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G = C5×Q8⋊C9order 360 = 23·32·5

Direct product of C5 and Q8⋊C9

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

C1C2Q8 — C5×Q8⋊C9
C1C2Q8C3×Q8Q8×C15 — C5×Q8⋊C9
Q8 — C5×Q8⋊C9
C1C30

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 >

3C4
4C9
3C12
4C18
3C20
4C45
3C60
4C90

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 78 160 234 170)(11 79 161 226 171)(12 80 162 227 163)(13 81 154 228 164)(14 73 155 229 165)(15 74 156 230 166)(16 75 157 231 167)(17 76 158 232 168)(18 77 159 233 169)(19 332 56 129 314)(20 333 57 130 315)(21 325 58 131 307)(22 326 59 132 308)(23 327 60 133 309)(24 328 61 134 310)(25 329 62 135 311)(26 330 63 127 312)(27 331 55 128 313)(28 102 72 239 194)(29 103 64 240 195)(30 104 65 241 196)(31 105 66 242 197)(32 106 67 243 198)(33 107 68 235 190)(34 108 69 236 191)(35 100 70 237 192)(36 101 71 238 193)(37 111 47 120 203)(38 112 48 121 204)(39 113 49 122 205)(40 114 50 123 206)(41 115 51 124 207)(42 116 52 125 199)(43 117 53 126 200)(44 109 54 118 201)(45 110 46 119 202)(82 266 340 174 247)(83 267 341 175 248)(84 268 342 176 249)(85 269 334 177 250)(86 270 335 178 251)(87 262 336 179 252)(88 263 337 180 244)(89 264 338 172 245)(90 265 339 173 246)(91 149 223 297 136)(92 150 224 289 137)(93 151 225 290 138)(94 152 217 291 139)(95 153 218 292 140)(96 145 219 293 141)(97 146 220 294 142)(98 147 221 295 143)(99 148 222 296 144)(181 253 321 275 349)(182 254 322 276 350)(183 255 323 277 351)(184 256 324 278 343)(185 257 316 279 344)(186 258 317 271 345)(187 259 318 272 346)(188 260 319 273 347)(189 261 320 274 348)
(1 294 27 184)(2 336 19 109)(3 233 20 69)(4 297 21 187)(5 339 22 112)(6 227 23 72)(7 291 24 181)(8 342 25 115)(9 230 26 66)(10 245 192 120)(11 319 193 92)(12 60 194 287)(13 248 195 123)(14 322 196 95)(15 63 197 281)(16 251 198 126)(17 316 190 98)(18 57 191 284)(28 352 80 133)(29 206 81 83)(30 153 73 276)(31 355 74 127)(32 200 75 86)(33 147 76 279)(34 358 77 130)(35 203 78 89)(36 150 79 273)(37 160 264 100)(38 305 265 308)(39 348 266 225)(40 154 267 103)(41 299 268 311)(42 351 269 219)(43 157 270 106)(44 302 262 314)(45 345 263 222)(46 258 180 144)(47 170 172 237)(48 212 173 326)(49 261 174 138)(50 164 175 240)(51 215 176 329)(52 255 177 141)(53 167 178 243)(54 209 179 332)(55 324 282 97)(56 118 283 252)(58 318 285 91)(59 121 286 246)(61 321 288 94)(62 124 280 249)(64 114 228 341)(65 292 229 182)(67 117 231 335)(68 295 232 185)(70 111 234 338)(71 289 226 188)(82 151 205 274)(84 135 207 354)(85 145 199 277)(87 129 201 357)(88 148 202 271)(90 132 204 360)(93 122 320 247)(96 125 323 250)(99 119 317 244)(101 224 161 347)(102 306 162 309)(104 218 155 350)(105 300 156 312)(107 221 158 344)(108 303 159 315)(110 186 337 296)(113 189 340 290)(116 183 334 293)(128 278 356 146)(131 272 359 149)(134 275 353 152)(136 325 259 211)(137 171 260 238)(139 328 253 214)(140 165 254 241)(142 331 256 208)(143 168 257 235)(163 327 239 213)(166 330 242 216)(169 333 236 210)(217 310 349 298)(220 313 343 301)(223 307 346 304)
(1 335 27 117)(2 232 19 68)(3 296 20 186)(4 338 21 111)(5 226 22 71)(6 290 23 189)(7 341 24 114)(8 229 25 65)(9 293 26 183)(10 318 192 91)(11 59 193 286)(12 247 194 122)(13 321 195 94)(14 62 196 280)(15 250 197 125)(16 324 198 97)(17 56 190 283)(18 244 191 119)(28 205 80 82)(29 152 81 275)(30 354 73 135)(31 199 74 85)(32 146 75 278)(33 357 76 129)(34 202 77 88)(35 149 78 272)(36 360 79 132)(37 304 264 307)(38 347 265 224)(39 162 266 102)(40 298 267 310)(41 350 268 218)(42 156 269 105)(43 301 270 313)(44 344 262 221)(45 159 263 108)(46 169 180 236)(47 211 172 325)(48 260 173 137)(49 163 174 239)(50 214 175 328)(51 254 176 140)(52 166 177 242)(53 208 178 331)(54 257 179 143)(55 126 282 251)(57 317 284 99)(58 120 285 245)(60 320 287 93)(61 123 288 248)(63 323 281 96)(64 291 228 181)(66 116 230 334)(67 294 231 184)(69 110 233 337)(70 297 234 187)(72 113 227 340)(83 134 206 353)(84 153 207 276)(86 128 200 356)(87 147 201 279)(89 131 203 359)(90 150 204 273)(92 121 319 246)(95 124 322 249)(98 118 316 252)(100 223 160 346)(101 305 161 308)(103 217 154 349)(104 299 155 311)(106 220 157 343)(107 302 158 314)(109 185 336 295)(112 188 339 289)(115 182 342 292)(127 277 355 145)(130 271 358 148)(133 274 352 151)(136 170 259 237)(138 327 261 213)(139 164 253 240)(141 330 255 216)(142 167 256 243)(144 333 258 210)(165 329 241 215)(168 332 235 209)(171 326 238 212)(219 312 351 300)(222 315 345 303)(225 309 348 306)
(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)(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)(10,78,160,234,170)(11,79,161,226,171)(12,80,162,227,163)(13,81,154,228,164)(14,73,155,229,165)(15,74,156,230,166)(16,75,157,231,167)(17,76,158,232,168)(18,77,159,233,169)(19,332,56,129,314)(20,333,57,130,315)(21,325,58,131,307)(22,326,59,132,308)(23,327,60,133,309)(24,328,61,134,310)(25,329,62,135,311)(26,330,63,127,312)(27,331,55,128,313)(28,102,72,239,194)(29,103,64,240,195)(30,104,65,241,196)(31,105,66,242,197)(32,106,67,243,198)(33,107,68,235,190)(34,108,69,236,191)(35,100,70,237,192)(36,101,71,238,193)(37,111,47,120,203)(38,112,48,121,204)(39,113,49,122,205)(40,114,50,123,206)(41,115,51,124,207)(42,116,52,125,199)(43,117,53,126,200)(44,109,54,118,201)(45,110,46,119,202)(82,266,340,174,247)(83,267,341,175,248)(84,268,342,176,249)(85,269,334,177,250)(86,270,335,178,251)(87,262,336,179,252)(88,263,337,180,244)(89,264,338,172,245)(90,265,339,173,246)(91,149,223,297,136)(92,150,224,289,137)(93,151,225,290,138)(94,152,217,291,139)(95,153,218,292,140)(96,145,219,293,141)(97,146,220,294,142)(98,147,221,295,143)(99,148,222,296,144)(181,253,321,275,349)(182,254,322,276,350)(183,255,323,277,351)(184,256,324,278,343)(185,257,316,279,344)(186,258,317,271,345)(187,259,318,272,346)(188,260,319,273,347)(189,261,320,274,348), (1,294,27,184)(2,336,19,109)(3,233,20,69)(4,297,21,187)(5,339,22,112)(6,227,23,72)(7,291,24,181)(8,342,25,115)(9,230,26,66)(10,245,192,120)(11,319,193,92)(12,60,194,287)(13,248,195,123)(14,322,196,95)(15,63,197,281)(16,251,198,126)(17,316,190,98)(18,57,191,284)(28,352,80,133)(29,206,81,83)(30,153,73,276)(31,355,74,127)(32,200,75,86)(33,147,76,279)(34,358,77,130)(35,203,78,89)(36,150,79,273)(37,160,264,100)(38,305,265,308)(39,348,266,225)(40,154,267,103)(41,299,268,311)(42,351,269,219)(43,157,270,106)(44,302,262,314)(45,345,263,222)(46,258,180,144)(47,170,172,237)(48,212,173,326)(49,261,174,138)(50,164,175,240)(51,215,176,329)(52,255,177,141)(53,167,178,243)(54,209,179,332)(55,324,282,97)(56,118,283,252)(58,318,285,91)(59,121,286,246)(61,321,288,94)(62,124,280,249)(64,114,228,341)(65,292,229,182)(67,117,231,335)(68,295,232,185)(70,111,234,338)(71,289,226,188)(82,151,205,274)(84,135,207,354)(85,145,199,277)(87,129,201,357)(88,148,202,271)(90,132,204,360)(93,122,320,247)(96,125,323,250)(99,119,317,244)(101,224,161,347)(102,306,162,309)(104,218,155,350)(105,300,156,312)(107,221,158,344)(108,303,159,315)(110,186,337,296)(113,189,340,290)(116,183,334,293)(128,278,356,146)(131,272,359,149)(134,275,353,152)(136,325,259,211)(137,171,260,238)(139,328,253,214)(140,165,254,241)(142,331,256,208)(143,168,257,235)(163,327,239,213)(166,330,242,216)(169,333,236,210)(217,310,349,298)(220,313,343,301)(223,307,346,304), (1,335,27,117)(2,232,19,68)(3,296,20,186)(4,338,21,111)(5,226,22,71)(6,290,23,189)(7,341,24,114)(8,229,25,65)(9,293,26,183)(10,318,192,91)(11,59,193,286)(12,247,194,122)(13,321,195,94)(14,62,196,280)(15,250,197,125)(16,324,198,97)(17,56,190,283)(18,244,191,119)(28,205,80,82)(29,152,81,275)(30,354,73,135)(31,199,74,85)(32,146,75,278)(33,357,76,129)(34,202,77,88)(35,149,78,272)(36,360,79,132)(37,304,264,307)(38,347,265,224)(39,162,266,102)(40,298,267,310)(41,350,268,218)(42,156,269,105)(43,301,270,313)(44,344,262,221)(45,159,263,108)(46,169,180,236)(47,211,172,325)(48,260,173,137)(49,163,174,239)(50,214,175,328)(51,254,176,140)(52,166,177,242)(53,208,178,331)(54,257,179,143)(55,126,282,251)(57,317,284,99)(58,120,285,245)(60,320,287,93)(61,123,288,248)(63,323,281,96)(64,291,228,181)(66,116,230,334)(67,294,231,184)(69,110,233,337)(70,297,234,187)(72,113,227,340)(83,134,206,353)(84,153,207,276)(86,128,200,356)(87,147,201,279)(89,131,203,359)(90,150,204,273)(92,121,319,246)(95,124,322,249)(98,118,316,252)(100,223,160,346)(101,305,161,308)(103,217,154,349)(104,299,155,311)(106,220,157,343)(107,302,158,314)(109,185,336,295)(112,188,339,289)(115,182,342,292)(127,277,355,145)(130,271,358,148)(133,274,352,151)(136,170,259,237)(138,327,261,213)(139,164,253,240)(141,330,255,216)(142,167,256,243)(144,333,258,210)(165,329,241,215)(168,332,235,209)(171,326,238,212)(219,312,351,300)(222,315,345,303)(225,309,348,306), (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)(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)(10,78,160,234,170)(11,79,161,226,171)(12,80,162,227,163)(13,81,154,228,164)(14,73,155,229,165)(15,74,156,230,166)(16,75,157,231,167)(17,76,158,232,168)(18,77,159,233,169)(19,332,56,129,314)(20,333,57,130,315)(21,325,58,131,307)(22,326,59,132,308)(23,327,60,133,309)(24,328,61,134,310)(25,329,62,135,311)(26,330,63,127,312)(27,331,55,128,313)(28,102,72,239,194)(29,103,64,240,195)(30,104,65,241,196)(31,105,66,242,197)(32,106,67,243,198)(33,107,68,235,190)(34,108,69,236,191)(35,100,70,237,192)(36,101,71,238,193)(37,111,47,120,203)(38,112,48,121,204)(39,113,49,122,205)(40,114,50,123,206)(41,115,51,124,207)(42,116,52,125,199)(43,117,53,126,200)(44,109,54,118,201)(45,110,46,119,202)(82,266,340,174,247)(83,267,341,175,248)(84,268,342,176,249)(85,269,334,177,250)(86,270,335,178,251)(87,262,336,179,252)(88,263,337,180,244)(89,264,338,172,245)(90,265,339,173,246)(91,149,223,297,136)(92,150,224,289,137)(93,151,225,290,138)(94,152,217,291,139)(95,153,218,292,140)(96,145,219,293,141)(97,146,220,294,142)(98,147,221,295,143)(99,148,222,296,144)(181,253,321,275,349)(182,254,322,276,350)(183,255,323,277,351)(184,256,324,278,343)(185,257,316,279,344)(186,258,317,271,345)(187,259,318,272,346)(188,260,319,273,347)(189,261,320,274,348), (1,294,27,184)(2,336,19,109)(3,233,20,69)(4,297,21,187)(5,339,22,112)(6,227,23,72)(7,291,24,181)(8,342,25,115)(9,230,26,66)(10,245,192,120)(11,319,193,92)(12,60,194,287)(13,248,195,123)(14,322,196,95)(15,63,197,281)(16,251,198,126)(17,316,190,98)(18,57,191,284)(28,352,80,133)(29,206,81,83)(30,153,73,276)(31,355,74,127)(32,200,75,86)(33,147,76,279)(34,358,77,130)(35,203,78,89)(36,150,79,273)(37,160,264,100)(38,305,265,308)(39,348,266,225)(40,154,267,103)(41,299,268,311)(42,351,269,219)(43,157,270,106)(44,302,262,314)(45,345,263,222)(46,258,180,144)(47,170,172,237)(48,212,173,326)(49,261,174,138)(50,164,175,240)(51,215,176,329)(52,255,177,141)(53,167,178,243)(54,209,179,332)(55,324,282,97)(56,118,283,252)(58,318,285,91)(59,121,286,246)(61,321,288,94)(62,124,280,249)(64,114,228,341)(65,292,229,182)(67,117,231,335)(68,295,232,185)(70,111,234,338)(71,289,226,188)(82,151,205,274)(84,135,207,354)(85,145,199,277)(87,129,201,357)(88,148,202,271)(90,132,204,360)(93,122,320,247)(96,125,323,250)(99,119,317,244)(101,224,161,347)(102,306,162,309)(104,218,155,350)(105,300,156,312)(107,221,158,344)(108,303,159,315)(110,186,337,296)(113,189,340,290)(116,183,334,293)(128,278,356,146)(131,272,359,149)(134,275,353,152)(136,325,259,211)(137,171,260,238)(139,328,253,214)(140,165,254,241)(142,331,256,208)(143,168,257,235)(163,327,239,213)(166,330,242,216)(169,333,236,210)(217,310,349,298)(220,313,343,301)(223,307,346,304), (1,335,27,117)(2,232,19,68)(3,296,20,186)(4,338,21,111)(5,226,22,71)(6,290,23,189)(7,341,24,114)(8,229,25,65)(9,293,26,183)(10,318,192,91)(11,59,193,286)(12,247,194,122)(13,321,195,94)(14,62,196,280)(15,250,197,125)(16,324,198,97)(17,56,190,283)(18,244,191,119)(28,205,80,82)(29,152,81,275)(30,354,73,135)(31,199,74,85)(32,146,75,278)(33,357,76,129)(34,202,77,88)(35,149,78,272)(36,360,79,132)(37,304,264,307)(38,347,265,224)(39,162,266,102)(40,298,267,310)(41,350,268,218)(42,156,269,105)(43,301,270,313)(44,344,262,221)(45,159,263,108)(46,169,180,236)(47,211,172,325)(48,260,173,137)(49,163,174,239)(50,214,175,328)(51,254,176,140)(52,166,177,242)(53,208,178,331)(54,257,179,143)(55,126,282,251)(57,317,284,99)(58,120,285,245)(60,320,287,93)(61,123,288,248)(63,323,281,96)(64,291,228,181)(66,116,230,334)(67,294,231,184)(69,110,233,337)(70,297,234,187)(72,113,227,340)(83,134,206,353)(84,153,207,276)(86,128,200,356)(87,147,201,279)(89,131,203,359)(90,150,204,273)(92,121,319,246)(95,124,322,249)(98,118,316,252)(100,223,160,346)(101,305,161,308)(103,217,154,349)(104,299,155,311)(106,220,157,343)(107,302,158,314)(109,185,336,295)(112,188,339,289)(115,182,342,292)(127,277,355,145)(130,271,358,148)(133,274,352,151)(136,170,259,237)(138,327,261,213)(139,164,253,240)(141,330,255,216)(142,167,256,243)(144,333,258,210)(165,329,241,215)(168,332,235,209)(171,326,238,212)(219,312,351,300)(222,315,345,303)(225,309,348,306), (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)(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),(10,78,160,234,170),(11,79,161,226,171),(12,80,162,227,163),(13,81,154,228,164),(14,73,155,229,165),(15,74,156,230,166),(16,75,157,231,167),(17,76,158,232,168),(18,77,159,233,169),(19,332,56,129,314),(20,333,57,130,315),(21,325,58,131,307),(22,326,59,132,308),(23,327,60,133,309),(24,328,61,134,310),(25,329,62,135,311),(26,330,63,127,312),(27,331,55,128,313),(28,102,72,239,194),(29,103,64,240,195),(30,104,65,241,196),(31,105,66,242,197),(32,106,67,243,198),(33,107,68,235,190),(34,108,69,236,191),(35,100,70,237,192),(36,101,71,238,193),(37,111,47,120,203),(38,112,48,121,204),(39,113,49,122,205),(40,114,50,123,206),(41,115,51,124,207),(42,116,52,125,199),(43,117,53,126,200),(44,109,54,118,201),(45,110,46,119,202),(82,266,340,174,247),(83,267,341,175,248),(84,268,342,176,249),(85,269,334,177,250),(86,270,335,178,251),(87,262,336,179,252),(88,263,337,180,244),(89,264,338,172,245),(90,265,339,173,246),(91,149,223,297,136),(92,150,224,289,137),(93,151,225,290,138),(94,152,217,291,139),(95,153,218,292,140),(96,145,219,293,141),(97,146,220,294,142),(98,147,221,295,143),(99,148,222,296,144),(181,253,321,275,349),(182,254,322,276,350),(183,255,323,277,351),(184,256,324,278,343),(185,257,316,279,344),(186,258,317,271,345),(187,259,318,272,346),(188,260,319,273,347),(189,261,320,274,348)], [(1,294,27,184),(2,336,19,109),(3,233,20,69),(4,297,21,187),(5,339,22,112),(6,227,23,72),(7,291,24,181),(8,342,25,115),(9,230,26,66),(10,245,192,120),(11,319,193,92),(12,60,194,287),(13,248,195,123),(14,322,196,95),(15,63,197,281),(16,251,198,126),(17,316,190,98),(18,57,191,284),(28,352,80,133),(29,206,81,83),(30,153,73,276),(31,355,74,127),(32,200,75,86),(33,147,76,279),(34,358,77,130),(35,203,78,89),(36,150,79,273),(37,160,264,100),(38,305,265,308),(39,348,266,225),(40,154,267,103),(41,299,268,311),(42,351,269,219),(43,157,270,106),(44,302,262,314),(45,345,263,222),(46,258,180,144),(47,170,172,237),(48,212,173,326),(49,261,174,138),(50,164,175,240),(51,215,176,329),(52,255,177,141),(53,167,178,243),(54,209,179,332),(55,324,282,97),(56,118,283,252),(58,318,285,91),(59,121,286,246),(61,321,288,94),(62,124,280,249),(64,114,228,341),(65,292,229,182),(67,117,231,335),(68,295,232,185),(70,111,234,338),(71,289,226,188),(82,151,205,274),(84,135,207,354),(85,145,199,277),(87,129,201,357),(88,148,202,271),(90,132,204,360),(93,122,320,247),(96,125,323,250),(99,119,317,244),(101,224,161,347),(102,306,162,309),(104,218,155,350),(105,300,156,312),(107,221,158,344),(108,303,159,315),(110,186,337,296),(113,189,340,290),(116,183,334,293),(128,278,356,146),(131,272,359,149),(134,275,353,152),(136,325,259,211),(137,171,260,238),(139,328,253,214),(140,165,254,241),(142,331,256,208),(143,168,257,235),(163,327,239,213),(166,330,242,216),(169,333,236,210),(217,310,349,298),(220,313,343,301),(223,307,346,304)], [(1,335,27,117),(2,232,19,68),(3,296,20,186),(4,338,21,111),(5,226,22,71),(6,290,23,189),(7,341,24,114),(8,229,25,65),(9,293,26,183),(10,318,192,91),(11,59,193,286),(12,247,194,122),(13,321,195,94),(14,62,196,280),(15,250,197,125),(16,324,198,97),(17,56,190,283),(18,244,191,119),(28,205,80,82),(29,152,81,275),(30,354,73,135),(31,199,74,85),(32,146,75,278),(33,357,76,129),(34,202,77,88),(35,149,78,272),(36,360,79,132),(37,304,264,307),(38,347,265,224),(39,162,266,102),(40,298,267,310),(41,350,268,218),(42,156,269,105),(43,301,270,313),(44,344,262,221),(45,159,263,108),(46,169,180,236),(47,211,172,325),(48,260,173,137),(49,163,174,239),(50,214,175,328),(51,254,176,140),(52,166,177,242),(53,208,178,331),(54,257,179,143),(55,126,282,251),(57,317,284,99),(58,120,285,245),(60,320,287,93),(61,123,288,248),(63,323,281,96),(64,291,228,181),(66,116,230,334),(67,294,231,184),(69,110,233,337),(70,297,234,187),(72,113,227,340),(83,134,206,353),(84,153,207,276),(86,128,200,356),(87,147,201,279),(89,131,203,359),(90,150,204,273),(92,121,319,246),(95,124,322,249),(98,118,316,252),(100,223,160,346),(101,305,161,308),(103,217,154,349),(104,299,155,311),(106,220,157,343),(107,302,158,314),(109,185,336,295),(112,188,339,289),(115,182,342,292),(127,277,355,145),(130,271,358,148),(133,274,352,151),(136,170,259,237),(138,327,261,213),(139,164,253,240),(141,330,255,216),(142,167,256,243),(144,333,258,210),(165,329,241,215),(168,332,235,209),(171,326,238,212),(219,312,351,300),(222,315,345,303),(225,309,348,306)], [(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),(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 3A3B 4 5A5B5C5D6A6B9A···9F10A10B10C10D12A12B15A···15H18A···18F20A20B20C20D30A···30H45A···45X60A···60H90A···90X
order123345555669···910101010121215···1518···182020202030···3045···4560···6090···90
size111161111114···41111661···14···466661···14···46···64···4

105 irreducible representations

dim111111222223333
type+-+
imageC1C3C5C9C15C45SL2(𝔽3)SL2(𝔽3)Q8⋊C9C5×SL2(𝔽3)C5×Q8⋊C9A4C3.A4C5×A4C5×C3.A4
kernelC5×Q8⋊C9Q8×C15Q8⋊C9C5×Q8C3×Q8Q8C15C15C5C3C1C30C10C6C2
# reps124682412612241248

Matrix representation of C5×Q8⋊C9 in GL3(𝔽181) generated by

4200
010
001
,
100
0113172
017268
,
100
00180
010
,
13200
04573
03497
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

Export

Subgroup lattice of C5×Q8⋊C9 in TeX

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