126,218)(72,127,219)(73,128,220)(74,129,221)(75,130,222)(76,131,223)(77,132,224)(78,133,225)(79,134,226)(80,135,227)(81,136,228)(82,137,229)(83,138,230)(84,139,231)(85,140,232)(86,141,233)(87,142,234)(88,143,235)(89,144,236)(90,145,237)(91,146,238)(92,147,239)(93,148,240)(94,149,241)(95,150,242)(96,151,243)(97,152,244)(98,153,245)(99,154,246), (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) );
G=PermutationGroup([[(1,155,247),(2,156,248),(3,157,249),(4,158,250),(5,159,251),(6,160,252),(7,161,253),(8,162,254),(9,163,255),(10,164,256),(11,165,257),(12,166,258),(13,167,259),(14,168,260),(15,169,261),(16,170,262),(17,171,263),(18,172,264),(19,173,265),(20,174,266),(21,175,267),(22,176,268),(23,177,269),(24,178,270),(25,179,271),(26,180,272),(27,181,273),(28,182,274),(29,183,275),(30,184,276),(31,185,277),(32,186,278),(33,187,279),(34,188,280),(35,189,281),(36,190,282),(37,191,283),(38,192,284),(39,193,285),(40,194,286),(41,195,287),(42,196,288),(43,197,289),(44,198,290),(45,100,291),(46,101,292),(47,102,293),(48,103,294),(49,104,295),(50,105,296),(51,106,297),(52,107,199),(53,108,200),(54,109,201),(55,110,202),(56,111,203),(57,112,204),(58,113,205),(59,114,206),(60,115,207),(61,116,208),(62,117,209),(63,118,210),(64,119,211),(65,120,212),(66,121,213),(67,122,214),(68,123,215),(69,124,216),(70,125,217),(71,126,218),(72,127,219),(73,128,220),(74,129,221),(75,130,222),(76,131,223),(77,132,224),(78,133,225),(79,134,226),(80,135,227),(81,136,228),(82,137,229),(83,138,230),(84,139,231),(85,140,232),(86,141,233),(87,142,234),(88,143,235),(89,144,236),(90,145,237),(91,146,238),(92,147,239),(93,148,240),(94,149,241),(95,150,242),(96,151,243),(97,152,244),(98,153,245),(99,154,246)], [(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)]])
297 conjugacy classes
| class | 1 | 3A | ··· | 3H | 9A | ··· | 9R | 11A | ··· | 11J | 33A | ··· | 33CB | 99A | ··· | 99FX |
| order | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 11 | ··· | 11 | 33 | ··· | 33 | 99 | ··· | 99 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
297 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||||
| image | C1 | C3 | C3 | C9 | C11 | C33 | C33 | C99 |
| kernel | C3×C99 | C99 | C3×C33 | C33 | C3×C9 | C9 | C32 | C3 |
| # reps | 1 | 6 | 2 | 18 | 10 | 60 | 20 | 180 |
Matrix representation of C3×C99 ►in GL2(𝔽199) generated by
| 106 | 0 |
| 0 | 1 |
| 52 | 0 |
| 0 | 180 |
G:=sub<GL(2,GF(199))| [106,0,0,1],[52,0,0,180] >;
C3×C99 in GAP, Magma, Sage, TeX
C_3\times C_{99} % in TeX
G:=Group("C3xC99"); // GroupNames label
G:=SmallGroup(297,2);
// by ID
G=gap.SmallGroup(297,2);
# by ID
G:=PCGroup([4,-3,-3,-11,-3,396]);
// Polycyclic
G:=Group<a,b|a^3=b^99=1,a*b=b*a>;
// generators/relations
Export