direct product, abelian, monomial, 3-elementary
Aliases: C3×C108, SmallGroup(324,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C108 |
| C1 — C3×C108 |
| C1 — C3×C108 |
Generators and relations for C3×C108
G = < a,b | a3=b108=1, ab=ba >
Smallest permutation representation of C3×C108
►Regular action on 324 points
Generators in S324
(1 127 317)(2 128 318)(3 129 319)(4 130 320)(5 131 321)(6 132 322)(7 133 323)(8 134 324)(9 135 217)(10 136 218)(11 137 219)(12 138 220)(13 139 221)(14 140 222)(15 141 223)(16 142 224)(17 143 225)(18 144 226)(19 145 227)(20 146 228)(21 147 229)(22 148 230)(23 149 231)(24 150 232)(25 151 233)(26 152 234)(27 153 235)(28 154 236)(29 155 237)(30 156 238)(31 157 239)(32 158 240)(33 159 241)(34 160 242)(35 161 243)(36 162 244)(37 163 245)(38 164 246)(39 165 247)(40 166 248)(41 167 249)(42 168 250)(43 169 251)(44 170 252)(45 171 253)(46 172 254)(47 173 255)(48 174 256)(49 175 257)(50 176 258)(51 177 259)(52 178 260)(53 179 261)(54 180 262)(55 181 263)(56 182 264)(57 183 265)(58 184 266)(59 185 267)(60 186 268)(61 187 269)(62 188 270)(63 189 271)(64 190 272)(65 191 273)(66 192 274)(67 193 275)(68 194 276)(69 195 277)(70 196 278)(71 197 279)(72 198 280)(73 199 281)(74 200 282)(75 201 283)(76 202 284)(77 203 285)(78 204 286)(79 205 287)(80 206 288)(81 207 289)(82 208 290)(83 209 291)(84 210 292)(85 211 293)(86 212 294)(87 213 295)(88 214 296)(89 215 297)(90 216 298)(91 109 299)(92 110 300)(93 111 301)(94 112 302)(95 113 303)(96 114 304)(97 115 305)(98 116 306)(99 117 307)(100 118 308)(101 119 309)(102 120 310)(103 121 311)(104 122 312)(105 123 313)(106 124 314)(107 125 315)(108 126 316)
(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)
G:=sub<Sym(324)| (1,127,317)(2,128,318)(3,129,319)(4,130,320)(5,131,321)(6,132,322)(7,133,323)(8,134,324)(9,135,217)(10,136,218)(11,137,219)(12,138,220)(13,139,221)(14,140,222)(15,141,223)(16,142,224)(17,143,225)(18,144,226)(19,145,227)(20,146,228)(21,147,229)(22,148,230)(23,149,231)(24,150,232)(25,151,233)(26,152,234)(27,153,235)(28,154,236)(29,155,237)(30,156,238)(31,157,239)(32,158,240)(33,159,241)(34,160,242)(35,161,243)(36,162,244)(37,163,245)(38,164,246)(39,165,247)(40,166,248)(41,167,249)(42,168,250)(43,169,251)(44,170,252)(45,171,253)(46,172,254)(47,173,255)(48,174,256)(49,175,257)(50,176,258)(51,177,259)(52,178,260)(53,179,261)(54,180,262)(55,181,263)(56,182,264)(57,183,265)(58,184,266)(59,185,267)(60,186,268)(61,187,269)(62,188,270)(63,189,271)(64,190,272)(65,191,273)(66,192,274)(67,193,275)(68,194,276)(69,195,277)(70,196,278)(71,197,279)(72,198,280)(73,199,281)(74,200,282)(75,201,283)(76,202,284)(77,203,285)(78,204,286)(79,205,287)(80,206,288)(81,207,289)(82,208,290)(83,209,291)(84,210,292)(85,211,293)(86,212,294)(87,213,295)(88,214,296)(89,215,297)(90,216,298)(91,109,299)(92,110,300)(93,111,301)(94,112,302)(95,113,303)(96,114,304)(97,115,305)(98,116,306)(99,117,307)(100,118,308)(101,119,309)(102,120,310)(103,121,311)(104,122,312)(105,123,313)(106,124,314)(107,125,315)(108,126,316), (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)>;
G:=Group( (1,127,317)(2,128,318)(3,129,319)(4,130,320)(5,131,321)(6,132,322)(7,133,323)(8,134,324)(9,135,217)(10,136,218)(11,137,219)(12,138,220)(13,139,221)(14,140,222)(15,141,223)(16,142,224)(17,143,225)(18,144,226)(19,145,227)(20,146,228)(21,147,229)(22,148,230)(23,149,231)(24,150,232)(25,151,233)(26,152,234)(27,153,235)(28,154,236)(29,155,237)(30,156,238)(31,157,239)(32,158,240)(33,159,241)(34,160,242)(35,161,243)(36,162,244)(37,163,245)(38,164,246)(39,165,247)(40,166,248)(41,167,249)(42,168,250)(43,169,251)(44,170,252)(45,171,253)(46,172,254)(47,173,255)(48,174,256)(49,175,257)(50,176,258)(51,177,259)(52,178,260)(53,179,261)(54,180,262)(55,181,263)(56,182,264)(57,183,265)(58,184,266)(59,185,267)(60,186,268)(61,187,269)(62,188,270)(63,189,271)(64,190,272)(65,191,273)(66,192,274)(67,193,275)(68,194,276)(69,195,277)(70,196,278)(71,197,279)(72,198,280)(73,199,281)(74,200,282)(75,201,283)(76,202,284)(77,203,285)(78,204,286)(79,205,287)(80,206,288)(81,207,289)(82,208,290)(83,209,291)(84,210,292)(85,211,293)(86,212,294)(87,213,295)(88,214,296)(89,215,297)(90,216,298)(91,109,299)(92,110,300)(93,111,301)(94,112,302)(95,113,303)(96,114,304)(97,115,305)(98,116,306)(99,117,307)(100,118,308)(101,119,309)(102,120,310)(103,121,311)(104,122,312)(105,123,313)(106,124,314)(107,125,315)(108,126,316), (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) );
G=PermutationGroup([[(1,127,317),(2,128,318),(3,129,319),(4,130,320),(5,131,321),(6,132,322),(7,133,323),(8,134,324),(9,135,217),(10,136,218),(11,137,219),(12,138,220),(13,139,221),(14,140,222),(15,141,223),(16,142,224),(17,143,225),(18,144,226),(19,145,227),(20,146,228),(21,147,229),(22,148,230),(23,149,231),(24,150,232),(25,151,233),(26,152,234),(27,153,235),(28,154,236),(29,155,237),(30,156,238),(31,157,239),(32,158,240),(33,159,241),(34,160,242),(35,161,243),(36,162,244),(37,163,245),(38,164,246),(39,165,247),(40,166,248),(41,167,249),(42,168,250),(43,169,251),(44,170,252),(45,171,253),(46,172,254),(47,173,255),(48,174,256),(49,175,257),(50,176,258),(51,177,259),(52,178,260),(53,179,261),(54,180,262),(55,181,263),(56,182,264),(57,183,265),(58,184,266),(59,185,267),(60,186,268),(61,187,269),(62,188,270),(63,189,271),(64,190,272),(65,191,273),(66,192,274),(67,193,275),(68,194,276),(69,195,277),(70,196,278),(71,197,279),(72,198,280),(73,199,281),(74,200,282),(75,201,283),(76,202,284),(77,203,285),(78,204,286),(79,205,287),(80,206,288),(81,207,289),(82,208,290),(83,209,291),(84,210,292),(85,211,293),(86,212,294),(87,213,295),(88,214,296),(89,215,297),(90,216,298),(91,109,299),(92,110,300),(93,111,301),(94,112,302),(95,113,303),(96,114,304),(97,115,305),(98,116,306),(99,117,307),(100,118,308),(101,119,309),(102,120,310),(103,121,311),(104,122,312),(105,123,313),(106,124,314),(107,125,315),(108,126,316)], [(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)]])
324 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 9A | ··· | 9R | 12A | ··· | 12P | 18A | ··· | 18R | 27A | ··· | 27BB | 36A | ··· | 36AJ | 54A | ··· | 54BB | 108A | ··· | 108DD |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
324 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C9 | C9 | C12 | C12 | C18 | C18 | C27 | C36 | C36 | C54 | C108 |
| kernel | C3×C108 | C3×C54 | C108 | C3×C36 | C3×C27 | C54 | C3×C18 | C36 | C3×C12 | C27 | C3×C9 | C18 | C3×C6 | C12 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 12 | 6 | 12 | 4 | 12 | 6 | 54 | 24 | 12 | 54 | 108 |
Matrix representation of C3×C108 ►in GL2(𝔽109) generated by
| 1 | 0 |
| 0 | 45 |
| 94 | 0 |
| 0 | 10 |
G:=sub<GL(2,GF(109))| [1,0,0,45],[94,0,0,10] >;
C3×C108 in GAP, Magma, Sage, TeX
C_3\times C_{108} % in TeX
G:=Group("C3xC108"); // GroupNames label
G:=SmallGroup(324,29);
// by ID
G=gap.SmallGroup(324,29);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,223,118]);
// Polycyclic
G:=Group<a,b|a^3=b^108=1,a*b=b*a>;
// generators/relations
Export