direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C5×C7⋊C9, C35⋊C9, C7⋊C45, C21.C15, C105.C3, C15.(C7⋊C3), C3.(C5×C7⋊C3), SmallGroup(315,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C5×C7⋊C9 |
Generators and relations for C5×C7⋊C9
G = < a,b,c | a5=b7=c9=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C5×C7⋊C9
►Regular action on 315 points
Generators in S315
(1 148 112 76 54)(2 149 113 77 46)(3 150 114 78 47)(4 151 115 79 48)(5 152 116 80 49)(6 153 117 81 50)(7 145 109 73 51)(8 146 110 74 52)(9 147 111 75 53)(10 163 19 84 99)(11 164 20 85 91)(12 165 21 86 92)(13 166 22 87 93)(14 167 23 88 94)(15 168 24 89 95)(16 169 25 90 96)(17 170 26 82 97)(18 171 27 83 98)(28 280 252 308 262)(29 281 244 309 263)(30 282 245 310 264)(31 283 246 311 265)(32 284 247 312 266)(33 285 248 313 267)(34 286 249 314 268)(35 287 250 315 269)(36 288 251 307 270)(37 101 65 227 291)(38 102 66 228 292)(39 103 67 229 293)(40 104 68 230 294)(41 105 69 231 295)(42 106 70 232 296)(43 107 71 233 297)(44 108 72 234 289)(45 100 64 226 290)(55 216 180 127 192)(56 208 172 128 193)(57 209 173 129 194)(58 210 174 130 195)(59 211 175 131 196)(60 212 176 132 197)(61 213 177 133 198)(62 214 178 134 190)(63 215 179 135 191)(118 271 136 201 155)(119 272 137 202 156)(120 273 138 203 157)(121 274 139 204 158)(122 275 140 205 159)(123 276 141 206 160)(124 277 142 207 161)(125 278 143 199 162)(126 279 144 200 154)(181 235 300 254 224)(182 236 301 255 225)(183 237 302 256 217)(184 238 303 257 218)(185 239 304 258 219)(186 240 305 259 220)(187 241 306 260 221)(188 242 298 261 222)(189 243 299 253 223)
(1 31 61 231 162 82 254)(2 154 32 83 62 255 232)(3 63 155 256 33 233 84)(4 34 55 234 156 85 257)(5 157 35 86 56 258 226)(6 57 158 259 36 227 87)(7 28 58 228 159 88 260)(8 160 29 89 59 261 229)(9 60 161 253 30 230 90)(10 114 179 271 183 248 43)(11 184 115 249 180 44 272)(12 172 185 45 116 273 250)(13 117 173 274 186 251 37)(14 187 109 252 174 38 275)(15 175 188 39 110 276 244)(16 111 176 277 189 245 40)(17 181 112 246 177 41 278)(18 178 182 42 113 279 247)(19 47 191 201 302 267 71)(20 303 48 268 192 72 202)(21 193 304 64 49 203 269)(22 50 194 204 305 270 65)(23 306 51 262 195 66 205)(24 196 298 67 52 206 263)(25 53 197 207 299 264 68)(26 300 54 265 198 69 199)(27 190 301 70 46 200 266)(73 308 130 102 140 167 241)(74 141 309 168 131 242 103)(75 132 142 243 310 104 169)(76 311 133 105 143 170 235)(77 144 312 171 134 236 106)(78 135 136 237 313 107 163)(79 314 127 108 137 164 238)(80 138 315 165 128 239 100)(81 129 139 240 307 101 166)(91 218 151 286 216 289 119)(92 208 219 290 152 120 287)(93 153 209 121 220 288 291)(94 221 145 280 210 292 122)(95 211 222 293 146 123 281)(96 147 212 124 223 282 294)(97 224 148 283 213 295 125)(98 214 225 296 149 126 284)(99 150 215 118 217 285 297)
(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)
G:=sub<Sym(315)| (1,148,112,76,54)(2,149,113,77,46)(3,150,114,78,47)(4,151,115,79,48)(5,152,116,80,49)(6,153,117,81,50)(7,145,109,73,51)(8,146,110,74,52)(9,147,111,75,53)(10,163,19,84,99)(11,164,20,85,91)(12,165,21,86,92)(13,166,22,87,93)(14,167,23,88,94)(15,168,24,89,95)(16,169,25,90,96)(17,170,26,82,97)(18,171,27,83,98)(28,280,252,308,262)(29,281,244,309,263)(30,282,245,310,264)(31,283,246,311,265)(32,284,247,312,266)(33,285,248,313,267)(34,286,249,314,268)(35,287,250,315,269)(36,288,251,307,270)(37,101,65,227,291)(38,102,66,228,292)(39,103,67,229,293)(40,104,68,230,294)(41,105,69,231,295)(42,106,70,232,296)(43,107,71,233,297)(44,108,72,234,289)(45,100,64,226,290)(55,216,180,127,192)(56,208,172,128,193)(57,209,173,129,194)(58,210,174,130,195)(59,211,175,131,196)(60,212,176,132,197)(61,213,177,133,198)(62,214,178,134,190)(63,215,179,135,191)(118,271,136,201,155)(119,272,137,202,156)(120,273,138,203,157)(121,274,139,204,158)(122,275,140,205,159)(123,276,141,206,160)(124,277,142,207,161)(125,278,143,199,162)(126,279,144,200,154)(181,235,300,254,224)(182,236,301,255,225)(183,237,302,256,217)(184,238,303,257,218)(185,239,304,258,219)(186,240,305,259,220)(187,241,306,260,221)(188,242,298,261,222)(189,243,299,253,223), (1,31,61,231,162,82,254)(2,154,32,83,62,255,232)(3,63,155,256,33,233,84)(4,34,55,234,156,85,257)(5,157,35,86,56,258,226)(6,57,158,259,36,227,87)(7,28,58,228,159,88,260)(8,160,29,89,59,261,229)(9,60,161,253,30,230,90)(10,114,179,271,183,248,43)(11,184,115,249,180,44,272)(12,172,185,45,116,273,250)(13,117,173,274,186,251,37)(14,187,109,252,174,38,275)(15,175,188,39,110,276,244)(16,111,176,277,189,245,40)(17,181,112,246,177,41,278)(18,178,182,42,113,279,247)(19,47,191,201,302,267,71)(20,303,48,268,192,72,202)(21,193,304,64,49,203,269)(22,50,194,204,305,270,65)(23,306,51,262,195,66,205)(24,196,298,67,52,206,263)(25,53,197,207,299,264,68)(26,300,54,265,198,69,199)(27,190,301,70,46,200,266)(73,308,130,102,140,167,241)(74,141,309,168,131,242,103)(75,132,142,243,310,104,169)(76,311,133,105,143,170,235)(77,144,312,171,134,236,106)(78,135,136,237,313,107,163)(79,314,127,108,137,164,238)(80,138,315,165,128,239,100)(81,129,139,240,307,101,166)(91,218,151,286,216,289,119)(92,208,219,290,152,120,287)(93,153,209,121,220,288,291)(94,221,145,280,210,292,122)(95,211,222,293,146,123,281)(96,147,212,124,223,282,294)(97,224,148,283,213,295,125)(98,214,225,296,149,126,284)(99,150,215,118,217,285,297), (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)>;
G:=Group( (1,148,112,76,54)(2,149,113,77,46)(3,150,114,78,47)(4,151,115,79,48)(5,152,116,80,49)(6,153,117,81,50)(7,145,109,73,51)(8,146,110,74,52)(9,147,111,75,53)(10,163,19,84,99)(11,164,20,85,91)(12,165,21,86,92)(13,166,22,87,93)(14,167,23,88,94)(15,168,24,89,95)(16,169,25,90,96)(17,170,26,82,97)(18,171,27,83,98)(28,280,252,308,262)(29,281,244,309,263)(30,282,245,310,264)(31,283,246,311,265)(32,284,247,312,266)(33,285,248,313,267)(34,286,249,314,268)(35,287,250,315,269)(36,288,251,307,270)(37,101,65,227,291)(38,102,66,228,292)(39,103,67,229,293)(40,104,68,230,294)(41,105,69,231,295)(42,106,70,232,296)(43,107,71,233,297)(44,108,72,234,289)(45,100,64,226,290)(55,216,180,127,192)(56,208,172,128,193)(57,209,173,129,194)(58,210,174,130,195)(59,211,175,131,196)(60,212,176,132,197)(61,213,177,133,198)(62,214,178,134,190)(63,215,179,135,191)(118,271,136,201,155)(119,272,137,202,156)(120,273,138,203,157)(121,274,139,204,158)(122,275,140,205,159)(123,276,141,206,160)(124,277,142,207,161)(125,278,143,199,162)(126,279,144,200,154)(181,235,300,254,224)(182,236,301,255,225)(183,237,302,256,217)(184,238,303,257,218)(185,239,304,258,219)(186,240,305,259,220)(187,241,306,260,221)(188,242,298,261,222)(189,243,299,253,223), (1,31,61,231,162,82,254)(2,154,32,83,62,255,232)(3,63,155,256,33,233,84)(4,34,55,234,156,85,257)(5,157,35,86,56,258,226)(6,57,158,259,36,227,87)(7,28,58,228,159,88,260)(8,160,29,89,59,261,229)(9,60,161,253,30,230,90)(10,114,179,271,183,248,43)(11,184,115,249,180,44,272)(12,172,185,45,116,273,250)(13,117,173,274,186,251,37)(14,187,109,252,174,38,275)(15,175,188,39,110,276,244)(16,111,176,277,189,245,40)(17,181,112,246,177,41,278)(18,178,182,42,113,279,247)(19,47,191,201,302,267,71)(20,303,48,268,192,72,202)(21,193,304,64,49,203,269)(22,50,194,204,305,270,65)(23,306,51,262,195,66,205)(24,196,298,67,52,206,263)(25,53,197,207,299,264,68)(26,300,54,265,198,69,199)(27,190,301,70,46,200,266)(73,308,130,102,140,167,241)(74,141,309,168,131,242,103)(75,132,142,243,310,104,169)(76,311,133,105,143,170,235)(77,144,312,171,134,236,106)(78,135,136,237,313,107,163)(79,314,127,108,137,164,238)(80,138,315,165,128,239,100)(81,129,139,240,307,101,166)(91,218,151,286,216,289,119)(92,208,219,290,152,120,287)(93,153,209,121,220,288,291)(94,221,145,280,210,292,122)(95,211,222,293,146,123,281)(96,147,212,124,223,282,294)(97,224,148,283,213,295,125)(98,214,225,296,149,126,284)(99,150,215,118,217,285,297), (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) );
G=PermutationGroup([[(1,148,112,76,54),(2,149,113,77,46),(3,150,114,78,47),(4,151,115,79,48),(5,152,116,80,49),(6,153,117,81,50),(7,145,109,73,51),(8,146,110,74,52),(9,147,111,75,53),(10,163,19,84,99),(11,164,20,85,91),(12,165,21,86,92),(13,166,22,87,93),(14,167,23,88,94),(15,168,24,89,95),(16,169,25,90,96),(17,170,26,82,97),(18,171,27,83,98),(28,280,252,308,262),(29,281,244,309,263),(30,282,245,310,264),(31,283,246,311,265),(32,284,247,312,266),(33,285,248,313,267),(34,286,249,314,268),(35,287,250,315,269),(36,288,251,307,270),(37,101,65,227,291),(38,102,66,228,292),(39,103,67,229,293),(40,104,68,230,294),(41,105,69,231,295),(42,106,70,232,296),(43,107,71,233,297),(44,108,72,234,289),(45,100,64,226,290),(55,216,180,127,192),(56,208,172,128,193),(57,209,173,129,194),(58,210,174,130,195),(59,211,175,131,196),(60,212,176,132,197),(61,213,177,133,198),(62,214,178,134,190),(63,215,179,135,191),(118,271,136,201,155),(119,272,137,202,156),(120,273,138,203,157),(121,274,139,204,158),(122,275,140,205,159),(123,276,141,206,160),(124,277,142,207,161),(125,278,143,199,162),(126,279,144,200,154),(181,235,300,254,224),(182,236,301,255,225),(183,237,302,256,217),(184,238,303,257,218),(185,239,304,258,219),(186,240,305,259,220),(187,241,306,260,221),(188,242,298,261,222),(189,243,299,253,223)], [(1,31,61,231,162,82,254),(2,154,32,83,62,255,232),(3,63,155,256,33,233,84),(4,34,55,234,156,85,257),(5,157,35,86,56,258,226),(6,57,158,259,36,227,87),(7,28,58,228,159,88,260),(8,160,29,89,59,261,229),(9,60,161,253,30,230,90),(10,114,179,271,183,248,43),(11,184,115,249,180,44,272),(12,172,185,45,116,273,250),(13,117,173,274,186,251,37),(14,187,109,252,174,38,275),(15,175,188,39,110,276,244),(16,111,176,277,189,245,40),(17,181,112,246,177,41,278),(18,178,182,42,113,279,247),(19,47,191,201,302,267,71),(20,303,48,268,192,72,202),(21,193,304,64,49,203,269),(22,50,194,204,305,270,65),(23,306,51,262,195,66,205),(24,196,298,67,52,206,263),(25,53,197,207,299,264,68),(26,300,54,265,198,69,199),(27,190,301,70,46,200,266),(73,308,130,102,140,167,241),(74,141,309,168,131,242,103),(75,132,142,243,310,104,169),(76,311,133,105,143,170,235),(77,144,312,171,134,236,106),(78,135,136,237,313,107,163),(79,314,127,108,137,164,238),(80,138,315,165,128,239,100),(81,129,139,240,307,101,166),(91,218,151,286,216,289,119),(92,208,219,290,152,120,287),(93,153,209,121,220,288,291),(94,221,145,280,210,292,122),(95,211,222,293,146,123,281),(96,147,212,124,223,282,294),(97,224,148,283,213,295,125),(98,214,225,296,149,126,284),(99,150,215,118,217,285,297)], [(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)]])
75 conjugacy classes
| class | 1 | 3A | 3B | 5A | 5B | 5C | 5D | 7A | 7B | 9A | ··· | 9F | 15A | ··· | 15H | 21A | 21B | 21C | 21D | 35A | ··· | 35H | 45A | ··· | 45X | 105A | ··· | 105P |
| order | 1 | 3 | 3 | 5 | 5 | 5 | 5 | 7 | 7 | 9 | ··· | 9 | 15 | ··· | 15 | 21 | 21 | 21 | 21 | 35 | ··· | 35 | 45 | ··· | 45 | 105 | ··· | 105 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 7 | ··· | 7 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 7 | ··· | 7 | 3 | ··· | 3 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | |||||||||
| image | C1 | C3 | C5 | C9 | C15 | C45 | C7⋊C3 | C7⋊C9 | C5×C7⋊C3 | C5×C7⋊C9 |
| kernel | C5×C7⋊C9 | C105 | C7⋊C9 | C35 | C21 | C7 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 4 | 6 | 8 | 24 | 2 | 4 | 8 | 16 |
Matrix representation of C5×C7⋊C9 ►in GL4(𝔽631) generated by
| 242 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 595 | 596 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 264 | 240 | 323 |
| 0 | 527 | 180 | 253 |
| 0 | 323 | 534 | 187 |
G:=sub<GL(4,GF(631))| [242,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,595,1,0,0,596,0,1,0,1,0,0],[1,0,0,0,0,264,527,323,0,240,180,534,0,323,253,187] >;
C5×C7⋊C9 in GAP, Magma, Sage, TeX
C_5\times C_7\rtimes C_9
% in TeX
G:=Group("C5xC7:C9"); // GroupNames label
G:=SmallGroup(315,1);
// by ID
G=gap.SmallGroup(315,1);
# by ID
G:=PCGroup([4,-3,-5,-3,-7,60,1443]);
// Polycyclic
G:=Group<a,b,c|a^5=b^7=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
Export