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## G = C5×C32⋊4C8order 360 = 23·32·5

### Direct product of C5 and C32⋊4C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×C32⋊4C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C60 — C5×C32⋊4C8
 Lower central C32 — C5×C32⋊4C8
 Upper central C1 — C20

Generators and relations for C5×C324C8
G = < a,b,c,d | a5=b3=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Smallest permutation representation of C5×C324C8
Regular action on 360 points
Generators in S360
(1 180 259 332 228)(2 181 260 333 229)(3 182 261 334 230)(4 183 262 335 231)(5 184 263 336 232)(6 177 264 329 225)(7 178 257 330 226)(8 179 258 331 227)(9 251 324 214 286)(10 252 325 215 287)(11 253 326 216 288)(12 254 327 209 281)(13 255 328 210 282)(14 256 321 211 283)(15 249 322 212 284)(16 250 323 213 285)(17 104 169 55 128)(18 97 170 56 121)(19 98 171 49 122)(20 99 172 50 123)(21 100 173 51 124)(22 101 174 52 125)(23 102 175 53 126)(24 103 176 54 127)(25 94 167 217 290)(26 95 168 218 291)(27 96 161 219 292)(28 89 162 220 293)(29 90 163 221 294)(30 91 164 222 295)(31 92 165 223 296)(32 93 166 224 289)(33 300 153 67 140)(34 301 154 68 141)(35 302 155 69 142)(36 303 156 70 143)(37 304 157 71 144)(38 297 158 72 137)(39 298 159 65 138)(40 299 160 66 139)(41 114 201 270 343)(42 115 202 271 344)(43 116 203 272 337)(44 117 204 265 338)(45 118 205 266 339)(46 119 206 267 340)(47 120 207 268 341)(48 113 208 269 342)(57 130 189 277 350)(58 131 190 278 351)(59 132 191 279 352)(60 133 192 280 345)(61 134 185 273 346)(62 135 186 274 347)(63 136 187 275 348)(64 129 188 276 349)(73 146 239 311 200)(74 147 240 312 193)(75 148 233 305 194)(76 149 234 306 195)(77 150 235 307 196)(78 151 236 308 197)(79 152 237 309 198)(80 145 238 310 199)(81 109 354 241 314)(82 110 355 242 315)(83 111 356 243 316)(84 112 357 244 317)(85 105 358 245 318)(86 106 359 246 319)(87 107 360 247 320)(88 108 353 248 313)
(1 84 132)(2 133 85)(3 86 134)(4 135 87)(5 88 136)(6 129 81)(7 82 130)(8 131 83)(9 99 272)(10 265 100)(11 101 266)(12 267 102)(13 103 268)(14 269 104)(15 97 270)(16 271 98)(17 283 208)(18 201 284)(19 285 202)(20 203 286)(21 287 204)(22 205 288)(23 281 206)(24 207 282)(25 303 311)(26 312 304)(27 297 305)(28 306 298)(29 299 307)(30 308 300)(31 301 309)(32 310 302)(33 295 236)(34 237 296)(35 289 238)(36 239 290)(37 291 240)(38 233 292)(39 293 234)(40 235 294)(41 322 56)(42 49 323)(43 324 50)(44 51 325)(45 326 52)(46 53 327)(47 328 54)(48 55 321)(57 226 315)(58 316 227)(59 228 317)(60 318 229)(61 230 319)(62 320 231)(63 232 313)(64 314 225)(65 162 76)(66 77 163)(67 164 78)(68 79 165)(69 166 80)(70 73 167)(71 168 74)(72 75 161)(89 195 159)(90 160 196)(91 197 153)(92 154 198)(93 199 155)(94 156 200)(95 193 157)(96 158 194)(105 181 192)(106 185 182)(107 183 186)(108 187 184)(109 177 188)(110 189 178)(111 179 190)(112 191 180)(113 128 211)(114 212 121)(115 122 213)(116 214 123)(117 124 215)(118 216 125)(119 126 209)(120 210 127)(137 148 219)(138 220 149)(139 150 221)(140 222 151)(141 152 223)(142 224 145)(143 146 217)(144 218 147)(169 256 342)(170 343 249)(171 250 344)(172 337 251)(173 252 338)(174 339 253)(175 254 340)(176 341 255)(241 329 349)(242 350 330)(243 331 351)(244 352 332)(245 333 345)(246 346 334)(247 335 347)(248 348 336)(257 355 277)(258 278 356)(259 357 279)(260 280 358)(261 359 273)(262 274 360)(263 353 275)(264 276 354)
(1 127 296)(2 289 128)(3 121 290)(4 291 122)(5 123 292)(6 293 124)(7 125 294)(8 295 126)(9 194 275)(10 276 195)(11 196 277)(12 278 197)(13 198 279)(14 280 199)(15 200 273)(16 274 193)(17 181 32)(18 25 182)(19 183 26)(20 27 184)(21 177 28)(22 29 178)(23 179 30)(24 31 180)(33 119 83)(34 84 120)(35 113 85)(36 86 114)(37 115 87)(38 88 116)(39 117 81)(40 82 118)(41 143 319)(42 320 144)(43 137 313)(44 314 138)(45 139 315)(46 316 140)(47 141 317)(48 318 142)(49 231 218)(50 219 232)(51 225 220)(52 221 226)(53 227 222)(54 223 228)(55 229 224)(56 217 230)(57 326 150)(58 151 327)(59 328 152)(60 145 321)(61 322 146)(62 147 323)(63 324 148)(64 149 325)(65 338 241)(66 242 339)(67 340 243)(68 244 341)(69 342 245)(70 246 343)(71 344 247)(72 248 337)(73 346 249)(74 250 347)(75 348 251)(76 252 349)(77 350 253)(78 254 351)(79 352 255)(80 256 345)(89 100 264)(90 257 101)(91 102 258)(92 259 103)(93 104 260)(94 261 97)(95 98 262)(96 263 99)(105 302 208)(106 201 303)(107 304 202)(108 203 297)(109 298 204)(110 205 299)(111 300 206)(112 207 301)(129 234 215)(130 216 235)(131 236 209)(132 210 237)(133 238 211)(134 212 239)(135 240 213)(136 214 233)(153 267 356)(154 357 268)(155 269 358)(156 359 270)(157 271 360)(158 353 272)(159 265 354)(160 355 266)(161 336 172)(162 173 329)(163 330 174)(164 175 331)(165 332 176)(166 169 333)(167 334 170)(168 171 335)(185 284 311)(186 312 285)(187 286 305)(188 306 287)(189 288 307)(190 308 281)(191 282 309)(192 310 283)
(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)| (1,180,259,332,228)(2,181,260,333,229)(3,182,261,334,230)(4,183,262,335,231)(5,184,263,336,232)(6,177,264,329,225)(7,178,257,330,226)(8,179,258,331,227)(9,251,324,214,286)(10,252,325,215,287)(11,253,326,216,288)(12,254,327,209,281)(13,255,328,210,282)(14,256,321,211,283)(15,249,322,212,284)(16,250,323,213,285)(17,104,169,55,128)(18,97,170,56,121)(19,98,171,49,122)(20,99,172,50,123)(21,100,173,51,124)(22,101,174,52,125)(23,102,175,53,126)(24,103,176,54,127)(25,94,167,217,290)(26,95,168,218,291)(27,96,161,219,292)(28,89,162,220,293)(29,90,163,221,294)(30,91,164,222,295)(31,92,165,223,296)(32,93,166,224,289)(33,300,153,67,140)(34,301,154,68,141)(35,302,155,69,142)(36,303,156,70,143)(37,304,157,71,144)(38,297,158,72,137)(39,298,159,65,138)(40,299,160,66,139)(41,114,201,270,343)(42,115,202,271,344)(43,116,203,272,337)(44,117,204,265,338)(45,118,205,266,339)(46,119,206,267,340)(47,120,207,268,341)(48,113,208,269,342)(57,130,189,277,350)(58,131,190,278,351)(59,132,191,279,352)(60,133,192,280,345)(61,134,185,273,346)(62,135,186,274,347)(63,136,187,275,348)(64,129,188,276,349)(73,146,239,311,200)(74,147,240,312,193)(75,148,233,305,194)(76,149,234,306,195)(77,150,235,307,196)(78,151,236,308,197)(79,152,237,309,198)(80,145,238,310,199)(81,109,354,241,314)(82,110,355,242,315)(83,111,356,243,316)(84,112,357,244,317)(85,105,358,245,318)(86,106,359,246,319)(87,107,360,247,320)(88,108,353,248,313), (1,84,132)(2,133,85)(3,86,134)(4,135,87)(5,88,136)(6,129,81)(7,82,130)(8,131,83)(9,99,272)(10,265,100)(11,101,266)(12,267,102)(13,103,268)(14,269,104)(15,97,270)(16,271,98)(17,283,208)(18,201,284)(19,285,202)(20,203,286)(21,287,204)(22,205,288)(23,281,206)(24,207,282)(25,303,311)(26,312,304)(27,297,305)(28,306,298)(29,299,307)(30,308,300)(31,301,309)(32,310,302)(33,295,236)(34,237,296)(35,289,238)(36,239,290)(37,291,240)(38,233,292)(39,293,234)(40,235,294)(41,322,56)(42,49,323)(43,324,50)(44,51,325)(45,326,52)(46,53,327)(47,328,54)(48,55,321)(57,226,315)(58,316,227)(59,228,317)(60,318,229)(61,230,319)(62,320,231)(63,232,313)(64,314,225)(65,162,76)(66,77,163)(67,164,78)(68,79,165)(69,166,80)(70,73,167)(71,168,74)(72,75,161)(89,195,159)(90,160,196)(91,197,153)(92,154,198)(93,199,155)(94,156,200)(95,193,157)(96,158,194)(105,181,192)(106,185,182)(107,183,186)(108,187,184)(109,177,188)(110,189,178)(111,179,190)(112,191,180)(113,128,211)(114,212,121)(115,122,213)(116,214,123)(117,124,215)(118,216,125)(119,126,209)(120,210,127)(137,148,219)(138,220,149)(139,150,221)(140,222,151)(141,152,223)(142,224,145)(143,146,217)(144,218,147)(169,256,342)(170,343,249)(171,250,344)(172,337,251)(173,252,338)(174,339,253)(175,254,340)(176,341,255)(241,329,349)(242,350,330)(243,331,351)(244,352,332)(245,333,345)(246,346,334)(247,335,347)(248,348,336)(257,355,277)(258,278,356)(259,357,279)(260,280,358)(261,359,273)(262,274,360)(263,353,275)(264,276,354), (1,127,296)(2,289,128)(3,121,290)(4,291,122)(5,123,292)(6,293,124)(7,125,294)(8,295,126)(9,194,275)(10,276,195)(11,196,277)(12,278,197)(13,198,279)(14,280,199)(15,200,273)(16,274,193)(17,181,32)(18,25,182)(19,183,26)(20,27,184)(21,177,28)(22,29,178)(23,179,30)(24,31,180)(33,119,83)(34,84,120)(35,113,85)(36,86,114)(37,115,87)(38,88,116)(39,117,81)(40,82,118)(41,143,319)(42,320,144)(43,137,313)(44,314,138)(45,139,315)(46,316,140)(47,141,317)(48,318,142)(49,231,218)(50,219,232)(51,225,220)(52,221,226)(53,227,222)(54,223,228)(55,229,224)(56,217,230)(57,326,150)(58,151,327)(59,328,152)(60,145,321)(61,322,146)(62,147,323)(63,324,148)(64,149,325)(65,338,241)(66,242,339)(67,340,243)(68,244,341)(69,342,245)(70,246,343)(71,344,247)(72,248,337)(73,346,249)(74,250,347)(75,348,251)(76,252,349)(77,350,253)(78,254,351)(79,352,255)(80,256,345)(89,100,264)(90,257,101)(91,102,258)(92,259,103)(93,104,260)(94,261,97)(95,98,262)(96,263,99)(105,302,208)(106,201,303)(107,304,202)(108,203,297)(109,298,204)(110,205,299)(111,300,206)(112,207,301)(129,234,215)(130,216,235)(131,236,209)(132,210,237)(133,238,211)(134,212,239)(135,240,213)(136,214,233)(153,267,356)(154,357,268)(155,269,358)(156,359,270)(157,271,360)(158,353,272)(159,265,354)(160,355,266)(161,336,172)(162,173,329)(163,330,174)(164,175,331)(165,332,176)(166,169,333)(167,334,170)(168,171,335)(185,284,311)(186,312,285)(187,286,305)(188,306,287)(189,288,307)(190,308,281)(191,282,309)(192,310,283), (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( (1,180,259,332,228)(2,181,260,333,229)(3,182,261,334,230)(4,183,262,335,231)(5,184,263,336,232)(6,177,264,329,225)(7,178,257,330,226)(8,179,258,331,227)(9,251,324,214,286)(10,252,325,215,287)(11,253,326,216,288)(12,254,327,209,281)(13,255,328,210,282)(14,256,321,211,283)(15,249,322,212,284)(16,250,323,213,285)(17,104,169,55,128)(18,97,170,56,121)(19,98,171,49,122)(20,99,172,50,123)(21,100,173,51,124)(22,101,174,52,125)(23,102,175,53,126)(24,103,176,54,127)(25,94,167,217,290)(26,95,168,218,291)(27,96,161,219,292)(28,89,162,220,293)(29,90,163,221,294)(30,91,164,222,295)(31,92,165,223,296)(32,93,166,224,289)(33,300,153,67,140)(34,301,154,68,141)(35,302,155,69,142)(36,303,156,70,143)(37,304,157,71,144)(38,297,158,72,137)(39,298,159,65,138)(40,299,160,66,139)(41,114,201,270,343)(42,115,202,271,344)(43,116,203,272,337)(44,117,204,265,338)(45,118,205,266,339)(46,119,206,267,340)(47,120,207,268,341)(48,113,208,269,342)(57,130,189,277,350)(58,131,190,278,351)(59,132,191,279,352)(60,133,192,280,345)(61,134,185,273,346)(62,135,186,274,347)(63,136,187,275,348)(64,129,188,276,349)(73,146,239,311,200)(74,147,240,312,193)(75,148,233,305,194)(76,149,234,306,195)(77,150,235,307,196)(78,151,236,308,197)(79,152,237,309,198)(80,145,238,310,199)(81,109,354,241,314)(82,110,355,242,315)(83,111,356,243,316)(84,112,357,244,317)(85,105,358,245,318)(86,106,359,246,319)(87,107,360,247,320)(88,108,353,248,313), (1,84,132)(2,133,85)(3,86,134)(4,135,87)(5,88,136)(6,129,81)(7,82,130)(8,131,83)(9,99,272)(10,265,100)(11,101,266)(12,267,102)(13,103,268)(14,269,104)(15,97,270)(16,271,98)(17,283,208)(18,201,284)(19,285,202)(20,203,286)(21,287,204)(22,205,288)(23,281,206)(24,207,282)(25,303,311)(26,312,304)(27,297,305)(28,306,298)(29,299,307)(30,308,300)(31,301,309)(32,310,302)(33,295,236)(34,237,296)(35,289,238)(36,239,290)(37,291,240)(38,233,292)(39,293,234)(40,235,294)(41,322,56)(42,49,323)(43,324,50)(44,51,325)(45,326,52)(46,53,327)(47,328,54)(48,55,321)(57,226,315)(58,316,227)(59,228,317)(60,318,229)(61,230,319)(62,320,231)(63,232,313)(64,314,225)(65,162,76)(66,77,163)(67,164,78)(68,79,165)(69,166,80)(70,73,167)(71,168,74)(72,75,161)(89,195,159)(90,160,196)(91,197,153)(92,154,198)(93,199,155)(94,156,200)(95,193,157)(96,158,194)(105,181,192)(106,185,182)(107,183,186)(108,187,184)(109,177,188)(110,189,178)(111,179,190)(112,191,180)(113,128,211)(114,212,121)(115,122,213)(116,214,123)(117,124,215)(118,216,125)(119,126,209)(120,210,127)(137,148,219)(138,220,149)(139,150,221)(140,222,151)(141,152,223)(142,224,145)(143,146,217)(144,218,147)(169,256,342)(170,343,249)(171,250,344)(172,337,251)(173,252,338)(174,339,253)(175,254,340)(176,341,255)(241,329,349)(242,350,330)(243,331,351)(244,352,332)(245,333,345)(246,346,334)(247,335,347)(248,348,336)(257,355,277)(258,278,356)(259,357,279)(260,280,358)(261,359,273)(262,274,360)(263,353,275)(264,276,354), (1,127,296)(2,289,128)(3,121,290)(4,291,122)(5,123,292)(6,293,124)(7,125,294)(8,295,126)(9,194,275)(10,276,195)(11,196,277)(12,278,197)(13,198,279)(14,280,199)(15,200,273)(16,274,193)(17,181,32)(18,25,182)(19,183,26)(20,27,184)(21,177,28)(22,29,178)(23,179,30)(24,31,180)(33,119,83)(34,84,120)(35,113,85)(36,86,114)(37,115,87)(38,88,116)(39,117,81)(40,82,118)(41,143,319)(42,320,144)(43,137,313)(44,314,138)(45,139,315)(46,316,140)(47,141,317)(48,318,142)(49,231,218)(50,219,232)(51,225,220)(52,221,226)(53,227,222)(54,223,228)(55,229,224)(56,217,230)(57,326,150)(58,151,327)(59,328,152)(60,145,321)(61,322,146)(62,147,323)(63,324,148)(64,149,325)(65,338,241)(66,242,339)(67,340,243)(68,244,341)(69,342,245)(70,246,343)(71,344,247)(72,248,337)(73,346,249)(74,250,347)(75,348,251)(76,252,349)(77,350,253)(78,254,351)(79,352,255)(80,256,345)(89,100,264)(90,257,101)(91,102,258)(92,259,103)(93,104,260)(94,261,97)(95,98,262)(96,263,99)(105,302,208)(106,201,303)(107,304,202)(108,203,297)(109,298,204)(110,205,299)(111,300,206)(112,207,301)(129,234,215)(130,216,235)(131,236,209)(132,210,237)(133,238,211)(134,212,239)(135,240,213)(136,214,233)(153,267,356)(154,357,268)(155,269,358)(156,359,270)(157,271,360)(158,353,272)(159,265,354)(160,355,266)(161,336,172)(162,173,329)(163,330,174)(164,175,331)(165,332,176)(166,169,333)(167,334,170)(168,171,335)(185,284,311)(186,312,285)(187,286,305)(188,306,287)(189,288,307)(190,308,281)(191,282,309)(192,310,283), (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([[(1,180,259,332,228),(2,181,260,333,229),(3,182,261,334,230),(4,183,262,335,231),(5,184,263,336,232),(6,177,264,329,225),(7,178,257,330,226),(8,179,258,331,227),(9,251,324,214,286),(10,252,325,215,287),(11,253,326,216,288),(12,254,327,209,281),(13,255,328,210,282),(14,256,321,211,283),(15,249,322,212,284),(16,250,323,213,285),(17,104,169,55,128),(18,97,170,56,121),(19,98,171,49,122),(20,99,172,50,123),(21,100,173,51,124),(22,101,174,52,125),(23,102,175,53,126),(24,103,176,54,127),(25,94,167,217,290),(26,95,168,218,291),(27,96,161,219,292),(28,89,162,220,293),(29,90,163,221,294),(30,91,164,222,295),(31,92,165,223,296),(32,93,166,224,289),(33,300,153,67,140),(34,301,154,68,141),(35,302,155,69,142),(36,303,156,70,143),(37,304,157,71,144),(38,297,158,72,137),(39,298,159,65,138),(40,299,160,66,139),(41,114,201,270,343),(42,115,202,271,344),(43,116,203,272,337),(44,117,204,265,338),(45,118,205,266,339),(46,119,206,267,340),(47,120,207,268,341),(48,113,208,269,342),(57,130,189,277,350),(58,131,190,278,351),(59,132,191,279,352),(60,133,192,280,345),(61,134,185,273,346),(62,135,186,274,347),(63,136,187,275,348),(64,129,188,276,349),(73,146,239,311,200),(74,147,240,312,193),(75,148,233,305,194),(76,149,234,306,195),(77,150,235,307,196),(78,151,236,308,197),(79,152,237,309,198),(80,145,238,310,199),(81,109,354,241,314),(82,110,355,242,315),(83,111,356,243,316),(84,112,357,244,317),(85,105,358,245,318),(86,106,359,246,319),(87,107,360,247,320),(88,108,353,248,313)], [(1,84,132),(2,133,85),(3,86,134),(4,135,87),(5,88,136),(6,129,81),(7,82,130),(8,131,83),(9,99,272),(10,265,100),(11,101,266),(12,267,102),(13,103,268),(14,269,104),(15,97,270),(16,271,98),(17,283,208),(18,201,284),(19,285,202),(20,203,286),(21,287,204),(22,205,288),(23,281,206),(24,207,282),(25,303,311),(26,312,304),(27,297,305),(28,306,298),(29,299,307),(30,308,300),(31,301,309),(32,310,302),(33,295,236),(34,237,296),(35,289,238),(36,239,290),(37,291,240),(38,233,292),(39,293,234),(40,235,294),(41,322,56),(42,49,323),(43,324,50),(44,51,325),(45,326,52),(46,53,327),(47,328,54),(48,55,321),(57,226,315),(58,316,227),(59,228,317),(60,318,229),(61,230,319),(62,320,231),(63,232,313),(64,314,225),(65,162,76),(66,77,163),(67,164,78),(68,79,165),(69,166,80),(70,73,167),(71,168,74),(72,75,161),(89,195,159),(90,160,196),(91,197,153),(92,154,198),(93,199,155),(94,156,200),(95,193,157),(96,158,194),(105,181,192),(106,185,182),(107,183,186),(108,187,184),(109,177,188),(110,189,178),(111,179,190),(112,191,180),(113,128,211),(114,212,121),(115,122,213),(116,214,123),(117,124,215),(118,216,125),(119,126,209),(120,210,127),(137,148,219),(138,220,149),(139,150,221),(140,222,151),(141,152,223),(142,224,145),(143,146,217),(144,218,147),(169,256,342),(170,343,249),(171,250,344),(172,337,251),(173,252,338),(174,339,253),(175,254,340),(176,341,255),(241,329,349),(242,350,330),(243,331,351),(244,352,332),(245,333,345),(246,346,334),(247,335,347),(248,348,336),(257,355,277),(258,278,356),(259,357,279),(260,280,358),(261,359,273),(262,274,360),(263,353,275),(264,276,354)], [(1,127,296),(2,289,128),(3,121,290),(4,291,122),(5,123,292),(6,293,124),(7,125,294),(8,295,126),(9,194,275),(10,276,195),(11,196,277),(12,278,197),(13,198,279),(14,280,199),(15,200,273),(16,274,193),(17,181,32),(18,25,182),(19,183,26),(20,27,184),(21,177,28),(22,29,178),(23,179,30),(24,31,180),(33,119,83),(34,84,120),(35,113,85),(36,86,114),(37,115,87),(38,88,116),(39,117,81),(40,82,118),(41,143,319),(42,320,144),(43,137,313),(44,314,138),(45,139,315),(46,316,140),(47,141,317),(48,318,142),(49,231,218),(50,219,232),(51,225,220),(52,221,226),(53,227,222),(54,223,228),(55,229,224),(56,217,230),(57,326,150),(58,151,327),(59,328,152),(60,145,321),(61,322,146),(62,147,323),(63,324,148),(64,149,325),(65,338,241),(66,242,339),(67,340,243),(68,244,341),(69,342,245),(70,246,343),(71,344,247),(72,248,337),(73,346,249),(74,250,347),(75,348,251),(76,252,349),(77,350,253),(78,254,351),(79,352,255),(80,256,345),(89,100,264),(90,257,101),(91,102,258),(92,259,103),(93,104,260),(94,261,97),(95,98,262),(96,263,99),(105,302,208),(106,201,303),(107,304,202),(108,203,297),(109,298,204),(110,205,299),(111,300,206),(112,207,301),(129,234,215),(130,216,235),(131,236,209),(132,210,237),(133,238,211),(134,212,239),(135,240,213),(136,214,233),(153,267,356),(154,357,268),(155,269,358),(156,359,270),(157,271,360),(158,353,272),(159,265,354),(160,355,266),(161,336,172),(162,173,329),(163,330,174),(164,175,331),(165,332,176),(166,169,333),(167,334,170),(168,171,335),(185,284,311),(186,312,285),(187,286,305),(188,306,287),(189,288,307),(190,308,281),(191,282,309),(192,310,283)], [(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)]])

120 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 5A 5B 5C 5D 6A 6B 6C 6D 8A 8B 8C 8D 10A 10B 10C 10D 12A ··· 12H 15A ··· 15P 20A ··· 20H 30A ··· 30P 40A ··· 40P 60A ··· 60AF order 1 2 3 3 3 3 4 4 5 5 5 5 6 6 6 6 8 8 8 8 10 10 10 10 12 ··· 12 15 ··· 15 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 2 2 2 2 1 1 1 1 1 1 2 2 2 2 9 9 9 9 1 1 1 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C4 C5 C8 C10 C20 C40 S3 Dic3 C3⋊C8 C5×S3 C5×Dic3 C5×C3⋊C8 kernel C5×C32⋊4C8 C3×C60 C3×C30 C32⋊4C8 C3×C15 C3×C12 C3×C6 C32 C60 C30 C15 C12 C6 C3 # reps 1 1 2 4 4 4 8 16 4 4 8 16 16 32

Matrix representation of C5×C324C8 in GL4(𝔽241) generated by

 205 0 0 0 0 205 0 0 0 0 1 0 0 0 0 1
,
 240 1 0 0 240 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 240 1 0 0 240 0
,
 0 240 0 0 240 0 0 0 0 0 98 156 0 0 13 143
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,1,0,0,0,0,1],[240,240,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,240,240,0,0,1,0],[0,240,0,0,240,0,0,0,0,0,98,13,0,0,156,143] >;

C5×C324C8 in GAP, Magma, Sage, TeX

C_5\times C_3^2\rtimes_4C_8
% in TeX

G:=Group("C5xC3^2:4C8");
// GroupNames label

G:=SmallGroup(360,36);
// by ID

G=gap.SmallGroup(360,36);
# by ID

G:=PCGroup([6,-2,-5,-2,-2,-3,-3,60,50,2404,8645]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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