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## G = C3×C6×C18order 324 = 22·34

### Abelian group of type [3,6,18]

Aliases: C3×C6×C18, SmallGroup(324,151)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6×C18
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C32×C18 — C3×C6×C18
 Lower central C1 — C3×C6×C18
 Upper central 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 [×3], C3, C3 [×12], C22, C6 [×39], C9 [×9], C32 [×13], C2×C6, C2×C6 [×12], C18 [×27], C3×C6 [×39], C3×C9 [×12], C33, C2×C18 [×9], C62 [×13], C3×C18 [×36], C32×C6 [×3], C32×C9, C6×C18 [×12], C3×C62, C32×C18 [×3], C3×C6×C18
Quotients: C1, C2 [×3], C3 [×13], C22, C6 [×39], C9 [×9], C32 [×13], C2×C6 [×13], C18 [×27], C3×C6 [×39], C3×C9 [×12], C33, C2×C18 [×9], C62 [×13], C3×C18 [×36], C32×C6 [×3], C32×C9, C6×C18 [×12], C3×C62, C32×C18 [×3], C3×C6×C18

Smallest permutation representation of C3×C6×C18
Regular action on 324 points
Generators in S324
(1 106 225)(2 107 226)(3 108 227)(4 91 228)(5 92 229)(6 93 230)(7 94 231)(8 95 232)(9 96 233)(10 97 234)(11 98 217)(12 99 218)(13 100 219)(14 101 220)(15 102 221)(16 103 222)(17 104 223)(18 105 224)(19 51 151)(20 52 152)(21 53 153)(22 54 154)(23 37 155)(24 38 156)(25 39 157)(26 40 158)(27 41 159)(28 42 160)(29 43 161)(30 44 162)(31 45 145)(32 46 146)(33 47 147)(34 48 148)(35 49 149)(36 50 150)(55 83 163)(56 84 164)(57 85 165)(58 86 166)(59 87 167)(60 88 168)(61 89 169)(62 90 170)(63 73 171)(64 74 172)(65 75 173)(66 76 174)(67 77 175)(68 78 176)(69 79 177)(70 80 178)(71 81 179)(72 82 180)(109 202 270)(110 203 253)(111 204 254)(112 205 255)(113 206 256)(114 207 257)(115 208 258)(116 209 259)(117 210 260)(118 211 261)(119 212 262)(120 213 263)(121 214 264)(122 215 265)(123 216 266)(124 199 267)(125 200 268)(126 201 269)(127 308 186)(128 309 187)(129 310 188)(130 311 189)(131 312 190)(132 313 191)(133 314 192)(134 315 193)(135 316 194)(136 317 195)(137 318 196)(138 319 197)(139 320 198)(140 321 181)(141 322 182)(142 323 183)(143 324 184)(144 307 185)(235 302 273)(236 303 274)(237 304 275)(238 305 276)(239 306 277)(240 289 278)(241 290 279)(242 291 280)(243 292 281)(244 293 282)(245 294 283)(246 295 284)(247 296 285)(248 297 286)(249 298 287)(250 299 288)(251 300 271)(252 301 272)
(1 246 322 175 205 154)(2 247 323 176 206 155)(3 248 324 177 207 156)(4 249 307 178 208 157)(5 250 308 179 209 158)(6 251 309 180 210 159)(7 252 310 163 211 160)(8 235 311 164 212 161)(9 236 312 165 213 162)(10 237 313 166 214 145)(11 238 314 167 215 146)(12 239 315 168 216 147)(13 240 316 169 199 148)(14 241 317 170 200 149)(15 242 318 171 201 150)(16 243 319 172 202 151)(17 244 320 173 203 152)(18 245 321 174 204 153)(19 103 292 197 64 270)(20 104 293 198 65 253)(21 105 294 181 66 254)(22 106 295 182 67 255)(23 107 296 183 68 256)(24 108 297 184 69 257)(25 91 298 185 70 258)(26 92 299 186 71 259)(27 93 300 187 72 260)(28 94 301 188 55 261)(29 95 302 189 56 262)(30 96 303 190 57 263)(31 97 304 191 58 264)(32 98 305 192 59 265)(33 99 306 193 60 266)(34 100 289 194 61 267)(35 101 290 195 62 268)(36 102 291 196 63 269)(37 226 285 142 78 113)(38 227 286 143 79 114)(39 228 287 144 80 115)(40 229 288 127 81 116)(41 230 271 128 82 117)(42 231 272 129 83 118)(43 232 273 130 84 119)(44 233 274 131 85 120)(45 234 275 132 86 121)(46 217 276 133 87 122)(47 218 277 134 88 123)(48 219 278 135 89 124)(49 220 279 136 90 125)(50 221 280 137 73 126)(51 222 281 138 74 109)(52 223 282 139 75 110)(53 224 283 140 76 111)(54 225 284 141 77 112)
(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)| (1,106,225)(2,107,226)(3,108,227)(4,91,228)(5,92,229)(6,93,230)(7,94,231)(8,95,232)(9,96,233)(10,97,234)(11,98,217)(12,99,218)(13,100,219)(14,101,220)(15,102,221)(16,103,222)(17,104,223)(18,105,224)(19,51,151)(20,52,152)(21,53,153)(22,54,154)(23,37,155)(24,38,156)(25,39,157)(26,40,158)(27,41,159)(28,42,160)(29,43,161)(30,44,162)(31,45,145)(32,46,146)(33,47,147)(34,48,148)(35,49,149)(36,50,150)(55,83,163)(56,84,164)(57,85,165)(58,86,166)(59,87,167)(60,88,168)(61,89,169)(62,90,170)(63,73,171)(64,74,172)(65,75,173)(66,76,174)(67,77,175)(68,78,176)(69,79,177)(70,80,178)(71,81,179)(72,82,180)(109,202,270)(110,203,253)(111,204,254)(112,205,255)(113,206,256)(114,207,257)(115,208,258)(116,209,259)(117,210,260)(118,211,261)(119,212,262)(120,213,263)(121,214,264)(122,215,265)(123,216,266)(124,199,267)(125,200,268)(126,201,269)(127,308,186)(128,309,187)(129,310,188)(130,311,189)(131,312,190)(132,313,191)(133,314,192)(134,315,193)(135,316,194)(136,317,195)(137,318,196)(138,319,197)(139,320,198)(140,321,181)(141,322,182)(142,323,183)(143,324,184)(144,307,185)(235,302,273)(236,303,274)(237,304,275)(238,305,276)(239,306,277)(240,289,278)(241,290,279)(242,291,280)(243,292,281)(244,293,282)(245,294,283)(246,295,284)(247,296,285)(248,297,286)(249,298,287)(250,299,288)(251,300,271)(252,301,272), (1,246,322,175,205,154)(2,247,323,176,206,155)(3,248,324,177,207,156)(4,249,307,178,208,157)(5,250,308,179,209,158)(6,251,309,180,210,159)(7,252,310,163,211,160)(8,235,311,164,212,161)(9,236,312,165,213,162)(10,237,313,166,214,145)(11,238,314,167,215,146)(12,239,315,168,216,147)(13,240,316,169,199,148)(14,241,317,170,200,149)(15,242,318,171,201,150)(16,243,319,172,202,151)(17,244,320,173,203,152)(18,245,321,174,204,153)(19,103,292,197,64,270)(20,104,293,198,65,253)(21,105,294,181,66,254)(22,106,295,182,67,255)(23,107,296,183,68,256)(24,108,297,184,69,257)(25,91,298,185,70,258)(26,92,299,186,71,259)(27,93,300,187,72,260)(28,94,301,188,55,261)(29,95,302,189,56,262)(30,96,303,190,57,263)(31,97,304,191,58,264)(32,98,305,192,59,265)(33,99,306,193,60,266)(34,100,289,194,61,267)(35,101,290,195,62,268)(36,102,291,196,63,269)(37,226,285,142,78,113)(38,227,286,143,79,114)(39,228,287,144,80,115)(40,229,288,127,81,116)(41,230,271,128,82,117)(42,231,272,129,83,118)(43,232,273,130,84,119)(44,233,274,131,85,120)(45,234,275,132,86,121)(46,217,276,133,87,122)(47,218,277,134,88,123)(48,219,278,135,89,124)(49,220,279,136,90,125)(50,221,280,137,73,126)(51,222,281,138,74,109)(52,223,282,139,75,110)(53,224,283,140,76,111)(54,225,284,141,77,112), (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( (1,106,225)(2,107,226)(3,108,227)(4,91,228)(5,92,229)(6,93,230)(7,94,231)(8,95,232)(9,96,233)(10,97,234)(11,98,217)(12,99,218)(13,100,219)(14,101,220)(15,102,221)(16,103,222)(17,104,223)(18,105,224)(19,51,151)(20,52,152)(21,53,153)(22,54,154)(23,37,155)(24,38,156)(25,39,157)(26,40,158)(27,41,159)(28,42,160)(29,43,161)(30,44,162)(31,45,145)(32,46,146)(33,47,147)(34,48,148)(35,49,149)(36,50,150)(55,83,163)(56,84,164)(57,85,165)(58,86,166)(59,87,167)(60,88,168)(61,89,169)(62,90,170)(63,73,171)(64,74,172)(65,75,173)(66,76,174)(67,77,175)(68,78,176)(69,79,177)(70,80,178)(71,81,179)(72,82,180)(109,202,270)(110,203,253)(111,204,254)(112,205,255)(113,206,256)(114,207,257)(115,208,258)(116,209,259)(117,210,260)(118,211,261)(119,212,262)(120,213,263)(121,214,264)(122,215,265)(123,216,266)(124,199,267)(125,200,268)(126,201,269)(127,308,186)(128,309,187)(129,310,188)(130,311,189)(131,312,190)(132,313,191)(133,314,192)(134,315,193)(135,316,194)(136,317,195)(137,318,196)(138,319,197)(139,320,198)(140,321,181)(141,322,182)(142,323,183)(143,324,184)(144,307,185)(235,302,273)(236,303,274)(237,304,275)(238,305,276)(239,306,277)(240,289,278)(241,290,279)(242,291,280)(243,292,281)(244,293,282)(245,294,283)(246,295,284)(247,296,285)(248,297,286)(249,298,287)(250,299,288)(251,300,271)(252,301,272), (1,246,322,175,205,154)(2,247,323,176,206,155)(3,248,324,177,207,156)(4,249,307,178,208,157)(5,250,308,179,209,158)(6,251,309,180,210,159)(7,252,310,163,211,160)(8,235,311,164,212,161)(9,236,312,165,213,162)(10,237,313,166,214,145)(11,238,314,167,215,146)(12,239,315,168,216,147)(13,240,316,169,199,148)(14,241,317,170,200,149)(15,242,318,171,201,150)(16,243,319,172,202,151)(17,244,320,173,203,152)(18,245,321,174,204,153)(19,103,292,197,64,270)(20,104,293,198,65,253)(21,105,294,181,66,254)(22,106,295,182,67,255)(23,107,296,183,68,256)(24,108,297,184,69,257)(25,91,298,185,70,258)(26,92,299,186,71,259)(27,93,300,187,72,260)(28,94,301,188,55,261)(29,95,302,189,56,262)(30,96,303,190,57,263)(31,97,304,191,58,264)(32,98,305,192,59,265)(33,99,306,193,60,266)(34,100,289,194,61,267)(35,101,290,195,62,268)(36,102,291,196,63,269)(37,226,285,142,78,113)(38,227,286,143,79,114)(39,228,287,144,80,115)(40,229,288,127,81,116)(41,230,271,128,82,117)(42,231,272,129,83,118)(43,232,273,130,84,119)(44,233,274,131,85,120)(45,234,275,132,86,121)(46,217,276,133,87,122)(47,218,277,134,88,123)(48,219,278,135,89,124)(49,220,279,136,90,125)(50,221,280,137,73,126)(51,222,281,138,74,109)(52,223,282,139,75,110)(53,224,283,140,76,111)(54,225,284,141,77,112), (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([(1,106,225),(2,107,226),(3,108,227),(4,91,228),(5,92,229),(6,93,230),(7,94,231),(8,95,232),(9,96,233),(10,97,234),(11,98,217),(12,99,218),(13,100,219),(14,101,220),(15,102,221),(16,103,222),(17,104,223),(18,105,224),(19,51,151),(20,52,152),(21,53,153),(22,54,154),(23,37,155),(24,38,156),(25,39,157),(26,40,158),(27,41,159),(28,42,160),(29,43,161),(30,44,162),(31,45,145),(32,46,146),(33,47,147),(34,48,148),(35,49,149),(36,50,150),(55,83,163),(56,84,164),(57,85,165),(58,86,166),(59,87,167),(60,88,168),(61,89,169),(62,90,170),(63,73,171),(64,74,172),(65,75,173),(66,76,174),(67,77,175),(68,78,176),(69,79,177),(70,80,178),(71,81,179),(72,82,180),(109,202,270),(110,203,253),(111,204,254),(112,205,255),(113,206,256),(114,207,257),(115,208,258),(116,209,259),(117,210,260),(118,211,261),(119,212,262),(120,213,263),(121,214,264),(122,215,265),(123,216,266),(124,199,267),(125,200,268),(126,201,269),(127,308,186),(128,309,187),(129,310,188),(130,311,189),(131,312,190),(132,313,191),(133,314,192),(134,315,193),(135,316,194),(136,317,195),(137,318,196),(138,319,197),(139,320,198),(140,321,181),(141,322,182),(142,323,183),(143,324,184),(144,307,185),(235,302,273),(236,303,274),(237,304,275),(238,305,276),(239,306,277),(240,289,278),(241,290,279),(242,291,280),(243,292,281),(244,293,282),(245,294,283),(246,295,284),(247,296,285),(248,297,286),(249,298,287),(250,299,288),(251,300,271),(252,301,272)], [(1,246,322,175,205,154),(2,247,323,176,206,155),(3,248,324,177,207,156),(4,249,307,178,208,157),(5,250,308,179,209,158),(6,251,309,180,210,159),(7,252,310,163,211,160),(8,235,311,164,212,161),(9,236,312,165,213,162),(10,237,313,166,214,145),(11,238,314,167,215,146),(12,239,315,168,216,147),(13,240,316,169,199,148),(14,241,317,170,200,149),(15,242,318,171,201,150),(16,243,319,172,202,151),(17,244,320,173,203,152),(18,245,321,174,204,153),(19,103,292,197,64,270),(20,104,293,198,65,253),(21,105,294,181,66,254),(22,106,295,182,67,255),(23,107,296,183,68,256),(24,108,297,184,69,257),(25,91,298,185,70,258),(26,92,299,186,71,259),(27,93,300,187,72,260),(28,94,301,188,55,261),(29,95,302,189,56,262),(30,96,303,190,57,263),(31,97,304,191,58,264),(32,98,305,192,59,265),(33,99,306,193,60,266),(34,100,289,194,61,267),(35,101,290,195,62,268),(36,102,291,196,63,269),(37,226,285,142,78,113),(38,227,286,143,79,114),(39,228,287,144,80,115),(40,229,288,127,81,116),(41,230,271,128,82,117),(42,231,272,129,83,118),(43,232,273,130,84,119),(44,233,274,131,85,120),(45,234,275,132,86,121),(46,217,276,133,87,122),(47,218,277,134,88,123),(48,219,278,135,89,124),(49,220,279,136,90,125),(50,221,280,137,73,126),(51,222,281,138,74,109),(52,223,282,139,75,110),(53,224,283,140,76,111),(54,225,284,141,77,112)], [(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

׿
×
𝔽