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G = C5×C7⋊C9order 315 = 32·5·7

Direct product of C5 and C7⋊C9

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

C1C7 — C5×C7⋊C9
C1C7C21C105 — C5×C7⋊C9
C7 — C5×C7⋊C9
C1C15

Generators and relations for C5×C7⋊C9
 G = < a,b,c | a5=b7=c9=1, ab=ba, ac=ca, cbc-1=b4 >

7C9
7C45

Smallest permutation representation of C5×C7⋊C9
Regular action on 315 points
Generators in S315
(1 253 119 76 137)(2 254 120 77 138)(3 255 121 78 139)(4 256 122 79 140)(5 257 123 80 141)(6 258 124 81 142)(7 259 125 73 143)(8 260 126 74 144)(9 261 118 75 136)(10 179 227 291 37)(11 180 228 292 38)(12 172 229 293 39)(13 173 230 294 40)(14 174 231 295 41)(15 175 232 296 42)(16 176 233 297 43)(17 177 234 289 44)(18 178 226 290 45)(19 84 204 101 277)(20 85 205 102 278)(21 86 206 103 279)(22 87 207 104 271)(23 88 199 105 272)(24 89 200 106 273)(25 90 201 107 274)(26 82 202 108 275)(27 83 203 100 276)(28 93 47 220 186)(29 94 48 221 187)(30 95 49 222 188)(31 96 50 223 189)(32 97 51 224 181)(33 98 52 225 182)(34 99 53 217 183)(35 91 54 218 184)(36 92 46 219 185)(55 216 164 241 306)(56 208 165 242 298)(57 209 166 243 299)(58 210 167 235 300)(59 211 168 236 301)(60 212 169 237 302)(61 213 170 238 303)(62 214 171 239 304)(63 215 163 240 305)(64 245 190 153 116)(65 246 191 145 117)(66 247 192 146 109)(67 248 193 147 110)(68 249 194 148 111)(69 250 195 149 112)(70 251 196 150 113)(71 252 197 151 114)(72 244 198 152 115)(127 287 161 315 269)(128 288 162 307 270)(129 280 154 308 262)(130 281 155 309 263)(131 282 156 310 264)(132 283 157 311 265)(133 284 158 312 266)(134 285 159 313 267)(135 286 160 314 268)
(1 132 61 231 146 82 51)(2 147 133 83 62 52 232)(3 63 148 53 134 233 84)(4 135 55 234 149 85 54)(5 150 127 86 56 46 226)(6 57 151 47 128 227 87)(7 129 58 228 152 88 48)(8 153 130 89 59 49 229)(9 60 145 50 131 230 90)(10 271 81 243 252 28 307)(11 244 272 29 73 308 235)(12 74 245 309 273 236 30)(13 274 75 237 246 31 310)(14 247 275 32 76 311 238)(15 77 248 312 276 239 33)(16 277 78 240 249 34 313)(17 250 278 35 79 314 241)(18 80 251 315 279 242 36)(19 139 305 194 99 267 176)(20 91 140 268 306 177 195)(21 298 92 178 141 196 269)(22 142 299 197 93 270 179)(23 94 143 262 300 180 198)(24 301 95 172 144 190 263)(25 136 302 191 96 264 173)(26 97 137 265 303 174 192)(27 304 98 175 138 193 266)(37 104 124 166 71 186 162)(38 72 105 187 125 154 167)(39 126 64 155 106 168 188)(40 107 118 169 65 189 156)(41 66 108 181 119 157 170)(42 120 67 158 100 171 182)(43 101 121 163 68 183 159)(44 69 102 184 122 160 164)(45 123 70 161 103 165 185)(109 202 224 253 283 213 295)(110 284 203 214 225 296 254)(111 217 285 297 204 255 215)(112 205 218 256 286 216 289)(113 287 206 208 219 290 257)(114 220 288 291 207 258 209)(115 199 221 259 280 210 292)(116 281 200 211 222 293 260)(117 223 282 294 201 261 212)
(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)| (1,253,119,76,137)(2,254,120,77,138)(3,255,121,78,139)(4,256,122,79,140)(5,257,123,80,141)(6,258,124,81,142)(7,259,125,73,143)(8,260,126,74,144)(9,261,118,75,136)(10,179,227,291,37)(11,180,228,292,38)(12,172,229,293,39)(13,173,230,294,40)(14,174,231,295,41)(15,175,232,296,42)(16,176,233,297,43)(17,177,234,289,44)(18,178,226,290,45)(19,84,204,101,277)(20,85,205,102,278)(21,86,206,103,279)(22,87,207,104,271)(23,88,199,105,272)(24,89,200,106,273)(25,90,201,107,274)(26,82,202,108,275)(27,83,203,100,276)(28,93,47,220,186)(29,94,48,221,187)(30,95,49,222,188)(31,96,50,223,189)(32,97,51,224,181)(33,98,52,225,182)(34,99,53,217,183)(35,91,54,218,184)(36,92,46,219,185)(55,216,164,241,306)(56,208,165,242,298)(57,209,166,243,299)(58,210,167,235,300)(59,211,168,236,301)(60,212,169,237,302)(61,213,170,238,303)(62,214,171,239,304)(63,215,163,240,305)(64,245,190,153,116)(65,246,191,145,117)(66,247,192,146,109)(67,248,193,147,110)(68,249,194,148,111)(69,250,195,149,112)(70,251,196,150,113)(71,252,197,151,114)(72,244,198,152,115)(127,287,161,315,269)(128,288,162,307,270)(129,280,154,308,262)(130,281,155,309,263)(131,282,156,310,264)(132,283,157,311,265)(133,284,158,312,266)(134,285,159,313,267)(135,286,160,314,268), (1,132,61,231,146,82,51)(2,147,133,83,62,52,232)(3,63,148,53,134,233,84)(4,135,55,234,149,85,54)(5,150,127,86,56,46,226)(6,57,151,47,128,227,87)(7,129,58,228,152,88,48)(8,153,130,89,59,49,229)(9,60,145,50,131,230,90)(10,271,81,243,252,28,307)(11,244,272,29,73,308,235)(12,74,245,309,273,236,30)(13,274,75,237,246,31,310)(14,247,275,32,76,311,238)(15,77,248,312,276,239,33)(16,277,78,240,249,34,313)(17,250,278,35,79,314,241)(18,80,251,315,279,242,36)(19,139,305,194,99,267,176)(20,91,140,268,306,177,195)(21,298,92,178,141,196,269)(22,142,299,197,93,270,179)(23,94,143,262,300,180,198)(24,301,95,172,144,190,263)(25,136,302,191,96,264,173)(26,97,137,265,303,174,192)(27,304,98,175,138,193,266)(37,104,124,166,71,186,162)(38,72,105,187,125,154,167)(39,126,64,155,106,168,188)(40,107,118,169,65,189,156)(41,66,108,181,119,157,170)(42,120,67,158,100,171,182)(43,101,121,163,68,183,159)(44,69,102,184,122,160,164)(45,123,70,161,103,165,185)(109,202,224,253,283,213,295)(110,284,203,214,225,296,254)(111,217,285,297,204,255,215)(112,205,218,256,286,216,289)(113,287,206,208,219,290,257)(114,220,288,291,207,258,209)(115,199,221,259,280,210,292)(116,281,200,211,222,293,260)(117,223,282,294,201,261,212), (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( (1,253,119,76,137)(2,254,120,77,138)(3,255,121,78,139)(4,256,122,79,140)(5,257,123,80,141)(6,258,124,81,142)(7,259,125,73,143)(8,260,126,74,144)(9,261,118,75,136)(10,179,227,291,37)(11,180,228,292,38)(12,172,229,293,39)(13,173,230,294,40)(14,174,231,295,41)(15,175,232,296,42)(16,176,233,297,43)(17,177,234,289,44)(18,178,226,290,45)(19,84,204,101,277)(20,85,205,102,278)(21,86,206,103,279)(22,87,207,104,271)(23,88,199,105,272)(24,89,200,106,273)(25,90,201,107,274)(26,82,202,108,275)(27,83,203,100,276)(28,93,47,220,186)(29,94,48,221,187)(30,95,49,222,188)(31,96,50,223,189)(32,97,51,224,181)(33,98,52,225,182)(34,99,53,217,183)(35,91,54,218,184)(36,92,46,219,185)(55,216,164,241,306)(56,208,165,242,298)(57,209,166,243,299)(58,210,167,235,300)(59,211,168,236,301)(60,212,169,237,302)(61,213,170,238,303)(62,214,171,239,304)(63,215,163,240,305)(64,245,190,153,116)(65,246,191,145,117)(66,247,192,146,109)(67,248,193,147,110)(68,249,194,148,111)(69,250,195,149,112)(70,251,196,150,113)(71,252,197,151,114)(72,244,198,152,115)(127,287,161,315,269)(128,288,162,307,270)(129,280,154,308,262)(130,281,155,309,263)(131,282,156,310,264)(132,283,157,311,265)(133,284,158,312,266)(134,285,159,313,267)(135,286,160,314,268), (1,132,61,231,146,82,51)(2,147,133,83,62,52,232)(3,63,148,53,134,233,84)(4,135,55,234,149,85,54)(5,150,127,86,56,46,226)(6,57,151,47,128,227,87)(7,129,58,228,152,88,48)(8,153,130,89,59,49,229)(9,60,145,50,131,230,90)(10,271,81,243,252,28,307)(11,244,272,29,73,308,235)(12,74,245,309,273,236,30)(13,274,75,237,246,31,310)(14,247,275,32,76,311,238)(15,77,248,312,276,239,33)(16,277,78,240,249,34,313)(17,250,278,35,79,314,241)(18,80,251,315,279,242,36)(19,139,305,194,99,267,176)(20,91,140,268,306,177,195)(21,298,92,178,141,196,269)(22,142,299,197,93,270,179)(23,94,143,262,300,180,198)(24,301,95,172,144,190,263)(25,136,302,191,96,264,173)(26,97,137,265,303,174,192)(27,304,98,175,138,193,266)(37,104,124,166,71,186,162)(38,72,105,187,125,154,167)(39,126,64,155,106,168,188)(40,107,118,169,65,189,156)(41,66,108,181,119,157,170)(42,120,67,158,100,171,182)(43,101,121,163,68,183,159)(44,69,102,184,122,160,164)(45,123,70,161,103,165,185)(109,202,224,253,283,213,295)(110,284,203,214,225,296,254)(111,217,285,297,204,255,215)(112,205,218,256,286,216,289)(113,287,206,208,219,290,257)(114,220,288,291,207,258,209)(115,199,221,259,280,210,292)(116,281,200,211,222,293,260)(117,223,282,294,201,261,212), (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([(1,253,119,76,137),(2,254,120,77,138),(3,255,121,78,139),(4,256,122,79,140),(5,257,123,80,141),(6,258,124,81,142),(7,259,125,73,143),(8,260,126,74,144),(9,261,118,75,136),(10,179,227,291,37),(11,180,228,292,38),(12,172,229,293,39),(13,173,230,294,40),(14,174,231,295,41),(15,175,232,296,42),(16,176,233,297,43),(17,177,234,289,44),(18,178,226,290,45),(19,84,204,101,277),(20,85,205,102,278),(21,86,206,103,279),(22,87,207,104,271),(23,88,199,105,272),(24,89,200,106,273),(25,90,201,107,274),(26,82,202,108,275),(27,83,203,100,276),(28,93,47,220,186),(29,94,48,221,187),(30,95,49,222,188),(31,96,50,223,189),(32,97,51,224,181),(33,98,52,225,182),(34,99,53,217,183),(35,91,54,218,184),(36,92,46,219,185),(55,216,164,241,306),(56,208,165,242,298),(57,209,166,243,299),(58,210,167,235,300),(59,211,168,236,301),(60,212,169,237,302),(61,213,170,238,303),(62,214,171,239,304),(63,215,163,240,305),(64,245,190,153,116),(65,246,191,145,117),(66,247,192,146,109),(67,248,193,147,110),(68,249,194,148,111),(69,250,195,149,112),(70,251,196,150,113),(71,252,197,151,114),(72,244,198,152,115),(127,287,161,315,269),(128,288,162,307,270),(129,280,154,308,262),(130,281,155,309,263),(131,282,156,310,264),(132,283,157,311,265),(133,284,158,312,266),(134,285,159,313,267),(135,286,160,314,268)], [(1,132,61,231,146,82,51),(2,147,133,83,62,52,232),(3,63,148,53,134,233,84),(4,135,55,234,149,85,54),(5,150,127,86,56,46,226),(6,57,151,47,128,227,87),(7,129,58,228,152,88,48),(8,153,130,89,59,49,229),(9,60,145,50,131,230,90),(10,271,81,243,252,28,307),(11,244,272,29,73,308,235),(12,74,245,309,273,236,30),(13,274,75,237,246,31,310),(14,247,275,32,76,311,238),(15,77,248,312,276,239,33),(16,277,78,240,249,34,313),(17,250,278,35,79,314,241),(18,80,251,315,279,242,36),(19,139,305,194,99,267,176),(20,91,140,268,306,177,195),(21,298,92,178,141,196,269),(22,142,299,197,93,270,179),(23,94,143,262,300,180,198),(24,301,95,172,144,190,263),(25,136,302,191,96,264,173),(26,97,137,265,303,174,192),(27,304,98,175,138,193,266),(37,104,124,166,71,186,162),(38,72,105,187,125,154,167),(39,126,64,155,106,168,188),(40,107,118,169,65,189,156),(41,66,108,181,119,157,170),(42,120,67,158,100,171,182),(43,101,121,163,68,183,159),(44,69,102,184,122,160,164),(45,123,70,161,103,165,185),(109,202,224,253,283,213,295),(110,284,203,214,225,296,254),(111,217,285,297,204,255,215),(112,205,218,256,286,216,289),(113,287,206,208,219,290,257),(114,220,288,291,207,258,209),(115,199,221,259,280,210,292),(116,281,200,211,222,293,260),(117,223,282,294,201,261,212)], [(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 3A3B5A5B5C5D7A7B9A···9F15A···15H21A21B21C21D35A···35H45A···45X105A···105P
order1335555779···915···152121212135···3545···45105···105
size1111111337···71···133333···37···73···3

75 irreducible representations

dim1111113333
type+
imageC1C3C5C9C15C45C7⋊C3C7⋊C9C5×C7⋊C3C5×C7⋊C9
kernelC5×C7⋊C9C105C7⋊C9C35C21C7C15C5C3C1
# reps124682424816

Matrix representation of C5×C7⋊C9 in GL4(𝔽631) generated by

242000
0100
0010
0001
,
1000
05955961
0100
0010
,
1000
0264240323
0527180253
0323534187
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

Subgroup lattice of C5×C7⋊C9 in TeX

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