direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C3⋊Dic15, C6⋊Dic15, C30⋊3Dic3, C6.17D30, C30.49D6, C62.2D5, (C3×C30)⋊7C4, (C2×C30).6S3, (C6×C30).1C2, (C3×C6)⋊3Dic5, (C2×C6).5D15, C3⋊2(C2×Dic15), (C3×C6).35D10, C22.(C3⋊D15), C32⋊7(C2×Dic5), C10⋊2(C3⋊Dic3), C15⋊10(C2×Dic3), (C3×C30).35C22, C5⋊3(C2×C3⋊Dic3), (C3×C15)⋊31(C2×C4), (C2×C10).(C3⋊S3), C2.2(C2×C3⋊D15), C10.11(C2×C3⋊S3), SmallGroup(360,113)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — C2×C3⋊Dic15 |
Generators and relations for C2×C3⋊Dic15
G = < a,b,c,d | a2=b3=c30=1, d2=c15, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 456 in 96 conjugacy classes, 63 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C2×C4, C32, C10, C10, Dic3, C2×C6, C15, C3×C6, C3×C6, Dic5, C2×C10, C2×Dic3, C30, C3⋊Dic3, C62, C2×Dic5, C3×C15, Dic15, C2×C30, C2×C3⋊Dic3, C3×C30, C3×C30, C2×Dic15, C3⋊Dic15, C6×C30, C2×C3⋊Dic15
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, C3⋊S3, Dic5, D10, C2×Dic3, D15, C3⋊Dic3, C2×C3⋊S3, C2×Dic5, Dic15, D30, C2×C3⋊Dic3, C3⋊D15, C2×Dic15, C3⋊Dic15, C2×C3⋊D15, C2×C3⋊Dic15
►Regular action on 360 points
Generators in S360
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 31)(27 32)(28 33)(29 34)(30 35)(61 306)(62 307)(63 308)(64 309)(65 310)(66 311)(67 312)(68 313)(69 314)(70 315)(71 316)(72 317)(73 318)(74 319)(75 320)(76 321)(77 322)(78 323)(79 324)(80 325)(81 326)(82 327)(83 328)(84 329)(85 330)(86 301)(87 302)(88 303)(89 304)(90 305)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 128)(106 129)(107 130)(108 131)(109 132)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(116 139)(117 140)(118 141)(119 142)(120 143)(151 211)(152 212)(153 213)(154 214)(155 215)(156 216)(157 217)(158 218)(159 219)(160 220)(161 221)(162 222)(163 223)(164 224)(165 225)(166 226)(167 227)(168 228)(169 229)(170 230)(171 231)(172 232)(173 233)(174 234)(175 235)(176 236)(177 237)(178 238)(179 239)(180 240)(181 275)(182 276)(183 277)(184 278)(185 279)(186 280)(187 281)(188 282)(189 283)(190 284)(191 285)(192 286)(193 287)(194 288)(195 289)(196 290)(197 291)(198 292)(199 293)(200 294)(201 295)(202 296)(203 297)(204 298)(205 299)(206 300)(207 271)(208 272)(209 273)(210 274)(241 342)(242 343)(243 344)(244 345)(245 346)(246 347)(247 348)(248 349)(249 350)(250 351)(251 352)(252 353)(253 354)(254 355)(255 356)(256 357)(257 358)(258 359)(259 360)(260 331)(261 332)(262 333)(263 334)(264 335)(265 336)(266 337)(267 338)(268 339)(269 340)(270 341)
(1 227 64)(2 228 65)(3 229 66)(4 230 67)(5 231 68)(6 232 69)(7 233 70)(8 234 71)(9 235 72)(10 236 73)(11 237 74)(12 238 75)(13 239 76)(14 240 77)(15 211 78)(16 212 79)(17 213 80)(18 214 81)(19 215 82)(20 216 83)(21 217 84)(22 218 85)(23 219 86)(24 220 87)(25 221 88)(26 222 89)(27 223 90)(28 224 61)(29 225 62)(30 226 63)(31 162 304)(32 163 305)(33 164 306)(34 165 307)(35 166 308)(36 167 309)(37 168 310)(38 169 311)(39 170 312)(40 171 313)(41 172 314)(42 173 315)(43 174 316)(44 175 317)(45 176 318)(46 177 319)(47 178 320)(48 179 321)(49 180 322)(50 151 323)(51 152 324)(52 153 325)(53 154 326)(54 155 327)(55 156 328)(56 157 329)(57 158 330)(58 159 301)(59 160 302)(60 161 303)(91 280 351)(92 281 352)(93 282 353)(94 283 354)(95 284 355)(96 285 356)(97 286 357)(98 287 358)(99 288 359)(100 289 360)(101 290 331)(102 291 332)(103 292 333)(104 293 334)(105 294 335)(106 295 336)(107 296 337)(108 297 338)(109 298 339)(110 299 340)(111 300 341)(112 271 342)(113 272 343)(114 273 344)(115 274 345)(116 275 346)(117 276 347)(118 277 348)(119 278 349)(120 279 350)(121 193 257)(122 194 258)(123 195 259)(124 196 260)(125 197 261)(126 198 262)(127 199 263)(128 200 264)(129 201 265)(130 202 266)(131 203 267)(132 204 268)(133 205 269)(134 206 270)(135 207 241)(136 208 242)(137 209 243)(138 210 244)(139 181 245)(140 182 246)(141 183 247)(142 184 248)(143 185 249)(144 186 250)(145 187 251)(146 188 252)(147 189 253)(148 190 254)(149 191 255)(150 192 256)
(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)
(1 332 16 347)(2 331 17 346)(3 360 18 345)(4 359 19 344)(5 358 20 343)(6 357 21 342)(7 356 22 341)(8 355 23 340)(9 354 24 339)(10 353 25 338)(11 352 26 337)(12 351 27 336)(13 350 28 335)(14 349 29 334)(15 348 30 333)(31 266 46 251)(32 265 47 250)(33 264 48 249)(34 263 49 248)(35 262 50 247)(36 261 51 246)(37 260 52 245)(38 259 53 244)(39 258 54 243)(40 257 55 242)(41 256 56 241)(42 255 57 270)(43 254 58 269)(44 253 59 268)(45 252 60 267)(61 105 76 120)(62 104 77 119)(63 103 78 118)(64 102 79 117)(65 101 80 116)(66 100 81 115)(67 99 82 114)(68 98 83 113)(69 97 84 112)(70 96 85 111)(71 95 86 110)(72 94 87 109)(73 93 88 108)(74 92 89 107)(75 91 90 106)(121 328 136 313)(122 327 137 312)(123 326 138 311)(124 325 139 310)(125 324 140 309)(126 323 141 308)(127 322 142 307)(128 321 143 306)(129 320 144 305)(130 319 145 304)(131 318 146 303)(132 317 147 302)(133 316 148 301)(134 315 149 330)(135 314 150 329)(151 183 166 198)(152 182 167 197)(153 181 168 196)(154 210 169 195)(155 209 170 194)(156 208 171 193)(157 207 172 192)(158 206 173 191)(159 205 174 190)(160 204 175 189)(161 203 176 188)(162 202 177 187)(163 201 178 186)(164 200 179 185)(165 199 180 184)(211 277 226 292)(212 276 227 291)(213 275 228 290)(214 274 229 289)(215 273 230 288)(216 272 231 287)(217 271 232 286)(218 300 233 285)(219 299 234 284)(220 298 235 283)(221 297 236 282)(222 296 237 281)(223 295 238 280)(224 294 239 279)(225 293 240 278)
G:=sub<Sym(360)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,31)(27,32)(28,33)(29,34)(30,35)(61,306)(62,307)(63,308)(64,309)(65,310)(66,311)(67,312)(68,313)(69,314)(70,315)(71,316)(72,317)(73,318)(74,319)(75,320)(76,321)(77,322)(78,323)(79,324)(80,325)(81,326)(82,327)(83,328)(84,329)(85,330)(86,301)(87,302)(88,303)(89,304)(90,305)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,141)(119,142)(120,143)(151,211)(152,212)(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240)(181,275)(182,276)(183,277)(184,278)(185,279)(186,280)(187,281)(188,282)(189,283)(190,284)(191,285)(192,286)(193,287)(194,288)(195,289)(196,290)(197,291)(198,292)(199,293)(200,294)(201,295)(202,296)(203,297)(204,298)(205,299)(206,300)(207,271)(208,272)(209,273)(210,274)(241,342)(242,343)(243,344)(244,345)(245,346)(246,347)(247,348)(248,349)(249,350)(250,351)(251,352)(252,353)(253,354)(254,355)(255,356)(256,357)(257,358)(258,359)(259,360)(260,331)(261,332)(262,333)(263,334)(264,335)(265,336)(266,337)(267,338)(268,339)(269,340)(270,341), (1,227,64)(2,228,65)(3,229,66)(4,230,67)(5,231,68)(6,232,69)(7,233,70)(8,234,71)(9,235,72)(10,236,73)(11,237,74)(12,238,75)(13,239,76)(14,240,77)(15,211,78)(16,212,79)(17,213,80)(18,214,81)(19,215,82)(20,216,83)(21,217,84)(22,218,85)(23,219,86)(24,220,87)(25,221,88)(26,222,89)(27,223,90)(28,224,61)(29,225,62)(30,226,63)(31,162,304)(32,163,305)(33,164,306)(34,165,307)(35,166,308)(36,167,309)(37,168,310)(38,169,311)(39,170,312)(40,171,313)(41,172,314)(42,173,315)(43,174,316)(44,175,317)(45,176,318)(46,177,319)(47,178,320)(48,179,321)(49,180,322)(50,151,323)(51,152,324)(52,153,325)(53,154,326)(54,155,327)(55,156,328)(56,157,329)(57,158,330)(58,159,301)(59,160,302)(60,161,303)(91,280,351)(92,281,352)(93,282,353)(94,283,354)(95,284,355)(96,285,356)(97,286,357)(98,287,358)(99,288,359)(100,289,360)(101,290,331)(102,291,332)(103,292,333)(104,293,334)(105,294,335)(106,295,336)(107,296,337)(108,297,338)(109,298,339)(110,299,340)(111,300,341)(112,271,342)(113,272,343)(114,273,344)(115,274,345)(116,275,346)(117,276,347)(118,277,348)(119,278,349)(120,279,350)(121,193,257)(122,194,258)(123,195,259)(124,196,260)(125,197,261)(126,198,262)(127,199,263)(128,200,264)(129,201,265)(130,202,266)(131,203,267)(132,204,268)(133,205,269)(134,206,270)(135,207,241)(136,208,242)(137,209,243)(138,210,244)(139,181,245)(140,182,246)(141,183,247)(142,184,248)(143,185,249)(144,186,250)(145,187,251)(146,188,252)(147,189,253)(148,190,254)(149,191,255)(150,192,256), (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), (1,332,16,347)(2,331,17,346)(3,360,18,345)(4,359,19,344)(5,358,20,343)(6,357,21,342)(7,356,22,341)(8,355,23,340)(9,354,24,339)(10,353,25,338)(11,352,26,337)(12,351,27,336)(13,350,28,335)(14,349,29,334)(15,348,30,333)(31,266,46,251)(32,265,47,250)(33,264,48,249)(34,263,49,248)(35,262,50,247)(36,261,51,246)(37,260,52,245)(38,259,53,244)(39,258,54,243)(40,257,55,242)(41,256,56,241)(42,255,57,270)(43,254,58,269)(44,253,59,268)(45,252,60,267)(61,105,76,120)(62,104,77,119)(63,103,78,118)(64,102,79,117)(65,101,80,116)(66,100,81,115)(67,99,82,114)(68,98,83,113)(69,97,84,112)(70,96,85,111)(71,95,86,110)(72,94,87,109)(73,93,88,108)(74,92,89,107)(75,91,90,106)(121,328,136,313)(122,327,137,312)(123,326,138,311)(124,325,139,310)(125,324,140,309)(126,323,141,308)(127,322,142,307)(128,321,143,306)(129,320,144,305)(130,319,145,304)(131,318,146,303)(132,317,147,302)(133,316,148,301)(134,315,149,330)(135,314,150,329)(151,183,166,198)(152,182,167,197)(153,181,168,196)(154,210,169,195)(155,209,170,194)(156,208,171,193)(157,207,172,192)(158,206,173,191)(159,205,174,190)(160,204,175,189)(161,203,176,188)(162,202,177,187)(163,201,178,186)(164,200,179,185)(165,199,180,184)(211,277,226,292)(212,276,227,291)(213,275,228,290)(214,274,229,289)(215,273,230,288)(216,272,231,287)(217,271,232,286)(218,300,233,285)(219,299,234,284)(220,298,235,283)(221,297,236,282)(222,296,237,281)(223,295,238,280)(224,294,239,279)(225,293,240,278)>;
G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,31)(27,32)(28,33)(29,34)(30,35)(61,306)(62,307)(63,308)(64,309)(65,310)(66,311)(67,312)(68,313)(69,314)(70,315)(71,316)(72,317)(73,318)(74,319)(75,320)(76,321)(77,322)(78,323)(79,324)(80,325)(81,326)(82,327)(83,328)(84,329)(85,330)(86,301)(87,302)(88,303)(89,304)(90,305)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,141)(119,142)(120,143)(151,211)(152,212)(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240)(181,275)(182,276)(183,277)(184,278)(185,279)(186,280)(187,281)(188,282)(189,283)(190,284)(191,285)(192,286)(193,287)(194,288)(195,289)(196,290)(197,291)(198,292)(199,293)(200,294)(201,295)(202,296)(203,297)(204,298)(205,299)(206,300)(207,271)(208,272)(209,273)(210,274)(241,342)(242,343)(243,344)(244,345)(245,346)(246,347)(247,348)(248,349)(249,350)(250,351)(251,352)(252,353)(253,354)(254,355)(255,356)(256,357)(257,358)(258,359)(259,360)(260,331)(261,332)(262,333)(263,334)(264,335)(265,336)(266,337)(267,338)(268,339)(269,340)(270,341), (1,227,64)(2,228,65)(3,229,66)(4,230,67)(5,231,68)(6,232,69)(7,233,70)(8,234,71)(9,235,72)(10,236,73)(11,237,74)(12,238,75)(13,239,76)(14,240,77)(15,211,78)(16,212,79)(17,213,80)(18,214,81)(19,215,82)(20,216,83)(21,217,84)(22,218,85)(23,219,86)(24,220,87)(25,221,88)(26,222,89)(27,223,90)(28,224,61)(29,225,62)(30,226,63)(31,162,304)(32,163,305)(33,164,306)(34,165,307)(35,166,308)(36,167,309)(37,168,310)(38,169,311)(39,170,312)(40,171,313)(41,172,314)(42,173,315)(43,174,316)(44,175,317)(45,176,318)(46,177,319)(47,178,320)(48,179,321)(49,180,322)(50,151,323)(51,152,324)(52,153,325)(53,154,326)(54,155,327)(55,156,328)(56,157,329)(57,158,330)(58,159,301)(59,160,302)(60,161,303)(91,280,351)(92,281,352)(93,282,353)(94,283,354)(95,284,355)(96,285,356)(97,286,357)(98,287,358)(99,288,359)(100,289,360)(101,290,331)(102,291,332)(103,292,333)(104,293,334)(105,294,335)(106,295,336)(107,296,337)(108,297,338)(109,298,339)(110,299,340)(111,300,341)(112,271,342)(113,272,343)(114,273,344)(115,274,345)(116,275,346)(117,276,347)(118,277,348)(119,278,349)(120,279,350)(121,193,257)(122,194,258)(123,195,259)(124,196,260)(125,197,261)(126,198,262)(127,199,263)(128,200,264)(129,201,265)(130,202,266)(131,203,267)(132,204,268)(133,205,269)(134,206,270)(135,207,241)(136,208,242)(137,209,243)(138,210,244)(139,181,245)(140,182,246)(141,183,247)(142,184,248)(143,185,249)(144,186,250)(145,187,251)(146,188,252)(147,189,253)(148,190,254)(149,191,255)(150,192,256), (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), (1,332,16,347)(2,331,17,346)(3,360,18,345)(4,359,19,344)(5,358,20,343)(6,357,21,342)(7,356,22,341)(8,355,23,340)(9,354,24,339)(10,353,25,338)(11,352,26,337)(12,351,27,336)(13,350,28,335)(14,349,29,334)(15,348,30,333)(31,266,46,251)(32,265,47,250)(33,264,48,249)(34,263,49,248)(35,262,50,247)(36,261,51,246)(37,260,52,245)(38,259,53,244)(39,258,54,243)(40,257,55,242)(41,256,56,241)(42,255,57,270)(43,254,58,269)(44,253,59,268)(45,252,60,267)(61,105,76,120)(62,104,77,119)(63,103,78,118)(64,102,79,117)(65,101,80,116)(66,100,81,115)(67,99,82,114)(68,98,83,113)(69,97,84,112)(70,96,85,111)(71,95,86,110)(72,94,87,109)(73,93,88,108)(74,92,89,107)(75,91,90,106)(121,328,136,313)(122,327,137,312)(123,326,138,311)(124,325,139,310)(125,324,140,309)(126,323,141,308)(127,322,142,307)(128,321,143,306)(129,320,144,305)(130,319,145,304)(131,318,146,303)(132,317,147,302)(133,316,148,301)(134,315,149,330)(135,314,150,329)(151,183,166,198)(152,182,167,197)(153,181,168,196)(154,210,169,195)(155,209,170,194)(156,208,171,193)(157,207,172,192)(158,206,173,191)(159,205,174,190)(160,204,175,189)(161,203,176,188)(162,202,177,187)(163,201,178,186)(164,200,179,185)(165,199,180,184)(211,277,226,292)(212,276,227,291)(213,275,228,290)(214,274,229,289)(215,273,230,288)(216,272,231,287)(217,271,232,286)(218,300,233,285)(219,299,234,284)(220,298,235,283)(221,297,236,282)(222,296,237,281)(223,295,238,280)(224,294,239,279)(225,293,240,278) );
G=PermutationGroup([[(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,57),(23,58),(24,59),(25,60),(26,31),(27,32),(28,33),(29,34),(30,35),(61,306),(62,307),(63,308),(64,309),(65,310),(66,311),(67,312),(68,313),(69,314),(70,315),(71,316),(72,317),(73,318),(74,319),(75,320),(76,321),(77,322),(78,323),(79,324),(80,325),(81,326),(82,327),(83,328),(84,329),(85,330),(86,301),(87,302),(88,303),(89,304),(90,305),(91,144),(92,145),(93,146),(94,147),(95,148),(96,149),(97,150),(98,121),(99,122),(100,123),(101,124),(102,125),(103,126),(104,127),(105,128),(106,129),(107,130),(108,131),(109,132),(110,133),(111,134),(112,135),(113,136),(114,137),(115,138),(116,139),(117,140),(118,141),(119,142),(120,143),(151,211),(152,212),(153,213),(154,214),(155,215),(156,216),(157,217),(158,218),(159,219),(160,220),(161,221),(162,222),(163,223),(164,224),(165,225),(166,226),(167,227),(168,228),(169,229),(170,230),(171,231),(172,232),(173,233),(174,234),(175,235),(176,236),(177,237),(178,238),(179,239),(180,240),(181,275),(182,276),(183,277),(184,278),(185,279),(186,280),(187,281),(188,282),(189,283),(190,284),(191,285),(192,286),(193,287),(194,288),(195,289),(196,290),(197,291),(198,292),(199,293),(200,294),(201,295),(202,296),(203,297),(204,298),(205,299),(206,300),(207,271),(208,272),(209,273),(210,274),(241,342),(242,343),(243,344),(244,345),(245,346),(246,347),(247,348),(248,349),(249,350),(250,351),(251,352),(252,353),(253,354),(254,355),(255,356),(256,357),(257,358),(258,359),(259,360),(260,331),(261,332),(262,333),(263,334),(264,335),(265,336),(266,337),(267,338),(268,339),(269,340),(270,341)], [(1,227,64),(2,228,65),(3,229,66),(4,230,67),(5,231,68),(6,232,69),(7,233,70),(8,234,71),(9,235,72),(10,236,73),(11,237,74),(12,238,75),(13,239,76),(14,240,77),(15,211,78),(16,212,79),(17,213,80),(18,214,81),(19,215,82),(20,216,83),(21,217,84),(22,218,85),(23,219,86),(24,220,87),(25,221,88),(26,222,89),(27,223,90),(28,224,61),(29,225,62),(30,226,63),(31,162,304),(32,163,305),(33,164,306),(34,165,307),(35,166,308),(36,167,309),(37,168,310),(38,169,311),(39,170,312),(40,171,313),(41,172,314),(42,173,315),(43,174,316),(44,175,317),(45,176,318),(46,177,319),(47,178,320),(48,179,321),(49,180,322),(50,151,323),(51,152,324),(52,153,325),(53,154,326),(54,155,327),(55,156,328),(56,157,329),(57,158,330),(58,159,301),(59,160,302),(60,161,303),(91,280,351),(92,281,352),(93,282,353),(94,283,354),(95,284,355),(96,285,356),(97,286,357),(98,287,358),(99,288,359),(100,289,360),(101,290,331),(102,291,332),(103,292,333),(104,293,334),(105,294,335),(106,295,336),(107,296,337),(108,297,338),(109,298,339),(110,299,340),(111,300,341),(112,271,342),(113,272,343),(114,273,344),(115,274,345),(116,275,346),(117,276,347),(118,277,348),(119,278,349),(120,279,350),(121,193,257),(122,194,258),(123,195,259),(124,196,260),(125,197,261),(126,198,262),(127,199,263),(128,200,264),(129,201,265),(130,202,266),(131,203,267),(132,204,268),(133,205,269),(134,206,270),(135,207,241),(136,208,242),(137,209,243),(138,210,244),(139,181,245),(140,182,246),(141,183,247),(142,184,248),(143,185,249),(144,186,250),(145,187,251),(146,188,252),(147,189,253),(148,190,254),(149,191,255),(150,192,256)], [(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)], [(1,332,16,347),(2,331,17,346),(3,360,18,345),(4,359,19,344),(5,358,20,343),(6,357,21,342),(7,356,22,341),(8,355,23,340),(9,354,24,339),(10,353,25,338),(11,352,26,337),(12,351,27,336),(13,350,28,335),(14,349,29,334),(15,348,30,333),(31,266,46,251),(32,265,47,250),(33,264,48,249),(34,263,49,248),(35,262,50,247),(36,261,51,246),(37,260,52,245),(38,259,53,244),(39,258,54,243),(40,257,55,242),(41,256,56,241),(42,255,57,270),(43,254,58,269),(44,253,59,268),(45,252,60,267),(61,105,76,120),(62,104,77,119),(63,103,78,118),(64,102,79,117),(65,101,80,116),(66,100,81,115),(67,99,82,114),(68,98,83,113),(69,97,84,112),(70,96,85,111),(71,95,86,110),(72,94,87,109),(73,93,88,108),(74,92,89,107),(75,91,90,106),(121,328,136,313),(122,327,137,312),(123,326,138,311),(124,325,139,310),(125,324,140,309),(126,323,141,308),(127,322,142,307),(128,321,143,306),(129,320,144,305),(130,319,145,304),(131,318,146,303),(132,317,147,302),(133,316,148,301),(134,315,149,330),(135,314,150,329),(151,183,166,198),(152,182,167,197),(153,181,168,196),(154,210,169,195),(155,209,170,194),(156,208,171,193),(157,207,172,192),(158,206,173,191),(159,205,174,190),(160,204,175,189),(161,203,176,188),(162,202,177,187),(163,201,178,186),(164,200,179,185),(165,199,180,184),(211,277,226,292),(212,276,227,291),(213,275,228,290),(214,274,229,289),(215,273,230,288),(216,272,231,287),(217,271,232,286),(218,300,233,285),(219,299,234,284),(220,298,235,283),(221,297,236,282),(222,296,237,281),(223,295,238,280),(224,294,239,279),(225,293,240,278)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6L | 10A | ··· | 10F | 15A | ··· | 15P | 30A | ··· | 30AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 45 | 45 | 45 | 45 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | D5 | Dic3 | D6 | Dic5 | D10 | D15 | Dic15 | D30 |
| kernel | C2×C3⋊Dic15 | C3⋊Dic15 | C6×C30 | C3×C30 | C2×C30 | C62 | C30 | C30 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 |
| # reps | 1 | 2 | 1 | 4 | 4 | 2 | 8 | 4 | 4 | 2 | 16 | 32 | 16 |
Matrix representation of C2×C3⋊Dic15 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 52 | 16 | 0 | 0 |
| 45 | 8 | 0 | 0 |
| 0 | 0 | 8 | 45 |
| 0 | 0 | 16 | 52 |
| 5 | 52 | 0 | 0 |
| 9 | 45 | 0 | 0 |
| 0 | 0 | 18 | 1 |
| 0 | 0 | 60 | 0 |
| 56 | 29 | 0 | 0 |
| 58 | 5 | 0 | 0 |
| 0 | 0 | 59 | 22 |
| 0 | 0 | 58 | 2 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[52,45,0,0,16,8,0,0,0,0,8,16,0,0,45,52],[5,9,0,0,52,45,0,0,0,0,18,60,0,0,1,0],[56,58,0,0,29,5,0,0,0,0,59,58,0,0,22,2] >;
C2×C3⋊Dic15 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes {\rm Dic}_{15} % in TeX
G:=Group("C2xC3:Dic15"); // GroupNames label
G:=SmallGroup(360,113);
// by ID
G=gap.SmallGroup(360,113);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,387,1444,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^30=1,d^2=c^15,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