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## G = C10×C3⋊Dic3order 360 = 23·32·5

### Direct product of C10 and C3⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C10×C3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C30 — C5×C3⋊Dic3 — C10×C3⋊Dic3
 Lower central C32 — C10×C3⋊Dic3
 Upper central C1 — C2×C10

Generators and relations for C10×C3⋊Dic3
G = < a,b,c,d | a10=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 192 in 96 conjugacy classes, 66 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C2×C4, C32, C10, C10, Dic3, C2×C6, C15, C3×C6, C3×C6, C20, C2×C10, C2×Dic3, C30, C3⋊Dic3, C62, C2×C20, C3×C15, C5×Dic3, C2×C30, C2×C3⋊Dic3, C3×C30, C3×C30, C10×Dic3, C5×C3⋊Dic3, C6×C30, C10×C3⋊Dic3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C10, Dic3, D6, C3⋊S3, C20, C2×C10, C2×Dic3, C5×S3, C3⋊Dic3, C2×C3⋊S3, C2×C20, C5×Dic3, S3×C10, C2×C3⋊Dic3, C5×C3⋊S3, C10×Dic3, C5×C3⋊Dic3, C10×C3⋊S3, C10×C3⋊Dic3

Smallest permutation representation of C10×C3⋊Dic3
Regular action on 360 points
Generators in S360
(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)
(1 258 110)(2 259 101)(3 260 102)(4 251 103)(5 252 104)(6 253 105)(7 254 106)(8 255 107)(9 256 108)(10 257 109)(11 242 176)(12 243 177)(13 244 178)(14 245 179)(15 246 180)(16 247 171)(17 248 172)(18 249 173)(19 250 174)(20 241 175)(21 214 146)(22 215 147)(23 216 148)(24 217 149)(25 218 150)(26 219 141)(27 220 142)(28 211 143)(29 212 144)(30 213 145)(31 343 305)(32 344 306)(33 345 307)(34 346 308)(35 347 309)(36 348 310)(37 349 301)(38 350 302)(39 341 303)(40 342 304)(41 234 125)(42 235 126)(43 236 127)(44 237 128)(45 238 129)(46 239 130)(47 240 121)(48 231 122)(49 232 123)(50 233 124)(51 94 323)(52 95 324)(53 96 325)(54 97 326)(55 98 327)(56 99 328)(57 100 329)(58 91 330)(59 92 321)(60 93 322)(61 275 317)(62 276 318)(63 277 319)(64 278 320)(65 279 311)(66 280 312)(67 271 313)(68 272 314)(69 273 315)(70 274 316)(71 202 136)(72 203 137)(73 204 138)(74 205 139)(75 206 140)(76 207 131)(77 208 132)(78 209 133)(79 210 134)(80 201 135)(81 291 339)(82 292 340)(83 293 331)(84 294 332)(85 295 333)(86 296 334)(87 297 335)(88 298 336)(89 299 337)(90 300 338)(111 358 185)(112 359 186)(113 360 187)(114 351 188)(115 352 189)(116 353 190)(117 354 181)(118 355 182)(119 356 183)(120 357 184)(151 263 225)(152 264 226)(153 265 227)(154 266 228)(155 267 229)(156 268 230)(157 269 221)(158 270 222)(159 261 223)(160 262 224)(161 199 282)(162 200 283)(163 191 284)(164 192 285)(165 193 286)(166 194 287)(167 195 288)(168 196 289)(169 197 290)(170 198 281)
(1 150 67 121 198 295)(2 141 68 122 199 296)(3 142 69 123 200 297)(4 143 70 124 191 298)(5 144 61 125 192 299)(6 145 62 126 193 300)(7 146 63 127 194 291)(8 147 64 128 195 292)(9 148 65 129 196 293)(10 149 66 130 197 294)(11 96 71 222 307 182)(12 97 72 223 308 183)(13 98 73 224 309 184)(14 99 74 225 310 185)(15 100 75 226 301 186)(16 91 76 227 302 187)(17 92 77 228 303 188)(18 93 78 229 304 189)(19 94 79 230 305 190)(20 95 80 221 306 181)(21 277 43 287 339 254)(22 278 44 288 340 255)(23 279 45 289 331 256)(24 280 46 290 332 257)(25 271 47 281 333 258)(26 272 48 282 334 259)(27 273 49 283 335 260)(28 274 50 284 336 251)(29 275 41 285 337 252)(30 276 42 286 338 253)(31 116 250 323 210 156)(32 117 241 324 201 157)(33 118 242 325 202 158)(34 119 243 326 203 159)(35 120 244 327 204 160)(36 111 245 328 205 151)(37 112 246 329 206 152)(38 113 247 330 207 153)(39 114 248 321 208 154)(40 115 249 322 209 155)(51 134 268 343 353 174)(52 135 269 344 354 175)(53 136 270 345 355 176)(54 137 261 346 356 177)(55 138 262 347 357 178)(56 139 263 348 358 179)(57 140 264 349 359 180)(58 131 265 350 360 171)(59 132 266 341 351 172)(60 133 267 342 352 173)(81 106 214 319 236 166)(82 107 215 320 237 167)(83 108 216 311 238 168)(84 109 217 312 239 169)(85 110 218 313 240 170)(86 101 219 314 231 161)(87 102 220 315 232 162)(88 103 211 316 233 163)(89 104 212 317 234 164)(90 105 213 318 235 165)
(1 225 121 14)(2 226 122 15)(3 227 123 16)(4 228 124 17)(5 229 125 18)(6 230 126 19)(7 221 127 20)(8 222 128 11)(9 223 129 12)(10 224 130 13)(21 135 287 354)(22 136 288 355)(23 137 289 356)(24 138 290 357)(25 139 281 358)(26 140 282 359)(27 131 283 360)(28 132 284 351)(29 133 285 352)(30 134 286 353)(31 318 323 90)(32 319 324 81)(33 320 325 82)(34 311 326 83)(35 312 327 84)(36 313 328 85)(37 314 329 86)(38 315 330 87)(39 316 321 88)(40 317 322 89)(41 173 252 267)(42 174 253 268)(43 175 254 269)(44 176 255 270)(45 177 256 261)(46 178 257 262)(47 179 258 263)(48 180 259 264)(49 171 260 265)(50 172 251 266)(51 338 343 276)(52 339 344 277)(53 340 345 278)(54 331 346 279)(55 332 347 280)(56 333 348 271)(57 334 349 272)(58 335 350 273)(59 336 341 274)(60 337 342 275)(61 93 299 304)(62 94 300 305)(63 95 291 306)(64 96 292 307)(65 97 293 308)(66 98 294 309)(67 99 295 310)(68 100 296 301)(69 91 297 302)(70 92 298 303)(71 195 182 147)(72 196 183 148)(73 197 184 149)(74 198 185 150)(75 199 186 141)(76 200 187 142)(77 191 188 143)(78 192 189 144)(79 193 190 145)(80 194 181 146)(101 152 231 246)(102 153 232 247)(103 154 233 248)(104 155 234 249)(105 156 235 250)(106 157 236 241)(107 158 237 242)(108 159 238 243)(109 160 239 244)(110 151 240 245)(111 218 205 170)(112 219 206 161)(113 220 207 162)(114 211 208 163)(115 212 209 164)(116 213 210 165)(117 214 201 166)(118 215 202 167)(119 216 203 168)(120 217 204 169)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 5A 5B 5C 5D 6A ··· 6L 10A ··· 10L 15A ··· 15P 20A ··· 20P 30A ··· 30AV order 1 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 ··· 6 10 ··· 10 15 ··· 15 20 ··· 20 30 ··· 30 size 1 1 1 1 2 2 2 2 9 9 9 9 1 1 1 1 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 C2 C4 C5 C10 C10 C20 S3 Dic3 D6 C5×S3 C5×Dic3 S3×C10 kernel C10×C3⋊Dic3 C5×C3⋊Dic3 C6×C30 C3×C30 C2×C3⋊Dic3 C3⋊Dic3 C62 C3×C6 C2×C30 C30 C30 C2×C6 C6 C6 # reps 1 2 1 4 4 8 4 16 4 8 4 16 32 16

Matrix representation of C10×C3⋊Dic3 in GL4(𝔽61) generated by

 27 0 0 0 0 27 0 0 0 0 3 0 0 0 0 3
,
 0 1 0 0 60 60 0 0 0 0 0 1 0 0 60 60
,
 0 60 0 0 1 1 0 0 0 0 60 60 0 0 1 0
,
 31 27 0 0 57 30 0 0 0 0 9 48 0 0 39 52
G:=sub<GL(4,GF(61))| [27,0,0,0,0,27,0,0,0,0,3,0,0,0,0,3],[0,60,0,0,1,60,0,0,0,0,0,60,0,0,1,60],[0,1,0,0,60,1,0,0,0,0,60,1,0,0,60,0],[31,57,0,0,27,30,0,0,0,0,9,39,0,0,48,52] >;

C10×C3⋊Dic3 in GAP, Magma, Sage, TeX

C_{10}\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("C10xC3:Dic3");
// GroupNames label

G:=SmallGroup(360,108);
// by ID

G=gap.SmallGroup(360,108);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,120,2404,8645]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^3=c^6=1,d^2=c^3,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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