direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C32×C5⋊2C8, C15⋊4C24, C60.9C6, C30.8C12, C5⋊2(C3×C24), (C3×C15)⋊12C8, C20.2(C3×C6), (C3×C60).8C2, (C3×C30).9C4, C12.8(C3×D5), (C3×C12).6D5, C10.2(C3×C12), C6.4(C3×Dic5), (C3×C6).4Dic5, C4.2(C32×D5), C2.(C32×Dic5), SmallGroup(360,33)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C32×C5⋊2C8 |
Generators and relations for C32×C5⋊2C8
G = < a,b,c,d | a3=b3=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of C32×C5⋊2C8
►Regular action on 360 points
Generators in S360
(1 99 59)(2 100 60)(3 101 61)(4 102 62)(5 103 63)(6 104 64)(7 97 57)(8 98 58)(9 327 287)(10 328 288)(11 321 281)(12 322 282)(13 323 283)(14 324 284)(15 325 285)(16 326 286)(17 279 239)(18 280 240)(19 273 233)(20 274 234)(21 275 235)(22 276 236)(23 277 237)(24 278 238)(25 165 65)(26 166 66)(27 167 67)(28 168 68)(29 161 69)(30 162 70)(31 163 71)(32 164 72)(33 348 73)(34 349 74)(35 350 75)(36 351 76)(37 352 77)(38 345 78)(39 346 79)(40 347 80)(41 122 81)(42 123 82)(43 124 83)(44 125 84)(45 126 85)(46 127 86)(47 128 87)(48 121 88)(49 130 89)(50 131 90)(51 132 91)(52 133 92)(53 134 93)(54 135 94)(55 136 95)(56 129 96)(105 185 145)(106 186 146)(107 187 147)(108 188 148)(109 189 149)(110 190 150)(111 191 151)(112 192 152)(113 193 153)(114 194 154)(115 195 155)(116 196 156)(117 197 157)(118 198 158)(119 199 159)(120 200 160)(137 217 177)(138 218 178)(139 219 179)(140 220 180)(141 221 181)(142 222 182)(143 223 183)(144 224 184)(169 244 204)(170 245 205)(171 246 206)(172 247 207)(173 248 208)(174 241 201)(175 242 202)(176 243 203)(209 354 314)(210 355 315)(211 356 316)(212 357 317)(213 358 318)(214 359 319)(215 360 320)(216 353 313)(225 306 265)(226 307 266)(227 308 267)(228 309 268)(229 310 269)(230 311 270)(231 312 271)(232 305 272)(249 330 290)(250 331 291)(251 332 292)(252 333 293)(253 334 294)(254 335 295)(255 336 296)(256 329 289)(257 338 298)(258 339 299)(259 340 300)(260 341 301)(261 342 302)(262 343 303)(263 344 304)(264 337 297)
(1 220 301)(2 221 302)(3 222 303)(4 223 304)(5 224 297)(6 217 298)(7 218 299)(8 219 300)(9 90 171)(10 91 172)(11 92 173)(12 93 174)(13 94 175)(14 95 176)(15 96 169)(16 89 170)(17 215 296)(18 216 289)(19 209 290)(20 210 291)(21 211 292)(22 212 293)(23 213 294)(24 214 295)(25 106 125)(26 107 126)(27 108 127)(28 109 128)(29 110 121)(30 111 122)(31 112 123)(32 105 124)(33 114 308)(34 115 309)(35 116 310)(36 117 311)(37 118 312)(38 119 305)(39 120 306)(40 113 307)(41 70 151)(42 71 152)(43 72 145)(44 65 146)(45 66 147)(46 67 148)(47 68 149)(48 69 150)(49 245 326)(50 246 327)(51 247 328)(52 248 321)(53 241 322)(54 242 323)(55 243 324)(56 244 325)(57 138 339)(58 139 340)(59 140 341)(60 141 342)(61 142 343)(62 143 344)(63 144 337)(64 137 338)(73 154 227)(74 155 228)(75 156 229)(76 157 230)(77 158 231)(78 159 232)(79 160 225)(80 153 226)(81 162 191)(82 163 192)(83 164 185)(84 165 186)(85 166 187)(86 167 188)(87 168 189)(88 161 190)(97 178 258)(98 179 259)(99 180 260)(100 181 261)(101 182 262)(102 183 263)(103 184 264)(104 177 257)(129 204 285)(130 205 286)(131 206 287)(132 207 288)(133 208 281)(134 201 282)(135 202 283)(136 203 284)(193 266 347)(194 267 348)(195 268 349)(196 269 350)(197 270 351)(198 271 352)(199 272 345)(200 265 346)(233 314 330)(234 315 331)(235 316 332)(236 317 333)(237 318 334)(238 319 335)(239 320 336)(240 313 329)(249 273 354)(250 274 355)(251 275 356)(252 276 357)(253 277 358)(254 278 359)(255 279 360)(256 280 353)
(1 281 22 107 115)(2 116 108 23 282)(3 283 24 109 117)(4 118 110 17 284)(5 285 18 111 119)(6 120 112 19 286)(7 287 20 105 113)(8 114 106 21 288)(9 274 185 193 97)(10 98 194 186 275)(11 276 187 195 99)(12 100 196 188 277)(13 278 189 197 101)(14 102 198 190 279)(15 280 191 199 103)(16 104 200 192 273)(25 292 207 300 33)(26 34 301 208 293)(27 294 201 302 35)(28 36 303 202 295)(29 296 203 304 37)(30 38 297 204 289)(31 290 205 298 39)(32 40 299 206 291)(41 232 144 56 313)(42 314 49 137 225)(43 226 138 50 315)(44 316 51 139 227)(45 228 140 52 317)(46 318 53 141 229)(47 230 142 54 319)(48 320 55 143 231)(57 327 234 145 153)(58 154 146 235 328)(59 321 236 147 155)(60 156 148 237 322)(61 323 238 149 157)(62 158 150 239 324)(63 325 240 151 159)(64 160 152 233 326)(65 332 247 340 73)(66 74 341 248 333)(67 334 241 342 75)(68 76 343 242 335)(69 336 243 344 77)(70 78 337 244 329)(71 330 245 338 79)(72 80 339 246 331)(81 272 184 96 353)(82 354 89 177 265)(83 266 178 90 355)(84 356 91 179 267)(85 268 180 92 357)(86 358 93 181 269)(87 270 182 94 359)(88 360 95 183 271)(121 215 136 223 312)(122 305 224 129 216)(123 209 130 217 306)(124 307 218 131 210)(125 211 132 219 308)(126 309 220 133 212)(127 213 134 221 310)(128 311 222 135 214)(161 255 176 263 352)(162 345 264 169 256)(163 249 170 257 346)(164 347 258 171 250)(165 251 172 259 348)(166 349 260 173 252)(167 253 174 261 350)(168 351 262 175 254)
(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)| (1,99,59)(2,100,60)(3,101,61)(4,102,62)(5,103,63)(6,104,64)(7,97,57)(8,98,58)(9,327,287)(10,328,288)(11,321,281)(12,322,282)(13,323,283)(14,324,284)(15,325,285)(16,326,286)(17,279,239)(18,280,240)(19,273,233)(20,274,234)(21,275,235)(22,276,236)(23,277,237)(24,278,238)(25,165,65)(26,166,66)(27,167,67)(28,168,68)(29,161,69)(30,162,70)(31,163,71)(32,164,72)(33,348,73)(34,349,74)(35,350,75)(36,351,76)(37,352,77)(38,345,78)(39,346,79)(40,347,80)(41,122,81)(42,123,82)(43,124,83)(44,125,84)(45,126,85)(46,127,86)(47,128,87)(48,121,88)(49,130,89)(50,131,90)(51,132,91)(52,133,92)(53,134,93)(54,135,94)(55,136,95)(56,129,96)(105,185,145)(106,186,146)(107,187,147)(108,188,148)(109,189,149)(110,190,150)(111,191,151)(112,192,152)(113,193,153)(114,194,154)(115,195,155)(116,196,156)(117,197,157)(118,198,158)(119,199,159)(120,200,160)(137,217,177)(138,218,178)(139,219,179)(140,220,180)(141,221,181)(142,222,182)(143,223,183)(144,224,184)(169,244,204)(170,245,205)(171,246,206)(172,247,207)(173,248,208)(174,241,201)(175,242,202)(176,243,203)(209,354,314)(210,355,315)(211,356,316)(212,357,317)(213,358,318)(214,359,319)(215,360,320)(216,353,313)(225,306,265)(226,307,266)(227,308,267)(228,309,268)(229,310,269)(230,311,270)(231,312,271)(232,305,272)(249,330,290)(250,331,291)(251,332,292)(252,333,293)(253,334,294)(254,335,295)(255,336,296)(256,329,289)(257,338,298)(258,339,299)(259,340,300)(260,341,301)(261,342,302)(262,343,303)(263,344,304)(264,337,297), (1,220,301)(2,221,302)(3,222,303)(4,223,304)(5,224,297)(6,217,298)(7,218,299)(8,219,300)(9,90,171)(10,91,172)(11,92,173)(12,93,174)(13,94,175)(14,95,176)(15,96,169)(16,89,170)(17,215,296)(18,216,289)(19,209,290)(20,210,291)(21,211,292)(22,212,293)(23,213,294)(24,214,295)(25,106,125)(26,107,126)(27,108,127)(28,109,128)(29,110,121)(30,111,122)(31,112,123)(32,105,124)(33,114,308)(34,115,309)(35,116,310)(36,117,311)(37,118,312)(38,119,305)(39,120,306)(40,113,307)(41,70,151)(42,71,152)(43,72,145)(44,65,146)(45,66,147)(46,67,148)(47,68,149)(48,69,150)(49,245,326)(50,246,327)(51,247,328)(52,248,321)(53,241,322)(54,242,323)(55,243,324)(56,244,325)(57,138,339)(58,139,340)(59,140,341)(60,141,342)(61,142,343)(62,143,344)(63,144,337)(64,137,338)(73,154,227)(74,155,228)(75,156,229)(76,157,230)(77,158,231)(78,159,232)(79,160,225)(80,153,226)(81,162,191)(82,163,192)(83,164,185)(84,165,186)(85,166,187)(86,167,188)(87,168,189)(88,161,190)(97,178,258)(98,179,259)(99,180,260)(100,181,261)(101,182,262)(102,183,263)(103,184,264)(104,177,257)(129,204,285)(130,205,286)(131,206,287)(132,207,288)(133,208,281)(134,201,282)(135,202,283)(136,203,284)(193,266,347)(194,267,348)(195,268,349)(196,269,350)(197,270,351)(198,271,352)(199,272,345)(200,265,346)(233,314,330)(234,315,331)(235,316,332)(236,317,333)(237,318,334)(238,319,335)(239,320,336)(240,313,329)(249,273,354)(250,274,355)(251,275,356)(252,276,357)(253,277,358)(254,278,359)(255,279,360)(256,280,353), (1,281,22,107,115)(2,116,108,23,282)(3,283,24,109,117)(4,118,110,17,284)(5,285,18,111,119)(6,120,112,19,286)(7,287,20,105,113)(8,114,106,21,288)(9,274,185,193,97)(10,98,194,186,275)(11,276,187,195,99)(12,100,196,188,277)(13,278,189,197,101)(14,102,198,190,279)(15,280,191,199,103)(16,104,200,192,273)(25,292,207,300,33)(26,34,301,208,293)(27,294,201,302,35)(28,36,303,202,295)(29,296,203,304,37)(30,38,297,204,289)(31,290,205,298,39)(32,40,299,206,291)(41,232,144,56,313)(42,314,49,137,225)(43,226,138,50,315)(44,316,51,139,227)(45,228,140,52,317)(46,318,53,141,229)(47,230,142,54,319)(48,320,55,143,231)(57,327,234,145,153)(58,154,146,235,328)(59,321,236,147,155)(60,156,148,237,322)(61,323,238,149,157)(62,158,150,239,324)(63,325,240,151,159)(64,160,152,233,326)(65,332,247,340,73)(66,74,341,248,333)(67,334,241,342,75)(68,76,343,242,335)(69,336,243,344,77)(70,78,337,244,329)(71,330,245,338,79)(72,80,339,246,331)(81,272,184,96,353)(82,354,89,177,265)(83,266,178,90,355)(84,356,91,179,267)(85,268,180,92,357)(86,358,93,181,269)(87,270,182,94,359)(88,360,95,183,271)(121,215,136,223,312)(122,305,224,129,216)(123,209,130,217,306)(124,307,218,131,210)(125,211,132,219,308)(126,309,220,133,212)(127,213,134,221,310)(128,311,222,135,214)(161,255,176,263,352)(162,345,264,169,256)(163,249,170,257,346)(164,347,258,171,250)(165,251,172,259,348)(166,349,260,173,252)(167,253,174,261,350)(168,351,262,175,254), (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( (1,99,59)(2,100,60)(3,101,61)(4,102,62)(5,103,63)(6,104,64)(7,97,57)(8,98,58)(9,327,287)(10,328,288)(11,321,281)(12,322,282)(13,323,283)(14,324,284)(15,325,285)(16,326,286)(17,279,239)(18,280,240)(19,273,233)(20,274,234)(21,275,235)(22,276,236)(23,277,237)(24,278,238)(25,165,65)(26,166,66)(27,167,67)(28,168,68)(29,161,69)(30,162,70)(31,163,71)(32,164,72)(33,348,73)(34,349,74)(35,350,75)(36,351,76)(37,352,77)(38,345,78)(39,346,79)(40,347,80)(41,122,81)(42,123,82)(43,124,83)(44,125,84)(45,126,85)(46,127,86)(47,128,87)(48,121,88)(49,130,89)(50,131,90)(51,132,91)(52,133,92)(53,134,93)(54,135,94)(55,136,95)(56,129,96)(105,185,145)(106,186,146)(107,187,147)(108,188,148)(109,189,149)(110,190,150)(111,191,151)(112,192,152)(113,193,153)(114,194,154)(115,195,155)(116,196,156)(117,197,157)(118,198,158)(119,199,159)(120,200,160)(137,217,177)(138,218,178)(139,219,179)(140,220,180)(141,221,181)(142,222,182)(143,223,183)(144,224,184)(169,244,204)(170,245,205)(171,246,206)(172,247,207)(173,248,208)(174,241,201)(175,242,202)(176,243,203)(209,354,314)(210,355,315)(211,356,316)(212,357,317)(213,358,318)(214,359,319)(215,360,320)(216,353,313)(225,306,265)(226,307,266)(227,308,267)(228,309,268)(229,310,269)(230,311,270)(231,312,271)(232,305,272)(249,330,290)(250,331,291)(251,332,292)(252,333,293)(253,334,294)(254,335,295)(255,336,296)(256,329,289)(257,338,298)(258,339,299)(259,340,300)(260,341,301)(261,342,302)(262,343,303)(263,344,304)(264,337,297), (1,220,301)(2,221,302)(3,222,303)(4,223,304)(5,224,297)(6,217,298)(7,218,299)(8,219,300)(9,90,171)(10,91,172)(11,92,173)(12,93,174)(13,94,175)(14,95,176)(15,96,169)(16,89,170)(17,215,296)(18,216,289)(19,209,290)(20,210,291)(21,211,292)(22,212,293)(23,213,294)(24,214,295)(25,106,125)(26,107,126)(27,108,127)(28,109,128)(29,110,121)(30,111,122)(31,112,123)(32,105,124)(33,114,308)(34,115,309)(35,116,310)(36,117,311)(37,118,312)(38,119,305)(39,120,306)(40,113,307)(41,70,151)(42,71,152)(43,72,145)(44,65,146)(45,66,147)(46,67,148)(47,68,149)(48,69,150)(49,245,326)(50,246,327)(51,247,328)(52,248,321)(53,241,322)(54,242,323)(55,243,324)(56,244,325)(57,138,339)(58,139,340)(59,140,341)(60,141,342)(61,142,343)(62,143,344)(63,144,337)(64,137,338)(73,154,227)(74,155,228)(75,156,229)(76,157,230)(77,158,231)(78,159,232)(79,160,225)(80,153,226)(81,162,191)(82,163,192)(83,164,185)(84,165,186)(85,166,187)(86,167,188)(87,168,189)(88,161,190)(97,178,258)(98,179,259)(99,180,260)(100,181,261)(101,182,262)(102,183,263)(103,184,264)(104,177,257)(129,204,285)(130,205,286)(131,206,287)(132,207,288)(133,208,281)(134,201,282)(135,202,283)(136,203,284)(193,266,347)(194,267,348)(195,268,349)(196,269,350)(197,270,351)(198,271,352)(199,272,345)(200,265,346)(233,314,330)(234,315,331)(235,316,332)(236,317,333)(237,318,334)(238,319,335)(239,320,336)(240,313,329)(249,273,354)(250,274,355)(251,275,356)(252,276,357)(253,277,358)(254,278,359)(255,279,360)(256,280,353), (1,281,22,107,115)(2,116,108,23,282)(3,283,24,109,117)(4,118,110,17,284)(5,285,18,111,119)(6,120,112,19,286)(7,287,20,105,113)(8,114,106,21,288)(9,274,185,193,97)(10,98,194,186,275)(11,276,187,195,99)(12,100,196,188,277)(13,278,189,197,101)(14,102,198,190,279)(15,280,191,199,103)(16,104,200,192,273)(25,292,207,300,33)(26,34,301,208,293)(27,294,201,302,35)(28,36,303,202,295)(29,296,203,304,37)(30,38,297,204,289)(31,290,205,298,39)(32,40,299,206,291)(41,232,144,56,313)(42,314,49,137,225)(43,226,138,50,315)(44,316,51,139,227)(45,228,140,52,317)(46,318,53,141,229)(47,230,142,54,319)(48,320,55,143,231)(57,327,234,145,153)(58,154,146,235,328)(59,321,236,147,155)(60,156,148,237,322)(61,323,238,149,157)(62,158,150,239,324)(63,325,240,151,159)(64,160,152,233,326)(65,332,247,340,73)(66,74,341,248,333)(67,334,241,342,75)(68,76,343,242,335)(69,336,243,344,77)(70,78,337,244,329)(71,330,245,338,79)(72,80,339,246,331)(81,272,184,96,353)(82,354,89,177,265)(83,266,178,90,355)(84,356,91,179,267)(85,268,180,92,357)(86,358,93,181,269)(87,270,182,94,359)(88,360,95,183,271)(121,215,136,223,312)(122,305,224,129,216)(123,209,130,217,306)(124,307,218,131,210)(125,211,132,219,308)(126,309,220,133,212)(127,213,134,221,310)(128,311,222,135,214)(161,255,176,263,352)(162,345,264,169,256)(163,249,170,257,346)(164,347,258,171,250)(165,251,172,259,348)(166,349,260,173,252)(167,253,174,261,350)(168,351,262,175,254), (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([[(1,99,59),(2,100,60),(3,101,61),(4,102,62),(5,103,63),(6,104,64),(7,97,57),(8,98,58),(9,327,287),(10,328,288),(11,321,281),(12,322,282),(13,323,283),(14,324,284),(15,325,285),(16,326,286),(17,279,239),(18,280,240),(19,273,233),(20,274,234),(21,275,235),(22,276,236),(23,277,237),(24,278,238),(25,165,65),(26,166,66),(27,167,67),(28,168,68),(29,161,69),(30,162,70),(31,163,71),(32,164,72),(33,348,73),(34,349,74),(35,350,75),(36,351,76),(37,352,77),(38,345,78),(39,346,79),(40,347,80),(41,122,81),(42,123,82),(43,124,83),(44,125,84),(45,126,85),(46,127,86),(47,128,87),(48,121,88),(49,130,89),(50,131,90),(51,132,91),(52,133,92),(53,134,93),(54,135,94),(55,136,95),(56,129,96),(105,185,145),(106,186,146),(107,187,147),(108,188,148),(109,189,149),(110,190,150),(111,191,151),(112,192,152),(113,193,153),(114,194,154),(115,195,155),(116,196,156),(117,197,157),(118,198,158),(119,199,159),(120,200,160),(137,217,177),(138,218,178),(139,219,179),(140,220,180),(141,221,181),(142,222,182),(143,223,183),(144,224,184),(169,244,204),(170,245,205),(171,246,206),(172,247,207),(173,248,208),(174,241,201),(175,242,202),(176,243,203),(209,354,314),(210,355,315),(211,356,316),(212,357,317),(213,358,318),(214,359,319),(215,360,320),(216,353,313),(225,306,265),(226,307,266),(227,308,267),(228,309,268),(229,310,269),(230,311,270),(231,312,271),(232,305,272),(249,330,290),(250,331,291),(251,332,292),(252,333,293),(253,334,294),(254,335,295),(255,336,296),(256,329,289),(257,338,298),(258,339,299),(259,340,300),(260,341,301),(261,342,302),(262,343,303),(263,344,304),(264,337,297)], [(1,220,301),(2,221,302),(3,222,303),(4,223,304),(5,224,297),(6,217,298),(7,218,299),(8,219,300),(9,90,171),(10,91,172),(11,92,173),(12,93,174),(13,94,175),(14,95,176),(15,96,169),(16,89,170),(17,215,296),(18,216,289),(19,209,290),(20,210,291),(21,211,292),(22,212,293),(23,213,294),(24,214,295),(25,106,125),(26,107,126),(27,108,127),(28,109,128),(29,110,121),(30,111,122),(31,112,123),(32,105,124),(33,114,308),(34,115,309),(35,116,310),(36,117,311),(37,118,312),(38,119,305),(39,120,306),(40,113,307),(41,70,151),(42,71,152),(43,72,145),(44,65,146),(45,66,147),(46,67,148),(47,68,149),(48,69,150),(49,245,326),(50,246,327),(51,247,328),(52,248,321),(53,241,322),(54,242,323),(55,243,324),(56,244,325),(57,138,339),(58,139,340),(59,140,341),(60,141,342),(61,142,343),(62,143,344),(63,144,337),(64,137,338),(73,154,227),(74,155,228),(75,156,229),(76,157,230),(77,158,231),(78,159,232),(79,160,225),(80,153,226),(81,162,191),(82,163,192),(83,164,185),(84,165,186),(85,166,187),(86,167,188),(87,168,189),(88,161,190),(97,178,258),(98,179,259),(99,180,260),(100,181,261),(101,182,262),(102,183,263),(103,184,264),(104,177,257),(129,204,285),(130,205,286),(131,206,287),(132,207,288),(133,208,281),(134,201,282),(135,202,283),(136,203,284),(193,266,347),(194,267,348),(195,268,349),(196,269,350),(197,270,351),(198,271,352),(199,272,345),(200,265,346),(233,314,330),(234,315,331),(235,316,332),(236,317,333),(237,318,334),(238,319,335),(239,320,336),(240,313,329),(249,273,354),(250,274,355),(251,275,356),(252,276,357),(253,277,358),(254,278,359),(255,279,360),(256,280,353)], [(1,281,22,107,115),(2,116,108,23,282),(3,283,24,109,117),(4,118,110,17,284),(5,285,18,111,119),(6,120,112,19,286),(7,287,20,105,113),(8,114,106,21,288),(9,274,185,193,97),(10,98,194,186,275),(11,276,187,195,99),(12,100,196,188,277),(13,278,189,197,101),(14,102,198,190,279),(15,280,191,199,103),(16,104,200,192,273),(25,292,207,300,33),(26,34,301,208,293),(27,294,201,302,35),(28,36,303,202,295),(29,296,203,304,37),(30,38,297,204,289),(31,290,205,298,39),(32,40,299,206,291),(41,232,144,56,313),(42,314,49,137,225),(43,226,138,50,315),(44,316,51,139,227),(45,228,140,52,317),(46,318,53,141,229),(47,230,142,54,319),(48,320,55,143,231),(57,327,234,145,153),(58,154,146,235,328),(59,321,236,147,155),(60,156,148,237,322),(61,323,238,149,157),(62,158,150,239,324),(63,325,240,151,159),(64,160,152,233,326),(65,332,247,340,73),(66,74,341,248,333),(67,334,241,342,75),(68,76,343,242,335),(69,336,243,344,77),(70,78,337,244,329),(71,330,245,338,79),(72,80,339,246,331),(81,272,184,96,353),(82,354,89,177,265),(83,266,178,90,355),(84,356,91,179,267),(85,268,180,92,357),(86,358,93,181,269),(87,270,182,94,359),(88,360,95,183,271),(121,215,136,223,312),(122,305,224,129,216),(123,209,130,217,306),(124,307,218,131,210),(125,211,132,219,308),(126,309,220,133,212),(127,213,134,221,310),(128,311,222,135,214),(161,255,176,263,352),(162,345,264,169,256),(163,249,170,257,346),(164,347,258,171,250),(165,251,172,259,348),(166,349,260,173,252),(167,253,174,261,350),(168,351,262,175,254)], [(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)]])
144 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 5A | 5B | 6A | ··· | 6H | 8A | 8B | 8C | 8D | 10A | 10B | 12A | ··· | 12P | 15A | ··· | 15P | 20A | 20B | 20C | 20D | 24A | ··· | 24AF | 30A | ··· | 30P | 60A | ··· | 60AF |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | D5 | Dic5 | C3×D5 | C5⋊2C8 | C3×Dic5 | C3×C5⋊2C8 |
| kernel | C32×C5⋊2C8 | C3×C60 | C3×C5⋊2C8 | C3×C30 | C60 | C3×C15 | C30 | C15 | C3×C12 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 4 | 16 | 32 | 2 | 2 | 16 | 4 | 16 | 32 |
Matrix representation of C32×C5⋊2C8 ►in GL3(𝔽241) generated by
| 15 | 0 | 0 |
| 0 | 225 | 0 |
| 0 | 0 | 225 |
| 225 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 52 | 240 |
| 0 | 53 | 240 |
| 240 | 0 | 0 |
| 0 | 91 | 174 |
| 0 | 67 | 150 |
G:=sub<GL(3,GF(241))| [15,0,0,0,225,0,0,0,225],[225,0,0,0,1,0,0,0,1],[1,0,0,0,52,53,0,240,240],[240,0,0,0,91,67,0,174,150] >;
C32×C5⋊2C8 in GAP, Magma, Sage, TeX
C_3^2\times C_5\rtimes_2C_8
% in TeX
G:=Group("C3^2xC5:2C8"); // GroupNames label
G:=SmallGroup(360,33);
// by ID
G=gap.SmallGroup(360,33);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-2,-5,108,69,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^5=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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