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## G = C27⋊Dic3order 324 = 22·34

### The semidirect product of C27 and Dic3 acting via Dic3/C6=C2

Aliases: C27⋊Dic3, C3⋊Dic27, C54.3S3, C6.3D27, C18.4D9, C9.2Dic9, C32.3Dic9, (C3×C27)⋊3C4, C2.(C27⋊S3), (C3×C6).6D9, C6.2(C9⋊S3), (C3×C54).3C2, C9.(C3⋊Dic3), C18.1(C3⋊S3), (C3×C18).20S3, (C3×C9).7Dic3, C3.2(C9⋊Dic3), SmallGroup(324,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C27 — C27⋊Dic3
 Chief series C1 — C3 — C9 — C3×C9 — C3×C27 — C3×C54 — C27⋊Dic3
 Lower central C3×C27 — C27⋊Dic3
 Upper central C1 — C2

Generators and relations for C27⋊Dic3
G = < a,b,c | a27=b6=1, c2=b3, ab=ba, cac-1=a-1, cbc-1=b-1 >

Smallest permutation representation of C27⋊Dic3
Regular action on 324 points
Generators in S324
(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)
(1 99 39 264 215 302)(2 100 40 265 216 303)(3 101 41 266 190 304)(4 102 42 267 191 305)(5 103 43 268 192 306)(6 104 44 269 193 307)(7 105 45 270 194 308)(8 106 46 244 195 309)(9 107 47 245 196 310)(10 108 48 246 197 311)(11 82 49 247 198 312)(12 83 50 248 199 313)(13 84 51 249 200 314)(14 85 52 250 201 315)(15 86 53 251 202 316)(16 87 54 252 203 317)(17 88 28 253 204 318)(18 89 29 254 205 319)(19 90 30 255 206 320)(20 91 31 256 207 321)(21 92 32 257 208 322)(22 93 33 258 209 323)(23 94 34 259 210 324)(24 95 35 260 211 298)(25 96 36 261 212 299)(26 97 37 262 213 300)(27 98 38 263 214 301)(55 284 234 164 156 109)(56 285 235 165 157 110)(57 286 236 166 158 111)(58 287 237 167 159 112)(59 288 238 168 160 113)(60 289 239 169 161 114)(61 290 240 170 162 115)(62 291 241 171 136 116)(63 292 242 172 137 117)(64 293 243 173 138 118)(65 294 217 174 139 119)(66 295 218 175 140 120)(67 296 219 176 141 121)(68 297 220 177 142 122)(69 271 221 178 143 123)(70 272 222 179 144 124)(71 273 223 180 145 125)(72 274 224 181 146 126)(73 275 225 182 147 127)(74 276 226 183 148 128)(75 277 227 184 149 129)(76 278 228 185 150 130)(77 279 229 186 151 131)(78 280 230 187 152 132)(79 281 231 188 153 133)(80 282 232 189 154 134)(81 283 233 163 155 135)
(1 291 264 136)(2 290 265 162)(3 289 266 161)(4 288 267 160)(5 287 268 159)(6 286 269 158)(7 285 270 157)(8 284 244 156)(9 283 245 155)(10 282 246 154)(11 281 247 153)(12 280 248 152)(13 279 249 151)(14 278 250 150)(15 277 251 149)(16 276 252 148)(17 275 253 147)(18 274 254 146)(19 273 255 145)(20 272 256 144)(21 271 257 143)(22 297 258 142)(23 296 259 141)(24 295 260 140)(25 294 261 139)(26 293 262 138)(27 292 263 137)(28 127 318 225)(29 126 319 224)(30 125 320 223)(31 124 321 222)(32 123 322 221)(33 122 323 220)(34 121 324 219)(35 120 298 218)(36 119 299 217)(37 118 300 243)(38 117 301 242)(39 116 302 241)(40 115 303 240)(41 114 304 239)(42 113 305 238)(43 112 306 237)(44 111 307 236)(45 110 308 235)(46 109 309 234)(47 135 310 233)(48 134 311 232)(49 133 312 231)(50 132 313 230)(51 131 314 229)(52 130 315 228)(53 129 316 227)(54 128 317 226)(55 195 164 106)(56 194 165 105)(57 193 166 104)(58 192 167 103)(59 191 168 102)(60 190 169 101)(61 216 170 100)(62 215 171 99)(63 214 172 98)(64 213 173 97)(65 212 174 96)(66 211 175 95)(67 210 176 94)(68 209 177 93)(69 208 178 92)(70 207 179 91)(71 206 180 90)(72 205 181 89)(73 204 182 88)(74 203 183 87)(75 202 184 86)(76 201 185 85)(77 200 186 84)(78 199 187 83)(79 198 188 82)(80 197 189 108)(81 196 163 107)

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

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

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

84 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 9A ··· 9I 18A ··· 18I 27A ··· 27AA 54A ··· 54AA order 1 2 3 3 3 3 4 4 6 6 6 6 9 ··· 9 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 2 2 2 2 81 81 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - - + + - - + - image C1 C2 C4 S3 S3 Dic3 Dic3 D9 D9 Dic9 Dic9 D27 Dic27 kernel C27⋊Dic3 C3×C54 C3×C27 C54 C3×C18 C27 C3×C9 C18 C3×C6 C9 C32 C6 C3 # reps 1 1 2 3 1 3 1 6 3 6 3 27 27

Matrix representation of C27⋊Dic3 in GL4(𝔽109) generated by

 32 82 0 0 27 59 0 0 0 0 87 51 0 0 58 29
,
 0 108 0 0 1 1 0 0 0 0 1 108 0 0 1 0
,
 37 61 0 0 24 72 0 0 0 0 74 53 0 0 18 35
G:=sub<GL(4,GF(109))| [32,27,0,0,82,59,0,0,0,0,87,58,0,0,51,29],[0,1,0,0,108,1,0,0,0,0,1,1,0,0,108,0],[37,24,0,0,61,72,0,0,0,0,74,18,0,0,53,35] >;

C27⋊Dic3 in GAP, Magma, Sage, TeX

C_{27}\rtimes {\rm Dic}_3
% in TeX

G:=Group("C27:Dic3");
// GroupNames label

G:=SmallGroup(324,21);
// by ID

G=gap.SmallGroup(324,21);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,794,824,579,5404,208,7781]);
// Polycyclic

G:=Group<a,b,c|a^27=b^6=1,c^2=b^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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