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## G = C9×C5⋊2C8order 360 = 23·32·5

### Direct product of C9 and C5⋊2C8

Aliases: C9×C52C8, C52C72, C455C8, C90.5C4, C60.7C6, C36.4D5, C20.2C18, C180.6C2, C30.6C12, C15.2C24, C10.2C36, C18.2Dic5, C4.2(C9×D5), C2.(C9×Dic5), C12.6(C3×D5), C6.2(C3×Dic5), C3.(C3×C52C8), (C3×C52C8).C3, SmallGroup(360,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C9×C5⋊2C8
 Chief series C1 — C5 — C10 — C30 — C60 — C180 — C9×C5⋊2C8
 Lower central C5 — C9×C5⋊2C8
 Upper central C1 — C36

Generators and relations for C9×C52C8
G = < a,b,c | a9=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C9×C52C8
Regular action on 360 points
Generators in S360
(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)(325 326 327 328 329 330 331 332 333)(334 335 336 337 338 339 340 341 342)(343 344 345 346 347 348 349 350 351)(352 353 354 355 356 357 358 359 360)
(1 44 171 203 189)(2 45 163 204 181)(3 37 164 205 182)(4 38 165 206 183)(5 39 166 207 184)(6 40 167 199 185)(7 41 168 200 186)(8 42 169 201 187)(9 43 170 202 188)(10 341 235 287 57)(11 342 236 288 58)(12 334 237 280 59)(13 335 238 281 60)(14 336 239 282 61)(15 337 240 283 62)(16 338 241 284 63)(17 339 242 285 55)(18 340 243 286 56)(19 233 216 330 273)(20 234 208 331 274)(21 226 209 332 275)(22 227 210 333 276)(23 228 211 325 277)(24 229 212 326 278)(25 230 213 327 279)(26 231 214 328 271)(27 232 215 329 272)(28 148 87 107 307)(29 149 88 108 308)(30 150 89 100 309)(31 151 90 101 310)(32 152 82 102 311)(33 153 83 103 312)(34 145 84 104 313)(35 146 85 105 314)(36 147 86 106 315)(46 318 154 256 179)(47 319 155 257 180)(48 320 156 258 172)(49 321 157 259 173)(50 322 158 260 174)(51 323 159 261 175)(52 324 160 253 176)(53 316 161 254 177)(54 317 162 255 178)(64 347 220 77 120)(65 348 221 78 121)(66 349 222 79 122)(67 350 223 80 123)(68 351 224 81 124)(69 343 225 73 125)(70 344 217 74 126)(71 345 218 75 118)(72 346 219 76 119)(91 269 196 252 113)(92 270 197 244 114)(93 262 198 245 115)(94 263 190 246 116)(95 264 191 247 117)(96 265 192 248 109)(97 266 193 249 110)(98 267 194 250 111)(99 268 195 251 112)(127 297 138 299 353)(128 289 139 300 354)(129 290 140 301 355)(130 291 141 302 356)(131 292 142 303 357)(132 293 143 304 358)(133 294 144 305 359)(134 295 136 306 360)(135 296 137 298 352)
(1 290 159 23 307 70 263 287)(2 291 160 24 308 71 264 288)(3 292 161 25 309 72 265 280)(4 293 162 26 310 64 266 281)(5 294 154 27 311 65 267 282)(6 295 155 19 312 66 268 283)(7 296 156 20 313 67 269 284)(8 297 157 21 314 68 270 285)(9 289 158 22 315 69 262 286)(10 203 301 51 211 87 217 116)(11 204 302 52 212 88 218 117)(12 205 303 53 213 89 219 109)(13 206 304 54 214 90 220 110)(14 207 305 46 215 82 221 111)(15 199 306 47 216 83 222 112)(16 200 298 48 208 84 223 113)(17 201 299 49 209 85 224 114)(18 202 300 50 210 86 225 115)(28 126 190 235 44 129 261 277)(29 118 191 236 45 130 253 278)(30 119 192 237 37 131 254 279)(31 120 193 238 38 132 255 271)(32 121 194 239 39 133 256 272)(33 122 195 240 40 134 257 273)(34 123 196 241 41 135 258 274)(35 124 197 242 42 127 259 275)(36 125 198 243 43 128 260 276)(55 187 138 321 226 105 351 92)(56 188 139 322 227 106 343 93)(57 189 140 323 228 107 344 94)(58 181 141 324 229 108 345 95)(59 182 142 316 230 100 346 96)(60 183 143 317 231 101 347 97)(61 184 144 318 232 102 348 98)(62 185 136 319 233 103 349 99)(63 186 137 320 234 104 350 91)(73 245 340 170 354 174 333 147)(74 246 341 171 355 175 325 148)(75 247 342 163 356 176 326 149)(76 248 334 164 357 177 327 150)(77 249 335 165 358 178 328 151)(78 250 336 166 359 179 329 152)(79 251 337 167 360 180 330 153)(80 252 338 168 352 172 331 145)(81 244 339 169 353 173 332 146)

G:=sub<Sym(360)| (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)(325,326,327,328,329,330,331,332,333)(334,335,336,337,338,339,340,341,342)(343,344,345,346,347,348,349,350,351)(352,353,354,355,356,357,358,359,360), (1,44,171,203,189)(2,45,163,204,181)(3,37,164,205,182)(4,38,165,206,183)(5,39,166,207,184)(6,40,167,199,185)(7,41,168,200,186)(8,42,169,201,187)(9,43,170,202,188)(10,341,235,287,57)(11,342,236,288,58)(12,334,237,280,59)(13,335,238,281,60)(14,336,239,282,61)(15,337,240,283,62)(16,338,241,284,63)(17,339,242,285,55)(18,340,243,286,56)(19,233,216,330,273)(20,234,208,331,274)(21,226,209,332,275)(22,227,210,333,276)(23,228,211,325,277)(24,229,212,326,278)(25,230,213,327,279)(26,231,214,328,271)(27,232,215,329,272)(28,148,87,107,307)(29,149,88,108,308)(30,150,89,100,309)(31,151,90,101,310)(32,152,82,102,311)(33,153,83,103,312)(34,145,84,104,313)(35,146,85,105,314)(36,147,86,106,315)(46,318,154,256,179)(47,319,155,257,180)(48,320,156,258,172)(49,321,157,259,173)(50,322,158,260,174)(51,323,159,261,175)(52,324,160,253,176)(53,316,161,254,177)(54,317,162,255,178)(64,347,220,77,120)(65,348,221,78,121)(66,349,222,79,122)(67,350,223,80,123)(68,351,224,81,124)(69,343,225,73,125)(70,344,217,74,126)(71,345,218,75,118)(72,346,219,76,119)(91,269,196,252,113)(92,270,197,244,114)(93,262,198,245,115)(94,263,190,246,116)(95,264,191,247,117)(96,265,192,248,109)(97,266,193,249,110)(98,267,194,250,111)(99,268,195,251,112)(127,297,138,299,353)(128,289,139,300,354)(129,290,140,301,355)(130,291,141,302,356)(131,292,142,303,357)(132,293,143,304,358)(133,294,144,305,359)(134,295,136,306,360)(135,296,137,298,352), (1,290,159,23,307,70,263,287)(2,291,160,24,308,71,264,288)(3,292,161,25,309,72,265,280)(4,293,162,26,310,64,266,281)(5,294,154,27,311,65,267,282)(6,295,155,19,312,66,268,283)(7,296,156,20,313,67,269,284)(8,297,157,21,314,68,270,285)(9,289,158,22,315,69,262,286)(10,203,301,51,211,87,217,116)(11,204,302,52,212,88,218,117)(12,205,303,53,213,89,219,109)(13,206,304,54,214,90,220,110)(14,207,305,46,215,82,221,111)(15,199,306,47,216,83,222,112)(16,200,298,48,208,84,223,113)(17,201,299,49,209,85,224,114)(18,202,300,50,210,86,225,115)(28,126,190,235,44,129,261,277)(29,118,191,236,45,130,253,278)(30,119,192,237,37,131,254,279)(31,120,193,238,38,132,255,271)(32,121,194,239,39,133,256,272)(33,122,195,240,40,134,257,273)(34,123,196,241,41,135,258,274)(35,124,197,242,42,127,259,275)(36,125,198,243,43,128,260,276)(55,187,138,321,226,105,351,92)(56,188,139,322,227,106,343,93)(57,189,140,323,228,107,344,94)(58,181,141,324,229,108,345,95)(59,182,142,316,230,100,346,96)(60,183,143,317,231,101,347,97)(61,184,144,318,232,102,348,98)(62,185,136,319,233,103,349,99)(63,186,137,320,234,104,350,91)(73,245,340,170,354,174,333,147)(74,246,341,171,355,175,325,148)(75,247,342,163,356,176,326,149)(76,248,334,164,357,177,327,150)(77,249,335,165,358,178,328,151)(78,250,336,166,359,179,329,152)(79,251,337,167,360,180,330,153)(80,252,338,168,352,172,331,145)(81,244,339,169,353,173,332,146)>;

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)(325,326,327,328,329,330,331,332,333)(334,335,336,337,338,339,340,341,342)(343,344,345,346,347,348,349,350,351)(352,353,354,355,356,357,358,359,360), (1,44,171,203,189)(2,45,163,204,181)(3,37,164,205,182)(4,38,165,206,183)(5,39,166,207,184)(6,40,167,199,185)(7,41,168,200,186)(8,42,169,201,187)(9,43,170,202,188)(10,341,235,287,57)(11,342,236,288,58)(12,334,237,280,59)(13,335,238,281,60)(14,336,239,282,61)(15,337,240,283,62)(16,338,241,284,63)(17,339,242,285,55)(18,340,243,286,56)(19,233,216,330,273)(20,234,208,331,274)(21,226,209,332,275)(22,227,210,333,276)(23,228,211,325,277)(24,229,212,326,278)(25,230,213,327,279)(26,231,214,328,271)(27,232,215,329,272)(28,148,87,107,307)(29,149,88,108,308)(30,150,89,100,309)(31,151,90,101,310)(32,152,82,102,311)(33,153,83,103,312)(34,145,84,104,313)(35,146,85,105,314)(36,147,86,106,315)(46,318,154,256,179)(47,319,155,257,180)(48,320,156,258,172)(49,321,157,259,173)(50,322,158,260,174)(51,323,159,261,175)(52,324,160,253,176)(53,316,161,254,177)(54,317,162,255,178)(64,347,220,77,120)(65,348,221,78,121)(66,349,222,79,122)(67,350,223,80,123)(68,351,224,81,124)(69,343,225,73,125)(70,344,217,74,126)(71,345,218,75,118)(72,346,219,76,119)(91,269,196,252,113)(92,270,197,244,114)(93,262,198,245,115)(94,263,190,246,116)(95,264,191,247,117)(96,265,192,248,109)(97,266,193,249,110)(98,267,194,250,111)(99,268,195,251,112)(127,297,138,299,353)(128,289,139,300,354)(129,290,140,301,355)(130,291,141,302,356)(131,292,142,303,357)(132,293,143,304,358)(133,294,144,305,359)(134,295,136,306,360)(135,296,137,298,352), (1,290,159,23,307,70,263,287)(2,291,160,24,308,71,264,288)(3,292,161,25,309,72,265,280)(4,293,162,26,310,64,266,281)(5,294,154,27,311,65,267,282)(6,295,155,19,312,66,268,283)(7,296,156,20,313,67,269,284)(8,297,157,21,314,68,270,285)(9,289,158,22,315,69,262,286)(10,203,301,51,211,87,217,116)(11,204,302,52,212,88,218,117)(12,205,303,53,213,89,219,109)(13,206,304,54,214,90,220,110)(14,207,305,46,215,82,221,111)(15,199,306,47,216,83,222,112)(16,200,298,48,208,84,223,113)(17,201,299,49,209,85,224,114)(18,202,300,50,210,86,225,115)(28,126,190,235,44,129,261,277)(29,118,191,236,45,130,253,278)(30,119,192,237,37,131,254,279)(31,120,193,238,38,132,255,271)(32,121,194,239,39,133,256,272)(33,122,195,240,40,134,257,273)(34,123,196,241,41,135,258,274)(35,124,197,242,42,127,259,275)(36,125,198,243,43,128,260,276)(55,187,138,321,226,105,351,92)(56,188,139,322,227,106,343,93)(57,189,140,323,228,107,344,94)(58,181,141,324,229,108,345,95)(59,182,142,316,230,100,346,96)(60,183,143,317,231,101,347,97)(61,184,144,318,232,102,348,98)(62,185,136,319,233,103,349,99)(63,186,137,320,234,104,350,91)(73,245,340,170,354,174,333,147)(74,246,341,171,355,175,325,148)(75,247,342,163,356,176,326,149)(76,248,334,164,357,177,327,150)(77,249,335,165,358,178,328,151)(78,250,336,166,359,179,329,152)(79,251,337,167,360,180,330,153)(80,252,338,168,352,172,331,145)(81,244,339,169,353,173,332,146) );

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),(325,326,327,328,329,330,331,332,333),(334,335,336,337,338,339,340,341,342),(343,344,345,346,347,348,349,350,351),(352,353,354,355,356,357,358,359,360)], [(1,44,171,203,189),(2,45,163,204,181),(3,37,164,205,182),(4,38,165,206,183),(5,39,166,207,184),(6,40,167,199,185),(7,41,168,200,186),(8,42,169,201,187),(9,43,170,202,188),(10,341,235,287,57),(11,342,236,288,58),(12,334,237,280,59),(13,335,238,281,60),(14,336,239,282,61),(15,337,240,283,62),(16,338,241,284,63),(17,339,242,285,55),(18,340,243,286,56),(19,233,216,330,273),(20,234,208,331,274),(21,226,209,332,275),(22,227,210,333,276),(23,228,211,325,277),(24,229,212,326,278),(25,230,213,327,279),(26,231,214,328,271),(27,232,215,329,272),(28,148,87,107,307),(29,149,88,108,308),(30,150,89,100,309),(31,151,90,101,310),(32,152,82,102,311),(33,153,83,103,312),(34,145,84,104,313),(35,146,85,105,314),(36,147,86,106,315),(46,318,154,256,179),(47,319,155,257,180),(48,320,156,258,172),(49,321,157,259,173),(50,322,158,260,174),(51,323,159,261,175),(52,324,160,253,176),(53,316,161,254,177),(54,317,162,255,178),(64,347,220,77,120),(65,348,221,78,121),(66,349,222,79,122),(67,350,223,80,123),(68,351,224,81,124),(69,343,225,73,125),(70,344,217,74,126),(71,345,218,75,118),(72,346,219,76,119),(91,269,196,252,113),(92,270,197,244,114),(93,262,198,245,115),(94,263,190,246,116),(95,264,191,247,117),(96,265,192,248,109),(97,266,193,249,110),(98,267,194,250,111),(99,268,195,251,112),(127,297,138,299,353),(128,289,139,300,354),(129,290,140,301,355),(130,291,141,302,356),(131,292,142,303,357),(132,293,143,304,358),(133,294,144,305,359),(134,295,136,306,360),(135,296,137,298,352)], [(1,290,159,23,307,70,263,287),(2,291,160,24,308,71,264,288),(3,292,161,25,309,72,265,280),(4,293,162,26,310,64,266,281),(5,294,154,27,311,65,267,282),(6,295,155,19,312,66,268,283),(7,296,156,20,313,67,269,284),(8,297,157,21,314,68,270,285),(9,289,158,22,315,69,262,286),(10,203,301,51,211,87,217,116),(11,204,302,52,212,88,218,117),(12,205,303,53,213,89,219,109),(13,206,304,54,214,90,220,110),(14,207,305,46,215,82,221,111),(15,199,306,47,216,83,222,112),(16,200,298,48,208,84,223,113),(17,201,299,49,209,85,224,114),(18,202,300,50,210,86,225,115),(28,126,190,235,44,129,261,277),(29,118,191,236,45,130,253,278),(30,119,192,237,37,131,254,279),(31,120,193,238,38,132,255,271),(32,121,194,239,39,133,256,272),(33,122,195,240,40,134,257,273),(34,123,196,241,41,135,258,274),(35,124,197,242,42,127,259,275),(36,125,198,243,43,128,260,276),(55,187,138,321,226,105,351,92),(56,188,139,322,227,106,343,93),(57,189,140,323,228,107,344,94),(58,181,141,324,229,108,345,95),(59,182,142,316,230,100,346,96),(60,183,143,317,231,101,347,97),(61,184,144,318,232,102,348,98),(62,185,136,319,233,103,349,99),(63,186,137,320,234,104,350,91),(73,245,340,170,354,174,333,147),(74,246,341,171,355,175,325,148),(75,247,342,163,356,176,326,149),(76,248,334,164,357,177,327,150),(77,249,335,165,358,178,328,151),(78,250,336,166,359,179,329,152),(79,251,337,167,360,180,330,153),(80,252,338,168,352,172,331,145),(81,244,339,169,353,173,332,146)]])

144 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A 5B 6A 6B 8A 8B 8C 8D 9A ··· 9F 10A 10B 12A 12B 12C 12D 15A 15B 15C 15D 18A ··· 18F 20A 20B 20C 20D 24A ··· 24H 30A 30B 30C 30D 36A ··· 36L 45A ··· 45L 60A ··· 60H 72A ··· 72X 90A ··· 90L 180A ··· 180X order 1 2 3 3 4 4 5 5 6 6 8 8 8 8 9 ··· 9 10 10 12 12 12 12 15 15 15 15 18 ··· 18 20 20 20 20 24 ··· 24 30 30 30 30 36 ··· 36 45 ··· 45 60 ··· 60 72 ··· 72 90 ··· 90 180 ··· 180 size 1 1 1 1 1 1 2 2 1 1 5 5 5 5 1 ··· 1 2 2 1 1 1 1 2 2 2 2 1 ··· 1 2 2 2 2 5 ··· 5 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 5 ··· 5 2 ··· 2 2 ··· 2

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C8 C9 C12 C18 C24 C36 C72 D5 Dic5 C3×D5 C5⋊2C8 C3×Dic5 C9×D5 C3×C5⋊2C8 C9×Dic5 C9×C5⋊2C8 kernel C9×C5⋊2C8 C180 C3×C5⋊2C8 C90 C60 C45 C5⋊2C8 C30 C20 C15 C10 C5 C36 C18 C12 C9 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 6 4 6 8 12 24 2 2 4 4 4 12 8 12 24

Matrix representation of C9×C52C8 in GL2(𝔽1801) generated by

 888 0 0 888
,
 1374 1800 1 0
,
 102 1557 1227 1699
G:=sub<GL(2,GF(1801))| [888,0,0,888],[1374,1,1800,0],[102,1227,1557,1699] >;

C9×C52C8 in GAP, Magma, Sage, TeX

C_9\times C_5\rtimes_2C_8
% in TeX

G:=Group("C9xC5:2C8");
// GroupNames label

G:=SmallGroup(360,2);
// by ID

G=gap.SmallGroup(360,2);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-2,-5,36,79,122,10373]);
// Polycyclic

G:=Group<a,b,c|a^9=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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