direct product, abelian, monomial
Aliases: C3×C6×C18, SmallGroup(324,151)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C6×C18 |
| C1 — C3×C6×C18 |
| C1 — C3×C6×C18 |
Generators and relations for C3×C6×C18
G = < a,b,c | a3=b6=c18=1, ab=ba, ac=ca, bc=cb >
Subgroups: 250, all normal (8 characteristic)
C1, C2, C3, C3, C22, C6, C9, C32, C2×C6, C2×C6, C18, C3×C6, C3×C9, C33, C2×C18, C62, C3×C18, C32×C6, C32×C9, C6×C18, C3×C62, C32×C18, C3×C6×C18
Quotients: C1, C2, C3, C22, C6, C9, C32, C2×C6, C18, C3×C6, C3×C9, C33, C2×C18, C62, C3×C18, C32×C6, C32×C9, C6×C18, C3×C62, C32×C18, C3×C6×C18
►Regular action on 324 points
Generators in S324
(1 248 66)(2 249 67)(3 250 68)(4 251 69)(5 252 70)(6 235 71)(7 236 72)(8 237 55)(9 238 56)(10 239 57)(11 240 58)(12 241 59)(13 242 60)(14 243 61)(15 244 62)(16 245 63)(17 246 64)(18 247 65)(19 76 173)(20 77 174)(21 78 175)(22 79 176)(23 80 177)(24 81 178)(25 82 179)(26 83 180)(27 84 163)(28 85 164)(29 86 165)(30 87 166)(31 88 167)(32 89 168)(33 90 169)(34 73 170)(35 74 171)(36 75 172)(37 126 225)(38 109 226)(39 110 227)(40 111 228)(41 112 229)(42 113 230)(43 114 231)(44 115 232)(45 116 233)(46 117 234)(47 118 217)(48 119 218)(49 120 219)(50 121 220)(51 122 221)(52 123 222)(53 124 223)(54 125 224)(91 127 261)(92 128 262)(93 129 263)(94 130 264)(95 131 265)(96 132 266)(97 133 267)(98 134 268)(99 135 269)(100 136 270)(101 137 253)(102 138 254)(103 139 255)(104 140 256)(105 141 257)(106 142 258)(107 143 259)(108 144 260)(145 277 294)(146 278 295)(147 279 296)(148 280 297)(149 281 298)(150 282 299)(151 283 300)(152 284 301)(153 285 302)(154 286 303)(155 287 304)(156 288 305)(157 271 306)(158 272 289)(159 273 290)(160 274 291)(161 275 292)(162 276 293)(181 216 309)(182 199 310)(183 200 311)(184 201 312)(185 202 313)(186 203 314)(187 204 315)(188 205 316)(189 206 317)(190 207 318)(191 208 319)(192 209 320)(193 210 321)(194 211 322)(195 212 323)(196 213 324)(197 214 307)(198 215 308)
(1 155 174 131 193 227)(2 156 175 132 194 228)(3 157 176 133 195 229)(4 158 177 134 196 230)(5 159 178 135 197 231)(6 160 179 136 198 232)(7 161 180 137 181 233)(8 162 163 138 182 234)(9 145 164 139 183 217)(10 146 165 140 184 218)(11 147 166 141 185 219)(12 148 167 142 186 220)(13 149 168 143 187 221)(14 150 169 144 188 222)(15 151 170 127 189 223)(16 152 171 128 190 224)(17 153 172 129 191 225)(18 154 173 130 192 226)(19 264 209 38 247 286)(20 265 210 39 248 287)(21 266 211 40 249 288)(22 267 212 41 250 271)(23 268 213 42 251 272)(24 269 214 43 252 273)(25 270 215 44 235 274)(26 253 216 45 236 275)(27 254 199 46 237 276)(28 255 200 47 238 277)(29 256 201 48 239 278)(30 257 202 49 240 279)(31 258 203 50 241 280)(32 259 204 51 242 281)(33 260 205 52 243 282)(34 261 206 53 244 283)(35 262 207 54 245 284)(36 263 208 37 246 285)(55 293 84 102 310 117)(56 294 85 103 311 118)(57 295 86 104 312 119)(58 296 87 105 313 120)(59 297 88 106 314 121)(60 298 89 107 315 122)(61 299 90 108 316 123)(62 300 73 91 317 124)(63 301 74 92 318 125)(64 302 75 93 319 126)(65 303 76 94 320 109)(66 304 77 95 321 110)(67 305 78 96 322 111)(68 306 79 97 323 112)(69 289 80 98 324 113)(70 290 81 99 307 114)(71 291 82 100 308 115)(72 292 83 101 309 116)
(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)| (1,248,66)(2,249,67)(3,250,68)(4,251,69)(5,252,70)(6,235,71)(7,236,72)(8,237,55)(9,238,56)(10,239,57)(11,240,58)(12,241,59)(13,242,60)(14,243,61)(15,244,62)(16,245,63)(17,246,64)(18,247,65)(19,76,173)(20,77,174)(21,78,175)(22,79,176)(23,80,177)(24,81,178)(25,82,179)(26,83,180)(27,84,163)(28,85,164)(29,86,165)(30,87,166)(31,88,167)(32,89,168)(33,90,169)(34,73,170)(35,74,171)(36,75,172)(37,126,225)(38,109,226)(39,110,227)(40,111,228)(41,112,229)(42,113,230)(43,114,231)(44,115,232)(45,116,233)(46,117,234)(47,118,217)(48,119,218)(49,120,219)(50,121,220)(51,122,221)(52,123,222)(53,124,223)(54,125,224)(91,127,261)(92,128,262)(93,129,263)(94,130,264)(95,131,265)(96,132,266)(97,133,267)(98,134,268)(99,135,269)(100,136,270)(101,137,253)(102,138,254)(103,139,255)(104,140,256)(105,141,257)(106,142,258)(107,143,259)(108,144,260)(145,277,294)(146,278,295)(147,279,296)(148,280,297)(149,281,298)(150,282,299)(151,283,300)(152,284,301)(153,285,302)(154,286,303)(155,287,304)(156,288,305)(157,271,306)(158,272,289)(159,273,290)(160,274,291)(161,275,292)(162,276,293)(181,216,309)(182,199,310)(183,200,311)(184,201,312)(185,202,313)(186,203,314)(187,204,315)(188,205,316)(189,206,317)(190,207,318)(191,208,319)(192,209,320)(193,210,321)(194,211,322)(195,212,323)(196,213,324)(197,214,307)(198,215,308), (1,155,174,131,193,227)(2,156,175,132,194,228)(3,157,176,133,195,229)(4,158,177,134,196,230)(5,159,178,135,197,231)(6,160,179,136,198,232)(7,161,180,137,181,233)(8,162,163,138,182,234)(9,145,164,139,183,217)(10,146,165,140,184,218)(11,147,166,141,185,219)(12,148,167,142,186,220)(13,149,168,143,187,221)(14,150,169,144,188,222)(15,151,170,127,189,223)(16,152,171,128,190,224)(17,153,172,129,191,225)(18,154,173,130,192,226)(19,264,209,38,247,286)(20,265,210,39,248,287)(21,266,211,40,249,288)(22,267,212,41,250,271)(23,268,213,42,251,272)(24,269,214,43,252,273)(25,270,215,44,235,274)(26,253,216,45,236,275)(27,254,199,46,237,276)(28,255,200,47,238,277)(29,256,201,48,239,278)(30,257,202,49,240,279)(31,258,203,50,241,280)(32,259,204,51,242,281)(33,260,205,52,243,282)(34,261,206,53,244,283)(35,262,207,54,245,284)(36,263,208,37,246,285)(55,293,84,102,310,117)(56,294,85,103,311,118)(57,295,86,104,312,119)(58,296,87,105,313,120)(59,297,88,106,314,121)(60,298,89,107,315,122)(61,299,90,108,316,123)(62,300,73,91,317,124)(63,301,74,92,318,125)(64,302,75,93,319,126)(65,303,76,94,320,109)(66,304,77,95,321,110)(67,305,78,96,322,111)(68,306,79,97,323,112)(69,289,80,98,324,113)(70,290,81,99,307,114)(71,291,82,100,308,115)(72,292,83,101,309,116), (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( (1,248,66)(2,249,67)(3,250,68)(4,251,69)(5,252,70)(6,235,71)(7,236,72)(8,237,55)(9,238,56)(10,239,57)(11,240,58)(12,241,59)(13,242,60)(14,243,61)(15,244,62)(16,245,63)(17,246,64)(18,247,65)(19,76,173)(20,77,174)(21,78,175)(22,79,176)(23,80,177)(24,81,178)(25,82,179)(26,83,180)(27,84,163)(28,85,164)(29,86,165)(30,87,166)(31,88,167)(32,89,168)(33,90,169)(34,73,170)(35,74,171)(36,75,172)(37,126,225)(38,109,226)(39,110,227)(40,111,228)(41,112,229)(42,113,230)(43,114,231)(44,115,232)(45,116,233)(46,117,234)(47,118,217)(48,119,218)(49,120,219)(50,121,220)(51,122,221)(52,123,222)(53,124,223)(54,125,224)(91,127,261)(92,128,262)(93,129,263)(94,130,264)(95,131,265)(96,132,266)(97,133,267)(98,134,268)(99,135,269)(100,136,270)(101,137,253)(102,138,254)(103,139,255)(104,140,256)(105,141,257)(106,142,258)(107,143,259)(108,144,260)(145,277,294)(146,278,295)(147,279,296)(148,280,297)(149,281,298)(150,282,299)(151,283,300)(152,284,301)(153,285,302)(154,286,303)(155,287,304)(156,288,305)(157,271,306)(158,272,289)(159,273,290)(160,274,291)(161,275,292)(162,276,293)(181,216,309)(182,199,310)(183,200,311)(184,201,312)(185,202,313)(186,203,314)(187,204,315)(188,205,316)(189,206,317)(190,207,318)(191,208,319)(192,209,320)(193,210,321)(194,211,322)(195,212,323)(196,213,324)(197,214,307)(198,215,308), (1,155,174,131,193,227)(2,156,175,132,194,228)(3,157,176,133,195,229)(4,158,177,134,196,230)(5,159,178,135,197,231)(6,160,179,136,198,232)(7,161,180,137,181,233)(8,162,163,138,182,234)(9,145,164,139,183,217)(10,146,165,140,184,218)(11,147,166,141,185,219)(12,148,167,142,186,220)(13,149,168,143,187,221)(14,150,169,144,188,222)(15,151,170,127,189,223)(16,152,171,128,190,224)(17,153,172,129,191,225)(18,154,173,130,192,226)(19,264,209,38,247,286)(20,265,210,39,248,287)(21,266,211,40,249,288)(22,267,212,41,250,271)(23,268,213,42,251,272)(24,269,214,43,252,273)(25,270,215,44,235,274)(26,253,216,45,236,275)(27,254,199,46,237,276)(28,255,200,47,238,277)(29,256,201,48,239,278)(30,257,202,49,240,279)(31,258,203,50,241,280)(32,259,204,51,242,281)(33,260,205,52,243,282)(34,261,206,53,244,283)(35,262,207,54,245,284)(36,263,208,37,246,285)(55,293,84,102,310,117)(56,294,85,103,311,118)(57,295,86,104,312,119)(58,296,87,105,313,120)(59,297,88,106,314,121)(60,298,89,107,315,122)(61,299,90,108,316,123)(62,300,73,91,317,124)(63,301,74,92,318,125)(64,302,75,93,319,126)(65,303,76,94,320,109)(66,304,77,95,321,110)(67,305,78,96,322,111)(68,306,79,97,323,112)(69,289,80,98,324,113)(70,290,81,99,307,114)(71,291,82,100,308,115)(72,292,83,101,309,116), (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([[(1,248,66),(2,249,67),(3,250,68),(4,251,69),(5,252,70),(6,235,71),(7,236,72),(8,237,55),(9,238,56),(10,239,57),(11,240,58),(12,241,59),(13,242,60),(14,243,61),(15,244,62),(16,245,63),(17,246,64),(18,247,65),(19,76,173),(20,77,174),(21,78,175),(22,79,176),(23,80,177),(24,81,178),(25,82,179),(26,83,180),(27,84,163),(28,85,164),(29,86,165),(30,87,166),(31,88,167),(32,89,168),(33,90,169),(34,73,170),(35,74,171),(36,75,172),(37,126,225),(38,109,226),(39,110,227),(40,111,228),(41,112,229),(42,113,230),(43,114,231),(44,115,232),(45,116,233),(46,117,234),(47,118,217),(48,119,218),(49,120,219),(50,121,220),(51,122,221),(52,123,222),(53,124,223),(54,125,224),(91,127,261),(92,128,262),(93,129,263),(94,130,264),(95,131,265),(96,132,266),(97,133,267),(98,134,268),(99,135,269),(100,136,270),(101,137,253),(102,138,254),(103,139,255),(104,140,256),(105,141,257),(106,142,258),(107,143,259),(108,144,260),(145,277,294),(146,278,295),(147,279,296),(148,280,297),(149,281,298),(150,282,299),(151,283,300),(152,284,301),(153,285,302),(154,286,303),(155,287,304),(156,288,305),(157,271,306),(158,272,289),(159,273,290),(160,274,291),(161,275,292),(162,276,293),(181,216,309),(182,199,310),(183,200,311),(184,201,312),(185,202,313),(186,203,314),(187,204,315),(188,205,316),(189,206,317),(190,207,318),(191,208,319),(192,209,320),(193,210,321),(194,211,322),(195,212,323),(196,213,324),(197,214,307),(198,215,308)], [(1,155,174,131,193,227),(2,156,175,132,194,228),(3,157,176,133,195,229),(4,158,177,134,196,230),(5,159,178,135,197,231),(6,160,179,136,198,232),(7,161,180,137,181,233),(8,162,163,138,182,234),(9,145,164,139,183,217),(10,146,165,140,184,218),(11,147,166,141,185,219),(12,148,167,142,186,220),(13,149,168,143,187,221),(14,150,169,144,188,222),(15,151,170,127,189,223),(16,152,171,128,190,224),(17,153,172,129,191,225),(18,154,173,130,192,226),(19,264,209,38,247,286),(20,265,210,39,248,287),(21,266,211,40,249,288),(22,267,212,41,250,271),(23,268,213,42,251,272),(24,269,214,43,252,273),(25,270,215,44,235,274),(26,253,216,45,236,275),(27,254,199,46,237,276),(28,255,200,47,238,277),(29,256,201,48,239,278),(30,257,202,49,240,279),(31,258,203,50,241,280),(32,259,204,51,242,281),(33,260,205,52,243,282),(34,261,206,53,244,283),(35,262,207,54,245,284),(36,263,208,37,246,285),(55,293,84,102,310,117),(56,294,85,103,311,118),(57,295,86,104,312,119),(58,296,87,105,313,120),(59,297,88,106,314,121),(60,298,89,107,315,122),(61,299,90,108,316,123),(62,300,73,91,317,124),(63,301,74,92,318,125),(64,302,75,93,319,126),(65,303,76,94,320,109),(66,304,77,95,321,110),(67,305,78,96,322,111),(68,306,79,97,323,112),(69,289,80,98,324,113),(70,290,81,99,307,114),(71,291,82,100,308,115),(72,292,83,101,309,116)], [(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 | 2A | 2B | 2C | 3A | ··· | 3Z | 6A | ··· | 6BZ | 9A | ··· | 9BB | 18A | ··· | 18FF |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
324 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 |
| kernel | C3×C6×C18 | C32×C18 | C6×C18 | C3×C62 | C3×C18 | C32×C6 | C62 | C3×C6 |
| # reps | 1 | 3 | 24 | 2 | 72 | 6 | 54 | 162 |
Matrix representation of C3×C6×C18 ►in GL3(𝔽19) generated by
| 11 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 1 |
| 2 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 4 |
G:=sub<GL(3,GF(19))| [11,0,0,0,1,0,0,0,1],[12,0,0,0,12,0,0,0,1],[2,0,0,0,7,0,0,0,4] >;
C3×C6×C18 in GAP, Magma, Sage, TeX
C_3\times C_6\times C_{18} % in TeX
G:=Group("C3xC6xC18"); // GroupNames label
G:=SmallGroup(324,151);
// by ID
G=gap.SmallGroup(324,151);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,500]);
// Polycyclic
G:=Group<a,b,c|a^3=b^6=c^18=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations