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## G = C32×C5⋊2C8order 360 = 23·32·5

### Direct product of C32 and C5⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C32×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C60 — C3×C60 — C32×C5⋊2C8
 Lower central C5 — C32×C5⋊2C8
 Upper central C1 — C3×C12

Generators and relations for C32×C52C8
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×C52C8
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)(17 278 238)(18 279 239)(19 280 240)(20 273 233)(21 274 234)(22 275 235)(23 276 236)(24 277 237)(25 345 65)(26 346 66)(27 347 67)(28 348 68)(29 349 69)(30 350 70)(31 351 71)(32 352 72)(33 173 73)(34 174 74)(35 175 75)(36 176 76)(37 169 77)(38 170 78)(39 171 79)(40 172 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)(161 241 201)(162 242 202)(163 243 203)(164 244 204)(165 245 205)(166 246 206)(167 247 207)(168 248 208)(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 85 166)(10 86 167)(11 87 168)(12 88 161)(13 81 162)(14 82 163)(15 83 164)(16 84 165)(17 214 295)(18 215 296)(19 216 289)(20 209 290)(21 210 291)(22 211 292)(23 212 293)(24 213 294)(25 106 305)(26 107 306)(27 108 307)(28 109 308)(29 110 309)(30 111 310)(31 112 311)(32 105 312)(33 114 133)(34 115 134)(35 116 135)(36 117 136)(37 118 129)(38 119 130)(39 120 131)(40 113 132)(41 242 323)(42 243 324)(43 244 325)(44 245 326)(45 246 327)(46 247 328)(47 248 321)(48 241 322)(49 78 159)(50 79 160)(51 80 153)(52 73 154)(53 74 155)(54 75 156)(55 76 157)(56 77 158)(57 138 339)(58 139 340)(59 140 341)(60 141 342)(61 142 343)(62 143 344)(63 144 337)(64 137 338)(65 146 232)(66 147 225)(67 148 226)(68 149 227)(69 150 228)(70 151 229)(71 152 230)(72 145 231)(89 170 199)(90 171 200)(91 172 193)(92 173 194)(93 174 195)(94 175 196)(95 176 197)(96 169 198)(97 178 258)(98 179 259)(99 180 260)(100 181 261)(101 182 262)(102 183 263)(103 184 264)(104 177 257)(121 201 282)(122 202 283)(123 203 284)(124 204 285)(125 205 286)(126 206 287)(127 207 288)(128 208 281)(185 271 352)(186 272 345)(187 265 346)(188 266 347)(189 267 348)(190 268 349)(191 269 350)(192 270 351)(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)
(1 106 23 287 115)(2 116 288 24 107)(3 108 17 281 117)(4 118 282 18 109)(5 110 19 283 119)(6 120 284 20 111)(7 112 21 285 113)(8 114 286 22 105)(9 195 99 186 276)(10 277 187 100 196)(11 197 101 188 278)(12 279 189 102 198)(13 199 103 190 280)(14 273 191 104 200)(15 193 97 192 274)(16 275 185 98 194)(25 293 206 34 301)(26 302 35 207 294)(27 295 208 36 303)(28 304 37 201 296)(29 289 202 38 297)(30 298 39 203 290)(31 291 204 40 299)(32 300 33 205 292)(41 49 144 228 313)(42 314 229 137 50)(43 51 138 230 315)(44 316 231 139 52)(45 53 140 232 317)(46 318 225 141 54)(47 55 142 226 319)(48 320 227 143 56)(57 152 234 325 153)(58 154 326 235 145)(59 146 236 327 155)(60 156 328 237 147)(61 148 238 321 157)(62 158 322 239 149)(63 150 240 323 159)(64 160 324 233 151)(65 333 246 74 341)(66 342 75 247 334)(67 335 248 76 343)(68 344 77 241 336)(69 329 242 78 337)(70 338 79 243 330)(71 331 244 80 339)(72 340 73 245 332)(81 89 184 268 353)(82 354 269 177 90)(83 91 178 270 355)(84 356 271 179 92)(85 93 180 272 357)(86 358 265 181 94)(87 95 182 266 359)(88 360 267 183 96)(121 215 308 223 129)(122 130 224 309 216)(123 209 310 217 131)(124 132 218 311 210)(125 211 312 219 133)(126 134 220 305 212)(127 213 306 221 135)(128 136 222 307 214)(161 255 348 263 169)(162 170 264 349 256)(163 249 350 257 171)(164 172 258 351 250)(165 251 352 259 173)(166 174 260 345 252)(167 253 346 261 175)(168 176 262 347 254)
(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)(17,278,238)(18,279,239)(19,280,240)(20,273,233)(21,274,234)(22,275,235)(23,276,236)(24,277,237)(25,345,65)(26,346,66)(27,347,67)(28,348,68)(29,349,69)(30,350,70)(31,351,71)(32,352,72)(33,173,73)(34,174,74)(35,175,75)(36,176,76)(37,169,77)(38,170,78)(39,171,79)(40,172,80)(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)(161,241,201)(162,242,202)(163,243,203)(164,244,204)(165,245,205)(166,246,206)(167,247,207)(168,248,208)(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)(9,85,166)(10,86,167)(11,87,168)(12,88,161)(13,81,162)(14,82,163)(15,83,164)(16,84,165)(17,214,295)(18,215,296)(19,216,289)(20,209,290)(21,210,291)(22,211,292)(23,212,293)(24,213,294)(25,106,305)(26,107,306)(27,108,307)(28,109,308)(29,110,309)(30,111,310)(31,112,311)(32,105,312)(33,114,133)(34,115,134)(35,116,135)(36,117,136)(37,118,129)(38,119,130)(39,120,131)(40,113,132)(41,242,323)(42,243,324)(43,244,325)(44,245,326)(45,246,327)(46,247,328)(47,248,321)(48,241,322)(49,78,159)(50,79,160)(51,80,153)(52,73,154)(53,74,155)(54,75,156)(55,76,157)(56,77,158)(57,138,339)(58,139,340)(59,140,341)(60,141,342)(61,142,343)(62,143,344)(63,144,337)(64,137,338)(65,146,232)(66,147,225)(67,148,226)(68,149,227)(69,150,228)(70,151,229)(71,152,230)(72,145,231)(89,170,199)(90,171,200)(91,172,193)(92,173,194)(93,174,195)(94,175,196)(95,176,197)(96,169,198)(97,178,258)(98,179,259)(99,180,260)(100,181,261)(101,182,262)(102,183,263)(103,184,264)(104,177,257)(121,201,282)(122,202,283)(123,203,284)(124,204,285)(125,205,286)(126,206,287)(127,207,288)(128,208,281)(185,271,352)(186,272,345)(187,265,346)(188,266,347)(189,267,348)(190,268,349)(191,269,350)(192,270,351)(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), (1,106,23,287,115)(2,116,288,24,107)(3,108,17,281,117)(4,118,282,18,109)(5,110,19,283,119)(6,120,284,20,111)(7,112,21,285,113)(8,114,286,22,105)(9,195,99,186,276)(10,277,187,100,196)(11,197,101,188,278)(12,279,189,102,198)(13,199,103,190,280)(14,273,191,104,200)(15,193,97,192,274)(16,275,185,98,194)(25,293,206,34,301)(26,302,35,207,294)(27,295,208,36,303)(28,304,37,201,296)(29,289,202,38,297)(30,298,39,203,290)(31,291,204,40,299)(32,300,33,205,292)(41,49,144,228,313)(42,314,229,137,50)(43,51,138,230,315)(44,316,231,139,52)(45,53,140,232,317)(46,318,225,141,54)(47,55,142,226,319)(48,320,227,143,56)(57,152,234,325,153)(58,154,326,235,145)(59,146,236,327,155)(60,156,328,237,147)(61,148,238,321,157)(62,158,322,239,149)(63,150,240,323,159)(64,160,324,233,151)(65,333,246,74,341)(66,342,75,247,334)(67,335,248,76,343)(68,344,77,241,336)(69,329,242,78,337)(70,338,79,243,330)(71,331,244,80,339)(72,340,73,245,332)(81,89,184,268,353)(82,354,269,177,90)(83,91,178,270,355)(84,356,271,179,92)(85,93,180,272,357)(86,358,265,181,94)(87,95,182,266,359)(88,360,267,183,96)(121,215,308,223,129)(122,130,224,309,216)(123,209,310,217,131)(124,132,218,311,210)(125,211,312,219,133)(126,134,220,305,212)(127,213,306,221,135)(128,136,222,307,214)(161,255,348,263,169)(162,170,264,349,256)(163,249,350,257,171)(164,172,258,351,250)(165,251,352,259,173)(166,174,260,345,252)(167,253,346,261,175)(168,176,262,347,254), (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)(17,278,238)(18,279,239)(19,280,240)(20,273,233)(21,274,234)(22,275,235)(23,276,236)(24,277,237)(25,345,65)(26,346,66)(27,347,67)(28,348,68)(29,349,69)(30,350,70)(31,351,71)(32,352,72)(33,173,73)(34,174,74)(35,175,75)(36,176,76)(37,169,77)(38,170,78)(39,171,79)(40,172,80)(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)(161,241,201)(162,242,202)(163,243,203)(164,244,204)(165,245,205)(166,246,206)(167,247,207)(168,248,208)(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)(9,85,166)(10,86,167)(11,87,168)(12,88,161)(13,81,162)(14,82,163)(15,83,164)(16,84,165)(17,214,295)(18,215,296)(19,216,289)(20,209,290)(21,210,291)(22,211,292)(23,212,293)(24,213,294)(25,106,305)(26,107,306)(27,108,307)(28,109,308)(29,110,309)(30,111,310)(31,112,311)(32,105,312)(33,114,133)(34,115,134)(35,116,135)(36,117,136)(37,118,129)(38,119,130)(39,120,131)(40,113,132)(41,242,323)(42,243,324)(43,244,325)(44,245,326)(45,246,327)(46,247,328)(47,248,321)(48,241,322)(49,78,159)(50,79,160)(51,80,153)(52,73,154)(53,74,155)(54,75,156)(55,76,157)(56,77,158)(57,138,339)(58,139,340)(59,140,341)(60,141,342)(61,142,343)(62,143,344)(63,144,337)(64,137,338)(65,146,232)(66,147,225)(67,148,226)(68,149,227)(69,150,228)(70,151,229)(71,152,230)(72,145,231)(89,170,199)(90,171,200)(91,172,193)(92,173,194)(93,174,195)(94,175,196)(95,176,197)(96,169,198)(97,178,258)(98,179,259)(99,180,260)(100,181,261)(101,182,262)(102,183,263)(103,184,264)(104,177,257)(121,201,282)(122,202,283)(123,203,284)(124,204,285)(125,205,286)(126,206,287)(127,207,288)(128,208,281)(185,271,352)(186,272,345)(187,265,346)(188,266,347)(189,267,348)(190,268,349)(191,269,350)(192,270,351)(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), (1,106,23,287,115)(2,116,288,24,107)(3,108,17,281,117)(4,118,282,18,109)(5,110,19,283,119)(6,120,284,20,111)(7,112,21,285,113)(8,114,286,22,105)(9,195,99,186,276)(10,277,187,100,196)(11,197,101,188,278)(12,279,189,102,198)(13,199,103,190,280)(14,273,191,104,200)(15,193,97,192,274)(16,275,185,98,194)(25,293,206,34,301)(26,302,35,207,294)(27,295,208,36,303)(28,304,37,201,296)(29,289,202,38,297)(30,298,39,203,290)(31,291,204,40,299)(32,300,33,205,292)(41,49,144,228,313)(42,314,229,137,50)(43,51,138,230,315)(44,316,231,139,52)(45,53,140,232,317)(46,318,225,141,54)(47,55,142,226,319)(48,320,227,143,56)(57,152,234,325,153)(58,154,326,235,145)(59,146,236,327,155)(60,156,328,237,147)(61,148,238,321,157)(62,158,322,239,149)(63,150,240,323,159)(64,160,324,233,151)(65,333,246,74,341)(66,342,75,247,334)(67,335,248,76,343)(68,344,77,241,336)(69,329,242,78,337)(70,338,79,243,330)(71,331,244,80,339)(72,340,73,245,332)(81,89,184,268,353)(82,354,269,177,90)(83,91,178,270,355)(84,356,271,179,92)(85,93,180,272,357)(86,358,265,181,94)(87,95,182,266,359)(88,360,267,183,96)(121,215,308,223,129)(122,130,224,309,216)(123,209,310,217,131)(124,132,218,311,210)(125,211,312,219,133)(126,134,220,305,212)(127,213,306,221,135)(128,136,222,307,214)(161,255,348,263,169)(162,170,264,349,256)(163,249,350,257,171)(164,172,258,351,250)(165,251,352,259,173)(166,174,260,345,252)(167,253,346,261,175)(168,176,262,347,254), (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),(17,278,238),(18,279,239),(19,280,240),(20,273,233),(21,274,234),(22,275,235),(23,276,236),(24,277,237),(25,345,65),(26,346,66),(27,347,67),(28,348,68),(29,349,69),(30,350,70),(31,351,71),(32,352,72),(33,173,73),(34,174,74),(35,175,75),(36,176,76),(37,169,77),(38,170,78),(39,171,79),(40,172,80),(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),(161,241,201),(162,242,202),(163,243,203),(164,244,204),(165,245,205),(166,246,206),(167,247,207),(168,248,208),(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),(9,85,166),(10,86,167),(11,87,168),(12,88,161),(13,81,162),(14,82,163),(15,83,164),(16,84,165),(17,214,295),(18,215,296),(19,216,289),(20,209,290),(21,210,291),(22,211,292),(23,212,293),(24,213,294),(25,106,305),(26,107,306),(27,108,307),(28,109,308),(29,110,309),(30,111,310),(31,112,311),(32,105,312),(33,114,133),(34,115,134),(35,116,135),(36,117,136),(37,118,129),(38,119,130),(39,120,131),(40,113,132),(41,242,323),(42,243,324),(43,244,325),(44,245,326),(45,246,327),(46,247,328),(47,248,321),(48,241,322),(49,78,159),(50,79,160),(51,80,153),(52,73,154),(53,74,155),(54,75,156),(55,76,157),(56,77,158),(57,138,339),(58,139,340),(59,140,341),(60,141,342),(61,142,343),(62,143,344),(63,144,337),(64,137,338),(65,146,232),(66,147,225),(67,148,226),(68,149,227),(69,150,228),(70,151,229),(71,152,230),(72,145,231),(89,170,199),(90,171,200),(91,172,193),(92,173,194),(93,174,195),(94,175,196),(95,176,197),(96,169,198),(97,178,258),(98,179,259),(99,180,260),(100,181,261),(101,182,262),(102,183,263),(103,184,264),(104,177,257),(121,201,282),(122,202,283),(123,203,284),(124,204,285),(125,205,286),(126,206,287),(127,207,288),(128,208,281),(185,271,352),(186,272,345),(187,265,346),(188,266,347),(189,267,348),(190,268,349),(191,269,350),(192,270,351),(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)], [(1,106,23,287,115),(2,116,288,24,107),(3,108,17,281,117),(4,118,282,18,109),(5,110,19,283,119),(6,120,284,20,111),(7,112,21,285,113),(8,114,286,22,105),(9,195,99,186,276),(10,277,187,100,196),(11,197,101,188,278),(12,279,189,102,198),(13,199,103,190,280),(14,273,191,104,200),(15,193,97,192,274),(16,275,185,98,194),(25,293,206,34,301),(26,302,35,207,294),(27,295,208,36,303),(28,304,37,201,296),(29,289,202,38,297),(30,298,39,203,290),(31,291,204,40,299),(32,300,33,205,292),(41,49,144,228,313),(42,314,229,137,50),(43,51,138,230,315),(44,316,231,139,52),(45,53,140,232,317),(46,318,225,141,54),(47,55,142,226,319),(48,320,227,143,56),(57,152,234,325,153),(58,154,326,235,145),(59,146,236,327,155),(60,156,328,237,147),(61,148,238,321,157),(62,158,322,239,149),(63,150,240,323,159),(64,160,324,233,151),(65,333,246,74,341),(66,342,75,247,334),(67,335,248,76,343),(68,344,77,241,336),(69,329,242,78,337),(70,338,79,243,330),(71,331,244,80,339),(72,340,73,245,332),(81,89,184,268,353),(82,354,269,177,90),(83,91,178,270,355),(84,356,271,179,92),(85,93,180,272,357),(86,358,265,181,94),(87,95,182,266,359),(88,360,267,183,96),(121,215,308,223,129),(122,130,224,309,216),(123,209,310,217,131),(124,132,218,311,210),(125,211,312,219,133),(126,134,220,305,212),(127,213,306,221,135),(128,136,222,307,214),(161,255,348,263,169),(162,170,264,349,256),(163,249,350,257,171),(164,172,258,351,250),(165,251,352,259,173),(166,174,260,345,252),(167,253,346,261,175),(168,176,262,347,254)], [(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×C52C8 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×C52C8 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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