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## G = C32×Dic10order 360 = 23·32·5

### Direct product of C32 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C32×Dic10
 Chief series C1 — C5 — C10 — C30 — C3×C30 — C32×Dic5 — C32×Dic10
 Lower central C5 — C10 — C32×Dic10
 Upper central C1 — C3×C6 — C3×C12

Generators and relations for C32×Dic10
G = < a,b,c,d | a3=b3=c20=1, d2=c10, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 144 in 72 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C3, C4, C4, C5, C6, Q8, C32, C10, C12, C12, C15, C3×C6, Dic5, C20, C3×Q8, C30, C3×C12, C3×C12, Dic10, C3×C15, C3×Dic5, C60, Q8×C32, C3×C30, C3×Dic10, C32×Dic5, C3×C60, C32×Dic10
Quotients: C1, C2, C3, C22, C6, Q8, C32, D5, C2×C6, C3×C6, D10, C3×Q8, C3×D5, C62, Dic10, C6×D5, Q8×C32, C32×D5, C3×Dic10, D5×C3×C6, C32×Dic10

Smallest permutation representation of C32×Dic10
Regular action on 360 points
Generators in S360
(1 82 239)(2 83 240)(3 84 221)(4 85 222)(5 86 223)(6 87 224)(7 88 225)(8 89 226)(9 90 227)(10 91 228)(11 92 229)(12 93 230)(13 94 231)(14 95 232)(15 96 233)(16 97 234)(17 98 235)(18 99 236)(19 100 237)(20 81 238)(21 266 146)(22 267 147)(23 268 148)(24 269 149)(25 270 150)(26 271 151)(27 272 152)(28 273 153)(29 274 154)(30 275 155)(31 276 156)(32 277 157)(33 278 158)(34 279 159)(35 280 160)(36 261 141)(37 262 142)(38 263 143)(39 264 144)(40 265 145)(41 116 135)(42 117 136)(43 118 137)(44 119 138)(45 120 139)(46 101 140)(47 102 121)(48 103 122)(49 104 123)(50 105 124)(51 106 125)(52 107 126)(53 108 127)(54 109 128)(55 110 129)(56 111 130)(57 112 131)(58 113 132)(59 114 133)(60 115 134)(61 178 352)(62 179 353)(63 180 354)(64 161 355)(65 162 356)(66 163 357)(67 164 358)(68 165 359)(69 166 360)(70 167 341)(71 168 342)(72 169 343)(73 170 344)(74 171 345)(75 172 346)(76 173 347)(77 174 348)(78 175 349)(79 176 350)(80 177 351)(181 318 253)(182 319 254)(183 320 255)(184 301 256)(185 302 257)(186 303 258)(187 304 259)(188 305 260)(189 306 241)(190 307 242)(191 308 243)(192 309 244)(193 310 245)(194 311 246)(195 312 247)(196 313 248)(197 314 249)(198 315 250)(199 316 251)(200 317 252)(201 299 339)(202 300 340)(203 281 321)(204 282 322)(205 283 323)(206 284 324)(207 285 325)(208 286 326)(209 287 327)(210 288 328)(211 289 329)(212 290 330)(213 291 331)(214 292 332)(215 293 333)(216 294 334)(217 295 335)(218 296 336)(219 297 337)(220 298 338)
(1 246 325)(2 247 326)(3 248 327)(4 249 328)(5 250 329)(6 251 330)(7 252 331)(8 253 332)(9 254 333)(10 255 334)(11 256 335)(12 257 336)(13 258 337)(14 259 338)(15 260 339)(16 241 340)(17 242 321)(18 243 322)(19 244 323)(20 245 324)(21 107 163)(22 108 164)(23 109 165)(24 110 166)(25 111 167)(26 112 168)(27 113 169)(28 114 170)(29 115 171)(30 116 172)(31 117 173)(32 118 174)(33 119 175)(34 120 176)(35 101 177)(36 102 178)(37 103 179)(38 104 180)(39 105 161)(40 106 162)(41 75 155)(42 76 156)(43 77 157)(44 78 158)(45 79 159)(46 80 160)(47 61 141)(48 62 142)(49 63 143)(50 64 144)(51 65 145)(52 66 146)(53 67 147)(54 68 148)(55 69 149)(56 70 150)(57 71 151)(58 72 152)(59 73 153)(60 74 154)(81 193 206)(82 194 207)(83 195 208)(84 196 209)(85 197 210)(86 198 211)(87 199 212)(88 200 213)(89 181 214)(90 182 215)(91 183 216)(92 184 217)(93 185 218)(94 186 219)(95 187 220)(96 188 201)(97 189 202)(98 190 203)(99 191 204)(100 192 205)(121 352 261)(122 353 262)(123 354 263)(124 355 264)(125 356 265)(126 357 266)(127 358 267)(128 359 268)(129 360 269)(130 341 270)(131 342 271)(132 343 272)(133 344 273)(134 345 274)(135 346 275)(136 347 276)(137 348 277)(138 349 278)(139 350 279)(140 351 280)(221 313 287)(222 314 288)(223 315 289)(224 316 290)(225 317 291)(226 318 292)(227 319 293)(228 320 294)(229 301 295)(230 302 296)(231 303 297)(232 304 298)(233 305 299)(234 306 300)(235 307 281)(236 308 282)(237 309 283)(238 310 284)(239 311 285)(240 312 286)
(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)
(1 39 11 29)(2 38 12 28)(3 37 13 27)(4 36 14 26)(5 35 15 25)(6 34 16 24)(7 33 17 23)(8 32 18 22)(9 31 19 21)(10 30 20 40)(41 310 51 320)(42 309 52 319)(43 308 53 318)(44 307 54 317)(45 306 55 316)(46 305 56 315)(47 304 57 314)(48 303 58 313)(49 302 59 312)(50 301 60 311)(61 298 71 288)(62 297 72 287)(63 296 73 286)(64 295 74 285)(65 294 75 284)(66 293 76 283)(67 292 77 282)(68 291 78 281)(69 290 79 300)(70 289 80 299)(81 265 91 275)(82 264 92 274)(83 263 93 273)(84 262 94 272)(85 261 95 271)(86 280 96 270)(87 279 97 269)(88 278 98 268)(89 277 99 267)(90 276 100 266)(101 260 111 250)(102 259 112 249)(103 258 113 248)(104 257 114 247)(105 256 115 246)(106 255 116 245)(107 254 117 244)(108 253 118 243)(109 252 119 242)(110 251 120 241)(121 187 131 197)(122 186 132 196)(123 185 133 195)(124 184 134 194)(125 183 135 193)(126 182 136 192)(127 181 137 191)(128 200 138 190)(129 199 139 189)(130 198 140 188)(141 232 151 222)(142 231 152 221)(143 230 153 240)(144 229 154 239)(145 228 155 238)(146 227 156 237)(147 226 157 236)(148 225 158 235)(149 224 159 234)(150 223 160 233)(161 335 171 325)(162 334 172 324)(163 333 173 323)(164 332 174 322)(165 331 175 321)(166 330 176 340)(167 329 177 339)(168 328 178 338)(169 327 179 337)(170 326 180 336)(201 341 211 351)(202 360 212 350)(203 359 213 349)(204 358 214 348)(205 357 215 347)(206 356 216 346)(207 355 217 345)(208 354 218 344)(209 353 219 343)(210 352 220 342)

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

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

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

117 conjugacy classes

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

117 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - + + - image C1 C2 C2 C3 C6 C6 Q8 D5 D10 C3×Q8 C3×D5 Dic10 C6×D5 C3×Dic10 kernel C32×Dic10 C32×Dic5 C3×C60 C3×Dic10 C3×Dic5 C60 C3×C15 C3×C12 C3×C6 C15 C12 C32 C6 C3 # reps 1 2 1 8 16 8 1 2 2 8 16 4 16 32

Matrix representation of C32×Dic10 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 13 0 0 0 0 13 0 0 0 0 1 0 0 0 0 1
,
 0 60 0 0 1 44 0 0 0 0 1 59 0 0 1 60
,
 31 36 0 0 14 30 0 0 0 0 42 11 0 0 17 19
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,60,44,0,0,0,0,1,1,0,0,59,60],[31,14,0,0,36,30,0,0,0,0,42,17,0,0,11,19] >;

C32×Dic10 in GAP, Magma, Sage, TeX

C_3^2\times {\rm Dic}_{10}
% in TeX

G:=Group("C3^2xDic10");
// GroupNames label

G:=SmallGroup(360,90);
// by ID

G=gap.SmallGroup(360,90);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-5,216,457,223,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^20=1,d^2=c^10,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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