metabelian, supersoluble, monomial, A-group
Aliases: C9⋊Dic9, C92⋊3C4, C18.3D9, C2.(C9⋊D9), C6.1(C9⋊S3), (C9×C18).2C2, (C3×C18).19S3, (C3×C9).6Dic3, C3.1(C9⋊Dic3), C32.7(C3⋊Dic3), (C3×C6).15(C3⋊S3), SmallGroup(324,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C92 — C9⋊Dic9 |
Generators and relations for C9⋊Dic9
G = < a,b,c | a9=b18=1, c2=b9, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 365 in 69 conjugacy classes, 47 normal (7 characteristic)
C1, C2, C3, C4, C6, C9, C32, Dic3, C18, C3×C6, C3×C9, Dic9, C3⋊Dic3, C3×C18, C92, C9⋊Dic3, C9×C18, C9⋊Dic9
Quotients: C1, C2, C4, S3, Dic3, D9, C3⋊S3, Dic9, C3⋊Dic3, C9⋊S3, C9⋊Dic3, C9⋊D9, C9⋊Dic9
►Regular action on 324 points
Generators in S324
(1 202 51 19 275 258 225 109 163)(2 203 52 20 276 259 226 110 164)(3 204 53 21 277 260 227 111 165)(4 205 54 22 278 261 228 112 166)(5 206 37 23 279 262 229 113 167)(6 207 38 24 280 263 230 114 168)(7 208 39 25 281 264 231 115 169)(8 209 40 26 282 265 232 116 170)(9 210 41 27 283 266 233 117 171)(10 211 42 28 284 267 234 118 172)(11 212 43 29 285 268 217 119 173)(12 213 44 30 286 269 218 120 174)(13 214 45 31 287 270 219 121 175)(14 215 46 32 288 253 220 122 176)(15 216 47 33 271 254 221 123 177)(16 199 48 34 272 255 222 124 178)(17 200 49 35 273 256 223 125 179)(18 201 50 36 274 257 224 126 180)(55 247 188 96 151 132 83 314 293)(56 248 189 97 152 133 84 315 294)(57 249 190 98 153 134 85 316 295)(58 250 191 99 154 135 86 317 296)(59 251 192 100 155 136 87 318 297)(60 252 193 101 156 137 88 319 298)(61 235 194 102 157 138 89 320 299)(62 236 195 103 158 139 90 321 300)(63 237 196 104 159 140 73 322 301)(64 238 197 105 160 141 74 323 302)(65 239 198 106 161 142 75 324 303)(66 240 181 107 162 143 76 307 304)(67 241 182 108 145 144 77 308 305)(68 242 183 91 146 127 78 309 306)(69 243 184 92 147 128 79 310 289)(70 244 185 93 148 129 80 311 290)(71 245 186 94 149 130 81 312 291)(72 246 187 95 150 131 82 313 292)
(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)
(1 194 10 185)(2 193 11 184)(3 192 12 183)(4 191 13 182)(5 190 14 181)(6 189 15 198)(7 188 16 197)(8 187 17 196)(9 186 18 195)(19 299 28 290)(20 298 29 289)(21 297 30 306)(22 296 31 305)(23 295 32 304)(24 294 33 303)(25 293 34 302)(26 292 35 301)(27 291 36 300)(37 57 46 66)(38 56 47 65)(39 55 48 64)(40 72 49 63)(41 71 50 62)(42 70 51 61)(43 69 52 60)(44 68 53 59)(45 67 54 58)(73 265 82 256)(74 264 83 255)(75 263 84 254)(76 262 85 253)(77 261 86 270)(78 260 87 269)(79 259 88 268)(80 258 89 267)(81 257 90 266)(91 165 100 174)(92 164 101 173)(93 163 102 172)(94 180 103 171)(95 179 104 170)(96 178 105 169)(97 177 106 168)(98 176 107 167)(99 175 108 166)(109 157 118 148)(110 156 119 147)(111 155 120 146)(112 154 121 145)(113 153 122 162)(114 152 123 161)(115 151 124 160)(116 150 125 159)(117 149 126 158)(127 227 136 218)(128 226 137 217)(129 225 138 234)(130 224 139 233)(131 223 140 232)(132 222 141 231)(133 221 142 230)(134 220 143 229)(135 219 144 228)(199 238 208 247)(200 237 209 246)(201 236 210 245)(202 235 211 244)(203 252 212 243)(204 251 213 242)(205 250 214 241)(206 249 215 240)(207 248 216 239)(271 324 280 315)(272 323 281 314)(273 322 282 313)(274 321 283 312)(275 320 284 311)(276 319 285 310)(277 318 286 309)(278 317 287 308)(279 316 288 307)
G:=sub<Sym(324)| (1,202,51,19,275,258,225,109,163)(2,203,52,20,276,259,226,110,164)(3,204,53,21,277,260,227,111,165)(4,205,54,22,278,261,228,112,166)(5,206,37,23,279,262,229,113,167)(6,207,38,24,280,263,230,114,168)(7,208,39,25,281,264,231,115,169)(8,209,40,26,282,265,232,116,170)(9,210,41,27,283,266,233,117,171)(10,211,42,28,284,267,234,118,172)(11,212,43,29,285,268,217,119,173)(12,213,44,30,286,269,218,120,174)(13,214,45,31,287,270,219,121,175)(14,215,46,32,288,253,220,122,176)(15,216,47,33,271,254,221,123,177)(16,199,48,34,272,255,222,124,178)(17,200,49,35,273,256,223,125,179)(18,201,50,36,274,257,224,126,180)(55,247,188,96,151,132,83,314,293)(56,248,189,97,152,133,84,315,294)(57,249,190,98,153,134,85,316,295)(58,250,191,99,154,135,86,317,296)(59,251,192,100,155,136,87,318,297)(60,252,193,101,156,137,88,319,298)(61,235,194,102,157,138,89,320,299)(62,236,195,103,158,139,90,321,300)(63,237,196,104,159,140,73,322,301)(64,238,197,105,160,141,74,323,302)(65,239,198,106,161,142,75,324,303)(66,240,181,107,162,143,76,307,304)(67,241,182,108,145,144,77,308,305)(68,242,183,91,146,127,78,309,306)(69,243,184,92,147,128,79,310,289)(70,244,185,93,148,129,80,311,290)(71,245,186,94,149,130,81,312,291)(72,246,187,95,150,131,82,313,292), (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), (1,194,10,185)(2,193,11,184)(3,192,12,183)(4,191,13,182)(5,190,14,181)(6,189,15,198)(7,188,16,197)(8,187,17,196)(9,186,18,195)(19,299,28,290)(20,298,29,289)(21,297,30,306)(22,296,31,305)(23,295,32,304)(24,294,33,303)(25,293,34,302)(26,292,35,301)(27,291,36,300)(37,57,46,66)(38,56,47,65)(39,55,48,64)(40,72,49,63)(41,71,50,62)(42,70,51,61)(43,69,52,60)(44,68,53,59)(45,67,54,58)(73,265,82,256)(74,264,83,255)(75,263,84,254)(76,262,85,253)(77,261,86,270)(78,260,87,269)(79,259,88,268)(80,258,89,267)(81,257,90,266)(91,165,100,174)(92,164,101,173)(93,163,102,172)(94,180,103,171)(95,179,104,170)(96,178,105,169)(97,177,106,168)(98,176,107,167)(99,175,108,166)(109,157,118,148)(110,156,119,147)(111,155,120,146)(112,154,121,145)(113,153,122,162)(114,152,123,161)(115,151,124,160)(116,150,125,159)(117,149,126,158)(127,227,136,218)(128,226,137,217)(129,225,138,234)(130,224,139,233)(131,223,140,232)(132,222,141,231)(133,221,142,230)(134,220,143,229)(135,219,144,228)(199,238,208,247)(200,237,209,246)(201,236,210,245)(202,235,211,244)(203,252,212,243)(204,251,213,242)(205,250,214,241)(206,249,215,240)(207,248,216,239)(271,324,280,315)(272,323,281,314)(273,322,282,313)(274,321,283,312)(275,320,284,311)(276,319,285,310)(277,318,286,309)(278,317,287,308)(279,316,288,307)>;
G:=Group( (1,202,51,19,275,258,225,109,163)(2,203,52,20,276,259,226,110,164)(3,204,53,21,277,260,227,111,165)(4,205,54,22,278,261,228,112,166)(5,206,37,23,279,262,229,113,167)(6,207,38,24,280,263,230,114,168)(7,208,39,25,281,264,231,115,169)(8,209,40,26,282,265,232,116,170)(9,210,41,27,283,266,233,117,171)(10,211,42,28,284,267,234,118,172)(11,212,43,29,285,268,217,119,173)(12,213,44,30,286,269,218,120,174)(13,214,45,31,287,270,219,121,175)(14,215,46,32,288,253,220,122,176)(15,216,47,33,271,254,221,123,177)(16,199,48,34,272,255,222,124,178)(17,200,49,35,273,256,223,125,179)(18,201,50,36,274,257,224,126,180)(55,247,188,96,151,132,83,314,293)(56,248,189,97,152,133,84,315,294)(57,249,190,98,153,134,85,316,295)(58,250,191,99,154,135,86,317,296)(59,251,192,100,155,136,87,318,297)(60,252,193,101,156,137,88,319,298)(61,235,194,102,157,138,89,320,299)(62,236,195,103,158,139,90,321,300)(63,237,196,104,159,140,73,322,301)(64,238,197,105,160,141,74,323,302)(65,239,198,106,161,142,75,324,303)(66,240,181,107,162,143,76,307,304)(67,241,182,108,145,144,77,308,305)(68,242,183,91,146,127,78,309,306)(69,243,184,92,147,128,79,310,289)(70,244,185,93,148,129,80,311,290)(71,245,186,94,149,130,81,312,291)(72,246,187,95,150,131,82,313,292), (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), (1,194,10,185)(2,193,11,184)(3,192,12,183)(4,191,13,182)(5,190,14,181)(6,189,15,198)(7,188,16,197)(8,187,17,196)(9,186,18,195)(19,299,28,290)(20,298,29,289)(21,297,30,306)(22,296,31,305)(23,295,32,304)(24,294,33,303)(25,293,34,302)(26,292,35,301)(27,291,36,300)(37,57,46,66)(38,56,47,65)(39,55,48,64)(40,72,49,63)(41,71,50,62)(42,70,51,61)(43,69,52,60)(44,68,53,59)(45,67,54,58)(73,265,82,256)(74,264,83,255)(75,263,84,254)(76,262,85,253)(77,261,86,270)(78,260,87,269)(79,259,88,268)(80,258,89,267)(81,257,90,266)(91,165,100,174)(92,164,101,173)(93,163,102,172)(94,180,103,171)(95,179,104,170)(96,178,105,169)(97,177,106,168)(98,176,107,167)(99,175,108,166)(109,157,118,148)(110,156,119,147)(111,155,120,146)(112,154,121,145)(113,153,122,162)(114,152,123,161)(115,151,124,160)(116,150,125,159)(117,149,126,158)(127,227,136,218)(128,226,137,217)(129,225,138,234)(130,224,139,233)(131,223,140,232)(132,222,141,231)(133,221,142,230)(134,220,143,229)(135,219,144,228)(199,238,208,247)(200,237,209,246)(201,236,210,245)(202,235,211,244)(203,252,212,243)(204,251,213,242)(205,250,214,241)(206,249,215,240)(207,248,216,239)(271,324,280,315)(272,323,281,314)(273,322,282,313)(274,321,283,312)(275,320,284,311)(276,319,285,310)(277,318,286,309)(278,317,287,308)(279,316,288,307) );
G=PermutationGroup([[(1,202,51,19,275,258,225,109,163),(2,203,52,20,276,259,226,110,164),(3,204,53,21,277,260,227,111,165),(4,205,54,22,278,261,228,112,166),(5,206,37,23,279,262,229,113,167),(6,207,38,24,280,263,230,114,168),(7,208,39,25,281,264,231,115,169),(8,209,40,26,282,265,232,116,170),(9,210,41,27,283,266,233,117,171),(10,211,42,28,284,267,234,118,172),(11,212,43,29,285,268,217,119,173),(12,213,44,30,286,269,218,120,174),(13,214,45,31,287,270,219,121,175),(14,215,46,32,288,253,220,122,176),(15,216,47,33,271,254,221,123,177),(16,199,48,34,272,255,222,124,178),(17,200,49,35,273,256,223,125,179),(18,201,50,36,274,257,224,126,180),(55,247,188,96,151,132,83,314,293),(56,248,189,97,152,133,84,315,294),(57,249,190,98,153,134,85,316,295),(58,250,191,99,154,135,86,317,296),(59,251,192,100,155,136,87,318,297),(60,252,193,101,156,137,88,319,298),(61,235,194,102,157,138,89,320,299),(62,236,195,103,158,139,90,321,300),(63,237,196,104,159,140,73,322,301),(64,238,197,105,160,141,74,323,302),(65,239,198,106,161,142,75,324,303),(66,240,181,107,162,143,76,307,304),(67,241,182,108,145,144,77,308,305),(68,242,183,91,146,127,78,309,306),(69,243,184,92,147,128,79,310,289),(70,244,185,93,148,129,80,311,290),(71,245,186,94,149,130,81,312,291),(72,246,187,95,150,131,82,313,292)], [(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)], [(1,194,10,185),(2,193,11,184),(3,192,12,183),(4,191,13,182),(5,190,14,181),(6,189,15,198),(7,188,16,197),(8,187,17,196),(9,186,18,195),(19,299,28,290),(20,298,29,289),(21,297,30,306),(22,296,31,305),(23,295,32,304),(24,294,33,303),(25,293,34,302),(26,292,35,301),(27,291,36,300),(37,57,46,66),(38,56,47,65),(39,55,48,64),(40,72,49,63),(41,71,50,62),(42,70,51,61),(43,69,52,60),(44,68,53,59),(45,67,54,58),(73,265,82,256),(74,264,83,255),(75,263,84,254),(76,262,85,253),(77,261,86,270),(78,260,87,269),(79,259,88,268),(80,258,89,267),(81,257,90,266),(91,165,100,174),(92,164,101,173),(93,163,102,172),(94,180,103,171),(95,179,104,170),(96,178,105,169),(97,177,106,168),(98,176,107,167),(99,175,108,166),(109,157,118,148),(110,156,119,147),(111,155,120,146),(112,154,121,145),(113,153,122,162),(114,152,123,161),(115,151,124,160),(116,150,125,159),(117,149,126,158),(127,227,136,218),(128,226,137,217),(129,225,138,234),(130,224,139,233),(131,223,140,232),(132,222,141,231),(133,221,142,230),(134,220,143,229),(135,219,144,228),(199,238,208,247),(200,237,209,246),(201,236,210,245),(202,235,211,244),(203,252,212,243),(204,251,213,242),(205,250,214,241),(206,249,215,240),(207,248,216,239),(271,324,280,315),(272,323,281,314),(273,322,282,313),(274,321,283,312),(275,320,284,311),(276,319,285,310),(277,318,286,309),(278,317,287,308),(279,316,288,307)]])
84 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 9A | ··· | 9AJ | 18A | ··· | 18AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 81 | 81 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D9 | Dic9 |
| kernel | C9⋊Dic9 | C9×C18 | C92 | C3×C18 | C3×C9 | C18 | C9 |
| # reps | 1 | 1 | 2 | 4 | 4 | 36 | 36 |
Matrix representation of C9⋊Dic9 ►in GL4(𝔽37) generated by
| 6 | 26 | 0 | 0 |
| 11 | 17 | 0 | 0 |
| 0 | 0 | 17 | 11 |
| 0 | 0 | 26 | 6 |
| 20 | 26 | 0 | 0 |
| 11 | 31 | 0 | 0 |
| 0 | 0 | 26 | 6 |
| 0 | 0 | 31 | 20 |
| 2 | 13 | 0 | 0 |
| 11 | 35 | 0 | 0 |
| 0 | 0 | 11 | 17 |
| 0 | 0 | 6 | 26 |
G:=sub<GL(4,GF(37))| [6,11,0,0,26,17,0,0,0,0,17,26,0,0,11,6],[20,11,0,0,26,31,0,0,0,0,26,31,0,0,6,20],[2,11,0,0,13,35,0,0,0,0,11,6,0,0,17,26] >;
C9⋊Dic9 in GAP, Magma, Sage, TeX
C_9\rtimes {\rm Dic}_9 % in TeX
G:=Group("C9:Dic9"); // GroupNames label
G:=SmallGroup(324,19);
// by ID
G=gap.SmallGroup(324,19);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,794,338,3171,453,2164,7781]);
// Polycyclic
G:=Group<a,b,c|a^9=b^18=1,c^2=b^9,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations