direct product, non-abelian, soluble
Aliases: C5×Q8⋊C9, Q8⋊C45, C30.1A4, C15.SL2(𝔽3), (C5×Q8)⋊C9, C6.1(C5×A4), (C3×Q8).C15, (Q8×C15).C3, C10.(C3.A4), C3.(C5×SL2(𝔽3)), C2.(C5×C3.A4), SmallGroup(360,14)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C5×Q8⋊C9 |
Generators and relations for C5×Q8⋊C9
G = < a,b,c,d | a5=b4=d9=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >
Smallest permutation representation of C5×Q8⋊C9
►Regular action on 360 points
Generators in S360
(1 208 282 356 301)(2 209 283 357 302)(3 210 284 358 303)(4 211 285 359 304)(5 212 286 360 305)(6 213 287 352 306)(7 214 288 353 298)(8 215 280 354 299)(9 216 281 355 300)(10 198 270 335 290)(11 190 262 336 291)(12 191 263 337 292)(13 192 264 338 293)(14 193 265 339 294)(15 194 266 340 295)(16 195 267 341 296)(17 196 268 342 297)(18 197 269 334 289)(19 332 168 241 314)(20 333 169 242 315)(21 325 170 243 307)(22 326 171 235 308)(23 327 163 236 309)(24 328 164 237 310)(25 329 165 238 311)(26 330 166 239 312)(27 331 167 240 313)(28 102 72 144 82)(29 103 64 136 83)(30 104 65 137 84)(31 105 66 138 85)(32 106 67 139 86)(33 107 68 140 87)(34 108 69 141 88)(35 100 70 142 89)(36 101 71 143 90)(37 111 176 248 91)(38 112 177 249 92)(39 113 178 250 93)(40 114 179 251 94)(41 115 180 252 95)(42 116 172 244 96)(43 117 173 245 97)(44 109 174 246 98)(45 110 175 247 99)(46 119 323 148 222)(47 120 324 149 223)(48 121 316 150 224)(49 122 317 151 225)(50 123 318 152 217)(51 124 319 153 218)(52 125 320 145 219)(53 126 321 146 220)(54 118 322 147 221)(55 128 75 278 343)(56 129 76 279 344)(57 130 77 271 345)(58 131 78 272 346)(59 132 79 273 347)(60 133 80 274 348)(61 134 81 275 349)(62 135 73 276 350)(63 127 74 277 351)(154 228 181 253 200)(155 229 182 254 201)(156 230 183 255 202)(157 231 184 256 203)(158 232 185 257 204)(159 233 186 258 205)(160 234 187 259 206)(161 226 188 260 207)(162 227 189 261 199)
(1 173 27 184)(2 336 19 221)(3 345 20 69)(4 176 21 187)(5 339 22 224)(6 348 23 72)(7 179 24 181)(8 342 25 218)(9 351 26 66)(10 86 126 128)(11 168 118 283)(12 99 119 205)(13 89 120 131)(14 171 121 286)(15 93 122 199)(16 83 123 134)(17 165 124 280)(18 96 125 202)(28 352 80 236)(29 318 81 195)(30 41 73 155)(31 355 74 239)(32 321 75 198)(33 44 76 158)(34 358 77 242)(35 324 78 192)(36 38 79 161)(37 243 160 359)(39 317 162 194)(40 237 154 353)(42 320 156 197)(43 240 157 356)(45 323 159 191)(46 258 292 247)(47 58 293 142)(48 212 294 326)(49 261 295 250)(50 61 296 136)(51 215 297 329)(52 255 289 244)(53 55 290 139)(54 209 291 332)(56 257 140 246)(57 333 141 210)(59 260 143 249)(60 327 144 213)(62 254 137 252)(63 330 138 216)(64 217 349 341)(65 180 350 182)(67 220 343 335)(68 174 344 185)(70 223 346 338)(71 177 347 188)(82 287 133 163)(84 95 135 201)(85 281 127 166)(87 98 129 204)(88 284 130 169)(90 92 132 207)(91 170 206 285)(94 164 200 288)(97 167 203 282)(100 149 272 264)(101 112 273 226)(102 306 274 309)(103 152 275 267)(104 115 276 229)(105 300 277 312)(106 146 278 270)(107 109 279 232)(108 303 271 315)(110 148 233 263)(111 307 234 304)(113 151 227 266)(114 310 228 298)(116 145 230 269)(117 313 231 301)(147 302 262 314)(150 305 265 308)(153 299 268 311)(172 219 183 334)(175 222 186 337)(178 225 189 340)(190 241 322 357)(193 235 316 360)(196 238 319 354)(208 245 331 256)(211 248 325 259)(214 251 328 253)
(1 335 27 220)(2 344 19 68)(3 175 20 186)(4 338 21 223)(5 347 22 71)(6 178 23 189)(7 341 24 217)(8 350 25 65)(9 172 26 183)(10 167 126 282)(11 98 118 204)(12 88 119 130)(13 170 120 285)(14 92 121 207)(15 82 122 133)(16 164 123 288)(17 95 124 201)(18 85 125 127)(28 317 80 194)(29 40 81 154)(30 354 73 238)(31 320 74 197)(32 43 75 157)(33 357 76 241)(34 323 77 191)(35 37 78 160)(36 360 79 235)(38 316 161 193)(39 236 162 352)(41 319 155 196)(42 239 156 355)(44 322 158 190)(45 242 159 358)(46 57 292 141)(47 211 293 325)(48 260 294 249)(49 60 295 144)(50 214 296 328)(51 254 297 252)(52 63 289 138)(53 208 290 331)(54 257 291 246)(55 256 139 245)(56 332 140 209)(58 259 142 248)(59 326 143 212)(61 253 136 251)(62 329 137 215)(64 179 349 181)(66 219 351 334)(67 173 343 184)(69 222 345 337)(70 176 346 187)(72 225 348 340)(83 94 134 200)(84 280 135 165)(86 97 128 203)(87 283 129 168)(89 91 131 206)(90 286 132 171)(93 163 199 287)(96 166 202 281)(99 169 205 284)(100 111 272 234)(101 305 273 308)(102 151 274 266)(103 114 275 228)(104 299 276 311)(105 145 277 269)(106 117 278 231)(107 302 279 314)(108 148 271 263)(109 147 232 262)(110 315 233 303)(112 150 226 265)(113 309 227 306)(115 153 229 268)(116 312 230 300)(146 301 270 313)(149 304 264 307)(152 298 267 310)(174 221 185 336)(177 224 188 339)(180 218 182 342)(192 243 324 359)(195 237 318 353)(198 240 321 356)(210 247 333 258)(213 250 327 261)(216 244 330 255)
(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)
G:=sub<Sym(360)| (1,208,282,356,301)(2,209,283,357,302)(3,210,284,358,303)(4,211,285,359,304)(5,212,286,360,305)(6,213,287,352,306)(7,214,288,353,298)(8,215,280,354,299)(9,216,281,355,300)(10,198,270,335,290)(11,190,262,336,291)(12,191,263,337,292)(13,192,264,338,293)(14,193,265,339,294)(15,194,266,340,295)(16,195,267,341,296)(17,196,268,342,297)(18,197,269,334,289)(19,332,168,241,314)(20,333,169,242,315)(21,325,170,243,307)(22,326,171,235,308)(23,327,163,236,309)(24,328,164,237,310)(25,329,165,238,311)(26,330,166,239,312)(27,331,167,240,313)(28,102,72,144,82)(29,103,64,136,83)(30,104,65,137,84)(31,105,66,138,85)(32,106,67,139,86)(33,107,68,140,87)(34,108,69,141,88)(35,100,70,142,89)(36,101,71,143,90)(37,111,176,248,91)(38,112,177,249,92)(39,113,178,250,93)(40,114,179,251,94)(41,115,180,252,95)(42,116,172,244,96)(43,117,173,245,97)(44,109,174,246,98)(45,110,175,247,99)(46,119,323,148,222)(47,120,324,149,223)(48,121,316,150,224)(49,122,317,151,225)(50,123,318,152,217)(51,124,319,153,218)(52,125,320,145,219)(53,126,321,146,220)(54,118,322,147,221)(55,128,75,278,343)(56,129,76,279,344)(57,130,77,271,345)(58,131,78,272,346)(59,132,79,273,347)(60,133,80,274,348)(61,134,81,275,349)(62,135,73,276,350)(63,127,74,277,351)(154,228,181,253,200)(155,229,182,254,201)(156,230,183,255,202)(157,231,184,256,203)(158,232,185,257,204)(159,233,186,258,205)(160,234,187,259,206)(161,226,188,260,207)(162,227,189,261,199), (1,173,27,184)(2,336,19,221)(3,345,20,69)(4,176,21,187)(5,339,22,224)(6,348,23,72)(7,179,24,181)(8,342,25,218)(9,351,26,66)(10,86,126,128)(11,168,118,283)(12,99,119,205)(13,89,120,131)(14,171,121,286)(15,93,122,199)(16,83,123,134)(17,165,124,280)(18,96,125,202)(28,352,80,236)(29,318,81,195)(30,41,73,155)(31,355,74,239)(32,321,75,198)(33,44,76,158)(34,358,77,242)(35,324,78,192)(36,38,79,161)(37,243,160,359)(39,317,162,194)(40,237,154,353)(42,320,156,197)(43,240,157,356)(45,323,159,191)(46,258,292,247)(47,58,293,142)(48,212,294,326)(49,261,295,250)(50,61,296,136)(51,215,297,329)(52,255,289,244)(53,55,290,139)(54,209,291,332)(56,257,140,246)(57,333,141,210)(59,260,143,249)(60,327,144,213)(62,254,137,252)(63,330,138,216)(64,217,349,341)(65,180,350,182)(67,220,343,335)(68,174,344,185)(70,223,346,338)(71,177,347,188)(82,287,133,163)(84,95,135,201)(85,281,127,166)(87,98,129,204)(88,284,130,169)(90,92,132,207)(91,170,206,285)(94,164,200,288)(97,167,203,282)(100,149,272,264)(101,112,273,226)(102,306,274,309)(103,152,275,267)(104,115,276,229)(105,300,277,312)(106,146,278,270)(107,109,279,232)(108,303,271,315)(110,148,233,263)(111,307,234,304)(113,151,227,266)(114,310,228,298)(116,145,230,269)(117,313,231,301)(147,302,262,314)(150,305,265,308)(153,299,268,311)(172,219,183,334)(175,222,186,337)(178,225,189,340)(190,241,322,357)(193,235,316,360)(196,238,319,354)(208,245,331,256)(211,248,325,259)(214,251,328,253), (1,335,27,220)(2,344,19,68)(3,175,20,186)(4,338,21,223)(5,347,22,71)(6,178,23,189)(7,341,24,217)(8,350,25,65)(9,172,26,183)(10,167,126,282)(11,98,118,204)(12,88,119,130)(13,170,120,285)(14,92,121,207)(15,82,122,133)(16,164,123,288)(17,95,124,201)(18,85,125,127)(28,317,80,194)(29,40,81,154)(30,354,73,238)(31,320,74,197)(32,43,75,157)(33,357,76,241)(34,323,77,191)(35,37,78,160)(36,360,79,235)(38,316,161,193)(39,236,162,352)(41,319,155,196)(42,239,156,355)(44,322,158,190)(45,242,159,358)(46,57,292,141)(47,211,293,325)(48,260,294,249)(49,60,295,144)(50,214,296,328)(51,254,297,252)(52,63,289,138)(53,208,290,331)(54,257,291,246)(55,256,139,245)(56,332,140,209)(58,259,142,248)(59,326,143,212)(61,253,136,251)(62,329,137,215)(64,179,349,181)(66,219,351,334)(67,173,343,184)(69,222,345,337)(70,176,346,187)(72,225,348,340)(83,94,134,200)(84,280,135,165)(86,97,128,203)(87,283,129,168)(89,91,131,206)(90,286,132,171)(93,163,199,287)(96,166,202,281)(99,169,205,284)(100,111,272,234)(101,305,273,308)(102,151,274,266)(103,114,275,228)(104,299,276,311)(105,145,277,269)(106,117,278,231)(107,302,279,314)(108,148,271,263)(109,147,232,262)(110,315,233,303)(112,150,226,265)(113,309,227,306)(115,153,229,268)(116,312,230,300)(146,301,270,313)(149,304,264,307)(152,298,267,310)(174,221,185,336)(177,224,188,339)(180,218,182,342)(192,243,324,359)(195,237,318,353)(198,240,321,356)(210,247,333,258)(213,250,327,261)(216,244,330,255), (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)>;
G:=Group( (1,208,282,356,301)(2,209,283,357,302)(3,210,284,358,303)(4,211,285,359,304)(5,212,286,360,305)(6,213,287,352,306)(7,214,288,353,298)(8,215,280,354,299)(9,216,281,355,300)(10,198,270,335,290)(11,190,262,336,291)(12,191,263,337,292)(13,192,264,338,293)(14,193,265,339,294)(15,194,266,340,295)(16,195,267,341,296)(17,196,268,342,297)(18,197,269,334,289)(19,332,168,241,314)(20,333,169,242,315)(21,325,170,243,307)(22,326,171,235,308)(23,327,163,236,309)(24,328,164,237,310)(25,329,165,238,311)(26,330,166,239,312)(27,331,167,240,313)(28,102,72,144,82)(29,103,64,136,83)(30,104,65,137,84)(31,105,66,138,85)(32,106,67,139,86)(33,107,68,140,87)(34,108,69,141,88)(35,100,70,142,89)(36,101,71,143,90)(37,111,176,248,91)(38,112,177,249,92)(39,113,178,250,93)(40,114,179,251,94)(41,115,180,252,95)(42,116,172,244,96)(43,117,173,245,97)(44,109,174,246,98)(45,110,175,247,99)(46,119,323,148,222)(47,120,324,149,223)(48,121,316,150,224)(49,122,317,151,225)(50,123,318,152,217)(51,124,319,153,218)(52,125,320,145,219)(53,126,321,146,220)(54,118,322,147,221)(55,128,75,278,343)(56,129,76,279,344)(57,130,77,271,345)(58,131,78,272,346)(59,132,79,273,347)(60,133,80,274,348)(61,134,81,275,349)(62,135,73,276,350)(63,127,74,277,351)(154,228,181,253,200)(155,229,182,254,201)(156,230,183,255,202)(157,231,184,256,203)(158,232,185,257,204)(159,233,186,258,205)(160,234,187,259,206)(161,226,188,260,207)(162,227,189,261,199), (1,173,27,184)(2,336,19,221)(3,345,20,69)(4,176,21,187)(5,339,22,224)(6,348,23,72)(7,179,24,181)(8,342,25,218)(9,351,26,66)(10,86,126,128)(11,168,118,283)(12,99,119,205)(13,89,120,131)(14,171,121,286)(15,93,122,199)(16,83,123,134)(17,165,124,280)(18,96,125,202)(28,352,80,236)(29,318,81,195)(30,41,73,155)(31,355,74,239)(32,321,75,198)(33,44,76,158)(34,358,77,242)(35,324,78,192)(36,38,79,161)(37,243,160,359)(39,317,162,194)(40,237,154,353)(42,320,156,197)(43,240,157,356)(45,323,159,191)(46,258,292,247)(47,58,293,142)(48,212,294,326)(49,261,295,250)(50,61,296,136)(51,215,297,329)(52,255,289,244)(53,55,290,139)(54,209,291,332)(56,257,140,246)(57,333,141,210)(59,260,143,249)(60,327,144,213)(62,254,137,252)(63,330,138,216)(64,217,349,341)(65,180,350,182)(67,220,343,335)(68,174,344,185)(70,223,346,338)(71,177,347,188)(82,287,133,163)(84,95,135,201)(85,281,127,166)(87,98,129,204)(88,284,130,169)(90,92,132,207)(91,170,206,285)(94,164,200,288)(97,167,203,282)(100,149,272,264)(101,112,273,226)(102,306,274,309)(103,152,275,267)(104,115,276,229)(105,300,277,312)(106,146,278,270)(107,109,279,232)(108,303,271,315)(110,148,233,263)(111,307,234,304)(113,151,227,266)(114,310,228,298)(116,145,230,269)(117,313,231,301)(147,302,262,314)(150,305,265,308)(153,299,268,311)(172,219,183,334)(175,222,186,337)(178,225,189,340)(190,241,322,357)(193,235,316,360)(196,238,319,354)(208,245,331,256)(211,248,325,259)(214,251,328,253), (1,335,27,220)(2,344,19,68)(3,175,20,186)(4,338,21,223)(5,347,22,71)(6,178,23,189)(7,341,24,217)(8,350,25,65)(9,172,26,183)(10,167,126,282)(11,98,118,204)(12,88,119,130)(13,170,120,285)(14,92,121,207)(15,82,122,133)(16,164,123,288)(17,95,124,201)(18,85,125,127)(28,317,80,194)(29,40,81,154)(30,354,73,238)(31,320,74,197)(32,43,75,157)(33,357,76,241)(34,323,77,191)(35,37,78,160)(36,360,79,235)(38,316,161,193)(39,236,162,352)(41,319,155,196)(42,239,156,355)(44,322,158,190)(45,242,159,358)(46,57,292,141)(47,211,293,325)(48,260,294,249)(49,60,295,144)(50,214,296,328)(51,254,297,252)(52,63,289,138)(53,208,290,331)(54,257,291,246)(55,256,139,245)(56,332,140,209)(58,259,142,248)(59,326,143,212)(61,253,136,251)(62,329,137,215)(64,179,349,181)(66,219,351,334)(67,173,343,184)(69,222,345,337)(70,176,346,187)(72,225,348,340)(83,94,134,200)(84,280,135,165)(86,97,128,203)(87,283,129,168)(89,91,131,206)(90,286,132,171)(93,163,199,287)(96,166,202,281)(99,169,205,284)(100,111,272,234)(101,305,273,308)(102,151,274,266)(103,114,275,228)(104,299,276,311)(105,145,277,269)(106,117,278,231)(107,302,279,314)(108,148,271,263)(109,147,232,262)(110,315,233,303)(112,150,226,265)(113,309,227,306)(115,153,229,268)(116,312,230,300)(146,301,270,313)(149,304,264,307)(152,298,267,310)(174,221,185,336)(177,224,188,339)(180,218,182,342)(192,243,324,359)(195,237,318,353)(198,240,321,356)(210,247,333,258)(213,250,327,261)(216,244,330,255), (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) );
G=PermutationGroup([[(1,208,282,356,301),(2,209,283,357,302),(3,210,284,358,303),(4,211,285,359,304),(5,212,286,360,305),(6,213,287,352,306),(7,214,288,353,298),(8,215,280,354,299),(9,216,281,355,300),(10,198,270,335,290),(11,190,262,336,291),(12,191,263,337,292),(13,192,264,338,293),(14,193,265,339,294),(15,194,266,340,295),(16,195,267,341,296),(17,196,268,342,297),(18,197,269,334,289),(19,332,168,241,314),(20,333,169,242,315),(21,325,170,243,307),(22,326,171,235,308),(23,327,163,236,309),(24,328,164,237,310),(25,329,165,238,311),(26,330,166,239,312),(27,331,167,240,313),(28,102,72,144,82),(29,103,64,136,83),(30,104,65,137,84),(31,105,66,138,85),(32,106,67,139,86),(33,107,68,140,87),(34,108,69,141,88),(35,100,70,142,89),(36,101,71,143,90),(37,111,176,248,91),(38,112,177,249,92),(39,113,178,250,93),(40,114,179,251,94),(41,115,180,252,95),(42,116,172,244,96),(43,117,173,245,97),(44,109,174,246,98),(45,110,175,247,99),(46,119,323,148,222),(47,120,324,149,223),(48,121,316,150,224),(49,122,317,151,225),(50,123,318,152,217),(51,124,319,153,218),(52,125,320,145,219),(53,126,321,146,220),(54,118,322,147,221),(55,128,75,278,343),(56,129,76,279,344),(57,130,77,271,345),(58,131,78,272,346),(59,132,79,273,347),(60,133,80,274,348),(61,134,81,275,349),(62,135,73,276,350),(63,127,74,277,351),(154,228,181,253,200),(155,229,182,254,201),(156,230,183,255,202),(157,231,184,256,203),(158,232,185,257,204),(159,233,186,258,205),(160,234,187,259,206),(161,226,188,260,207),(162,227,189,261,199)], [(1,173,27,184),(2,336,19,221),(3,345,20,69),(4,176,21,187),(5,339,22,224),(6,348,23,72),(7,179,24,181),(8,342,25,218),(9,351,26,66),(10,86,126,128),(11,168,118,283),(12,99,119,205),(13,89,120,131),(14,171,121,286),(15,93,122,199),(16,83,123,134),(17,165,124,280),(18,96,125,202),(28,352,80,236),(29,318,81,195),(30,41,73,155),(31,355,74,239),(32,321,75,198),(33,44,76,158),(34,358,77,242),(35,324,78,192),(36,38,79,161),(37,243,160,359),(39,317,162,194),(40,237,154,353),(42,320,156,197),(43,240,157,356),(45,323,159,191),(46,258,292,247),(47,58,293,142),(48,212,294,326),(49,261,295,250),(50,61,296,136),(51,215,297,329),(52,255,289,244),(53,55,290,139),(54,209,291,332),(56,257,140,246),(57,333,141,210),(59,260,143,249),(60,327,144,213),(62,254,137,252),(63,330,138,216),(64,217,349,341),(65,180,350,182),(67,220,343,335),(68,174,344,185),(70,223,346,338),(71,177,347,188),(82,287,133,163),(84,95,135,201),(85,281,127,166),(87,98,129,204),(88,284,130,169),(90,92,132,207),(91,170,206,285),(94,164,200,288),(97,167,203,282),(100,149,272,264),(101,112,273,226),(102,306,274,309),(103,152,275,267),(104,115,276,229),(105,300,277,312),(106,146,278,270),(107,109,279,232),(108,303,271,315),(110,148,233,263),(111,307,234,304),(113,151,227,266),(114,310,228,298),(116,145,230,269),(117,313,231,301),(147,302,262,314),(150,305,265,308),(153,299,268,311),(172,219,183,334),(175,222,186,337),(178,225,189,340),(190,241,322,357),(193,235,316,360),(196,238,319,354),(208,245,331,256),(211,248,325,259),(214,251,328,253)], [(1,335,27,220),(2,344,19,68),(3,175,20,186),(4,338,21,223),(5,347,22,71),(6,178,23,189),(7,341,24,217),(8,350,25,65),(9,172,26,183),(10,167,126,282),(11,98,118,204),(12,88,119,130),(13,170,120,285),(14,92,121,207),(15,82,122,133),(16,164,123,288),(17,95,124,201),(18,85,125,127),(28,317,80,194),(29,40,81,154),(30,354,73,238),(31,320,74,197),(32,43,75,157),(33,357,76,241),(34,323,77,191),(35,37,78,160),(36,360,79,235),(38,316,161,193),(39,236,162,352),(41,319,155,196),(42,239,156,355),(44,322,158,190),(45,242,159,358),(46,57,292,141),(47,211,293,325),(48,260,294,249),(49,60,295,144),(50,214,296,328),(51,254,297,252),(52,63,289,138),(53,208,290,331),(54,257,291,246),(55,256,139,245),(56,332,140,209),(58,259,142,248),(59,326,143,212),(61,253,136,251),(62,329,137,215),(64,179,349,181),(66,219,351,334),(67,173,343,184),(69,222,345,337),(70,176,346,187),(72,225,348,340),(83,94,134,200),(84,280,135,165),(86,97,128,203),(87,283,129,168),(89,91,131,206),(90,286,132,171),(93,163,199,287),(96,166,202,281),(99,169,205,284),(100,111,272,234),(101,305,273,308),(102,151,274,266),(103,114,275,228),(104,299,276,311),(105,145,277,269),(106,117,278,231),(107,302,279,314),(108,148,271,263),(109,147,232,262),(110,315,233,303),(112,150,226,265),(113,309,227,306),(115,153,229,268),(116,312,230,300),(146,301,270,313),(149,304,264,307),(152,298,267,310),(174,221,185,336),(177,224,188,339),(180,218,182,342),(192,243,324,359),(195,237,318,353),(198,240,321,356),(210,247,333,258),(213,250,327,261),(216,244,330,255)], [(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)]])
105 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4 | 5A | 5B | 5C | 5D | 6A | 6B | 9A | ··· | 9F | 10A | 10B | 10C | 10D | 12A | 12B | 15A | ··· | 15H | 18A | ··· | 18F | 20A | 20B | 20C | 20D | 30A | ··· | 30H | 45A | ··· | 45X | 60A | ··· | 60H | 90A | ··· | 90X |
| order | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 9 | ··· | 9 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | ··· | 15 | 18 | ··· | 18 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 45 | ··· | 45 | 60 | ··· | 60 | 90 | ··· | 90 |
| size | 1 | 1 | 1 | 1 | 6 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | - | + | ||||||||||||
| image | C1 | C3 | C5 | C9 | C15 | C45 | SL2(𝔽3) | SL2(𝔽3) | Q8⋊C9 | C5×SL2(𝔽3) | C5×Q8⋊C9 | A4 | C3.A4 | C5×A4 | C5×C3.A4 |
| kernel | C5×Q8⋊C9 | Q8×C15 | Q8⋊C9 | C5×Q8 | C3×Q8 | Q8 | C15 | C15 | C5 | C3 | C1 | C30 | C10 | C6 | C2 |
| # reps | 1 | 2 | 4 | 6 | 8 | 24 | 1 | 2 | 6 | 12 | 24 | 1 | 2 | 4 | 8 |
Matrix representation of C5×Q8⋊C9 ►in GL3(𝔽181) generated by
| 42 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 113 | 172 |
| 0 | 172 | 68 |
| 1 | 0 | 0 |
| 0 | 0 | 180 |
| 0 | 1 | 0 |
| 132 | 0 | 0 |
| 0 | 45 | 73 |
| 0 | 34 | 97 |
G:=sub<GL(3,GF(181))| [42,0,0,0,1,0,0,0,1],[1,0,0,0,113,172,0,172,68],[1,0,0,0,0,1,0,180,0],[132,0,0,0,45,34,0,73,97] >;
C5×Q8⋊C9 in GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes C_9
% in TeX
G:=Group("C5xQ8:C9"); // GroupNames label
G:=SmallGroup(360,14);
// by ID
G=gap.SmallGroup(360,14);
# by ID
G:=PCGroup([6,-3,-5,-3,-2,2,-2,90,2163,117,4054,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=d^9=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations
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