direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary
Aliases: C4×C9⋊C9, C36⋊C9, C9⋊2C36, C18.2C18, C12.23- 1+2, C12.2(C3×C9), (C3×C9).2C12, C6.3(C3×C18), (C3×C36).2C3, C3.2(C3×C36), (C3×C18).11C6, (C3×C12).7C32, C32.10(C3×C12), C3.2(C4×3- 1+2), C6.3(C2×3- 1+2), C2.(C2×C9⋊C9), (C2×C9⋊C9).2C2, (C3×C6).24(C3×C6), SmallGroup(324,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C9⋊C9
G = < a,b,c | a4=b9=c9=1, ab=ba, ac=ca, cbc-1=b7 >
Smallest permutation representation of C4×C9⋊C9
►Regular action on 324 points
Generators in S324
(1 214 52 194)(2 215 53 195)(3 216 54 196)(4 208 46 197)(5 209 47 198)(6 210 48 190)(7 211 49 191)(8 212 50 192)(9 213 51 193)(10 174 31 202)(11 175 32 203)(12 176 33 204)(13 177 34 205)(14 178 35 206)(15 179 36 207)(16 180 28 199)(17 172 29 200)(18 173 30 201)(19 171 45 188)(20 163 37 189)(21 164 38 181)(22 165 39 182)(23 166 40 183)(24 167 41 184)(25 168 42 185)(26 169 43 186)(27 170 44 187)(55 237 75 217)(56 238 76 218)(57 239 77 219)(58 240 78 220)(59 241 79 221)(60 242 80 222)(61 243 81 223)(62 235 73 224)(63 236 74 225)(64 260 98 226)(65 261 99 227)(66 253 91 228)(67 254 92 229)(68 255 93 230)(69 256 94 231)(70 257 95 232)(71 258 96 233)(72 259 97 234)(82 267 105 244)(83 268 106 245)(84 269 107 246)(85 270 108 247)(86 262 100 248)(87 263 101 249)(88 264 102 250)(89 265 103 251)(90 266 104 252)(109 294 132 271)(110 295 133 272)(111 296 134 273)(112 297 135 274)(113 289 127 275)(114 290 128 276)(115 291 129 277)(116 292 130 278)(117 293 131 279)(118 311 149 280)(119 312 150 281)(120 313 151 282)(121 314 152 283)(122 315 153 284)(123 307 145 285)(124 308 146 286)(125 309 147 287)(126 310 148 288)(136 324 162 298)(137 316 154 299)(138 317 155 300)(139 318 156 301)(140 319 157 302)(141 320 158 303)(142 321 159 304)(143 322 160 305)(144 323 161 306)
(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)
(1 150 96 12 166 115 55 137 89)(2 145 94 13 170 113 56 141 87)(3 149 92 14 165 111 57 136 85)(4 153 99 15 169 109 58 140 83)(5 148 97 16 164 116 59 144 90)(6 152 95 17 168 114 60 139 88)(7 147 93 18 163 112 61 143 86)(8 151 91 10 167 110 62 138 84)(9 146 98 11 171 117 63 142 82)(19 279 225 304 244 193 308 260 203)(20 274 223 305 248 191 309 255 201)(21 278 221 306 252 198 310 259 199)(22 273 219 298 247 196 311 254 206)(23 277 217 299 251 194 312 258 204)(24 272 224 300 246 192 313 253 202)(25 276 222 301 250 190 314 257 200)(26 271 220 302 245 197 315 261 207)(27 275 218 303 249 195 307 256 205)(28 181 130 79 161 104 47 126 72)(29 185 128 80 156 102 48 121 70)(30 189 135 81 160 100 49 125 68)(31 184 133 73 155 107 50 120 66)(32 188 131 74 159 105 51 124 64)(33 183 129 75 154 103 52 119 71)(34 187 127 76 158 101 53 123 69)(35 182 134 77 162 108 54 118 67)(36 186 132 78 157 106 46 122 65)(37 297 243 322 262 211 287 230 173)(38 292 241 323 266 209 288 234 180)(39 296 239 324 270 216 280 229 178)(40 291 237 316 265 214 281 233 176)(41 295 235 317 269 212 282 228 174)(42 290 242 318 264 210 283 232 172)(43 294 240 319 268 208 284 227 179)(44 289 238 320 263 215 285 231 177)(45 293 236 321 267 213 286 226 175)
G:=sub<Sym(324)| (1,214,52,194)(2,215,53,195)(3,216,54,196)(4,208,46,197)(5,209,47,198)(6,210,48,190)(7,211,49,191)(8,212,50,192)(9,213,51,193)(10,174,31,202)(11,175,32,203)(12,176,33,204)(13,177,34,205)(14,178,35,206)(15,179,36,207)(16,180,28,199)(17,172,29,200)(18,173,30,201)(19,171,45,188)(20,163,37,189)(21,164,38,181)(22,165,39,182)(23,166,40,183)(24,167,41,184)(25,168,42,185)(26,169,43,186)(27,170,44,187)(55,237,75,217)(56,238,76,218)(57,239,77,219)(58,240,78,220)(59,241,79,221)(60,242,80,222)(61,243,81,223)(62,235,73,224)(63,236,74,225)(64,260,98,226)(65,261,99,227)(66,253,91,228)(67,254,92,229)(68,255,93,230)(69,256,94,231)(70,257,95,232)(71,258,96,233)(72,259,97,234)(82,267,105,244)(83,268,106,245)(84,269,107,246)(85,270,108,247)(86,262,100,248)(87,263,101,249)(88,264,102,250)(89,265,103,251)(90,266,104,252)(109,294,132,271)(110,295,133,272)(111,296,134,273)(112,297,135,274)(113,289,127,275)(114,290,128,276)(115,291,129,277)(116,292,130,278)(117,293,131,279)(118,311,149,280)(119,312,150,281)(120,313,151,282)(121,314,152,283)(122,315,153,284)(123,307,145,285)(124,308,146,286)(125,309,147,287)(126,310,148,288)(136,324,162,298)(137,316,154,299)(138,317,155,300)(139,318,156,301)(140,319,157,302)(141,320,158,303)(142,321,159,304)(143,322,160,305)(144,323,161,306), (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), (1,150,96,12,166,115,55,137,89)(2,145,94,13,170,113,56,141,87)(3,149,92,14,165,111,57,136,85)(4,153,99,15,169,109,58,140,83)(5,148,97,16,164,116,59,144,90)(6,152,95,17,168,114,60,139,88)(7,147,93,18,163,112,61,143,86)(8,151,91,10,167,110,62,138,84)(9,146,98,11,171,117,63,142,82)(19,279,225,304,244,193,308,260,203)(20,274,223,305,248,191,309,255,201)(21,278,221,306,252,198,310,259,199)(22,273,219,298,247,196,311,254,206)(23,277,217,299,251,194,312,258,204)(24,272,224,300,246,192,313,253,202)(25,276,222,301,250,190,314,257,200)(26,271,220,302,245,197,315,261,207)(27,275,218,303,249,195,307,256,205)(28,181,130,79,161,104,47,126,72)(29,185,128,80,156,102,48,121,70)(30,189,135,81,160,100,49,125,68)(31,184,133,73,155,107,50,120,66)(32,188,131,74,159,105,51,124,64)(33,183,129,75,154,103,52,119,71)(34,187,127,76,158,101,53,123,69)(35,182,134,77,162,108,54,118,67)(36,186,132,78,157,106,46,122,65)(37,297,243,322,262,211,287,230,173)(38,292,241,323,266,209,288,234,180)(39,296,239,324,270,216,280,229,178)(40,291,237,316,265,214,281,233,176)(41,295,235,317,269,212,282,228,174)(42,290,242,318,264,210,283,232,172)(43,294,240,319,268,208,284,227,179)(44,289,238,320,263,215,285,231,177)(45,293,236,321,267,213,286,226,175)>;
G:=Group( (1,214,52,194)(2,215,53,195)(3,216,54,196)(4,208,46,197)(5,209,47,198)(6,210,48,190)(7,211,49,191)(8,212,50,192)(9,213,51,193)(10,174,31,202)(11,175,32,203)(12,176,33,204)(13,177,34,205)(14,178,35,206)(15,179,36,207)(16,180,28,199)(17,172,29,200)(18,173,30,201)(19,171,45,188)(20,163,37,189)(21,164,38,181)(22,165,39,182)(23,166,40,183)(24,167,41,184)(25,168,42,185)(26,169,43,186)(27,170,44,187)(55,237,75,217)(56,238,76,218)(57,239,77,219)(58,240,78,220)(59,241,79,221)(60,242,80,222)(61,243,81,223)(62,235,73,224)(63,236,74,225)(64,260,98,226)(65,261,99,227)(66,253,91,228)(67,254,92,229)(68,255,93,230)(69,256,94,231)(70,257,95,232)(71,258,96,233)(72,259,97,234)(82,267,105,244)(83,268,106,245)(84,269,107,246)(85,270,108,247)(86,262,100,248)(87,263,101,249)(88,264,102,250)(89,265,103,251)(90,266,104,252)(109,294,132,271)(110,295,133,272)(111,296,134,273)(112,297,135,274)(113,289,127,275)(114,290,128,276)(115,291,129,277)(116,292,130,278)(117,293,131,279)(118,311,149,280)(119,312,150,281)(120,313,151,282)(121,314,152,283)(122,315,153,284)(123,307,145,285)(124,308,146,286)(125,309,147,287)(126,310,148,288)(136,324,162,298)(137,316,154,299)(138,317,155,300)(139,318,156,301)(140,319,157,302)(141,320,158,303)(142,321,159,304)(143,322,160,305)(144,323,161,306), (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), (1,150,96,12,166,115,55,137,89)(2,145,94,13,170,113,56,141,87)(3,149,92,14,165,111,57,136,85)(4,153,99,15,169,109,58,140,83)(5,148,97,16,164,116,59,144,90)(6,152,95,17,168,114,60,139,88)(7,147,93,18,163,112,61,143,86)(8,151,91,10,167,110,62,138,84)(9,146,98,11,171,117,63,142,82)(19,279,225,304,244,193,308,260,203)(20,274,223,305,248,191,309,255,201)(21,278,221,306,252,198,310,259,199)(22,273,219,298,247,196,311,254,206)(23,277,217,299,251,194,312,258,204)(24,272,224,300,246,192,313,253,202)(25,276,222,301,250,190,314,257,200)(26,271,220,302,245,197,315,261,207)(27,275,218,303,249,195,307,256,205)(28,181,130,79,161,104,47,126,72)(29,185,128,80,156,102,48,121,70)(30,189,135,81,160,100,49,125,68)(31,184,133,73,155,107,50,120,66)(32,188,131,74,159,105,51,124,64)(33,183,129,75,154,103,52,119,71)(34,187,127,76,158,101,53,123,69)(35,182,134,77,162,108,54,118,67)(36,186,132,78,157,106,46,122,65)(37,297,243,322,262,211,287,230,173)(38,292,241,323,266,209,288,234,180)(39,296,239,324,270,216,280,229,178)(40,291,237,316,265,214,281,233,176)(41,295,235,317,269,212,282,228,174)(42,290,242,318,264,210,283,232,172)(43,294,240,319,268,208,284,227,179)(44,289,238,320,263,215,285,231,177)(45,293,236,321,267,213,286,226,175) );
G=PermutationGroup([[(1,214,52,194),(2,215,53,195),(3,216,54,196),(4,208,46,197),(5,209,47,198),(6,210,48,190),(7,211,49,191),(8,212,50,192),(9,213,51,193),(10,174,31,202),(11,175,32,203),(12,176,33,204),(13,177,34,205),(14,178,35,206),(15,179,36,207),(16,180,28,199),(17,172,29,200),(18,173,30,201),(19,171,45,188),(20,163,37,189),(21,164,38,181),(22,165,39,182),(23,166,40,183),(24,167,41,184),(25,168,42,185),(26,169,43,186),(27,170,44,187),(55,237,75,217),(56,238,76,218),(57,239,77,219),(58,240,78,220),(59,241,79,221),(60,242,80,222),(61,243,81,223),(62,235,73,224),(63,236,74,225),(64,260,98,226),(65,261,99,227),(66,253,91,228),(67,254,92,229),(68,255,93,230),(69,256,94,231),(70,257,95,232),(71,258,96,233),(72,259,97,234),(82,267,105,244),(83,268,106,245),(84,269,107,246),(85,270,108,247),(86,262,100,248),(87,263,101,249),(88,264,102,250),(89,265,103,251),(90,266,104,252),(109,294,132,271),(110,295,133,272),(111,296,134,273),(112,297,135,274),(113,289,127,275),(114,290,128,276),(115,291,129,277),(116,292,130,278),(117,293,131,279),(118,311,149,280),(119,312,150,281),(120,313,151,282),(121,314,152,283),(122,315,153,284),(123,307,145,285),(124,308,146,286),(125,309,147,287),(126,310,148,288),(136,324,162,298),(137,316,154,299),(138,317,155,300),(139,318,156,301),(140,319,157,302),(141,320,158,303),(142,321,159,304),(143,322,160,305),(144,323,161,306)], [(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)], [(1,150,96,12,166,115,55,137,89),(2,145,94,13,170,113,56,141,87),(3,149,92,14,165,111,57,136,85),(4,153,99,15,169,109,58,140,83),(5,148,97,16,164,116,59,144,90),(6,152,95,17,168,114,60,139,88),(7,147,93,18,163,112,61,143,86),(8,151,91,10,167,110,62,138,84),(9,146,98,11,171,117,63,142,82),(19,279,225,304,244,193,308,260,203),(20,274,223,305,248,191,309,255,201),(21,278,221,306,252,198,310,259,199),(22,273,219,298,247,196,311,254,206),(23,277,217,299,251,194,312,258,204),(24,272,224,300,246,192,313,253,202),(25,276,222,301,250,190,314,257,200),(26,271,220,302,245,197,315,261,207),(27,275,218,303,249,195,307,256,205),(28,181,130,79,161,104,47,126,72),(29,185,128,80,156,102,48,121,70),(30,189,135,81,160,100,49,125,68),(31,184,133,73,155,107,50,120,66),(32,188,131,74,159,105,51,124,64),(33,183,129,75,154,103,52,119,71),(34,187,127,76,158,101,53,123,69),(35,182,134,77,162,108,54,118,67),(36,186,132,78,157,106,46,122,65),(37,297,243,322,262,211,287,230,173),(38,292,241,323,266,209,288,234,180),(39,296,239,324,270,216,280,229,178),(40,291,237,316,265,214,281,233,176),(41,295,235,317,269,212,282,228,174),(42,290,242,318,264,210,283,232,172),(43,294,240,319,268,208,284,227,179),(44,289,238,320,263,215,285,231,177),(45,293,236,321,267,213,286,226,175)]])
132 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 9A | ··· | 9X | 12A | ··· | 12P | 18A | ··· | 18X | 36A | ··· | 36AV |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | 3- 1+2 | C2×3- 1+2 | C4×3- 1+2 |
| kernel | C4×C9⋊C9 | C2×C9⋊C9 | C3×C36 | C9⋊C9 | C3×C18 | C36 | C3×C9 | C18 | C9 | C12 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 18 | 16 | 18 | 36 | 6 | 6 | 12 |
Matrix representation of C4×C9⋊C9 ►in GL5(𝔽37)
| 36 | 0 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 30 | 25 | 0 |
| 0 | 0 | 34 | 22 | 12 |
| 33 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 1 |
| 0 | 0 | 34 | 34 | 12 |
| 0 | 0 | 14 | 27 | 24 |
G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,31,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,0,30,34,0,0,10,25,22,0,0,0,0,12],[33,0,0,0,0,0,1,0,0,0,0,0,16,34,14,0,0,0,34,27,0,0,1,12,24] >;
C4×C9⋊C9 in GAP, Magma, Sage, TeX
C_4\times C_9\rtimes C_9
% in TeX
G:=Group("C4xC9:C9"); // GroupNames label
G:=SmallGroup(324,28);
// by ID
G=gap.SmallGroup(324,28);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,655,386,122]);
// Polycyclic
G:=Group<a,b,c|a^4=b^9=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^7>;
// generators/relations
Export