Copied to
clipboard

## G = C3×C6×Dic5order 360 = 23·32·5

### Direct product of C3×C6 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C3×C6×Dic5
 Chief series C1 — C5 — C10 — C30 — C3×C30 — C32×Dic5 — C3×C6×Dic5
 Lower central C5 — C3×C6×Dic5
 Upper central C1 — C62

Generators and relations for C3×C6×Dic5
G = < a,b,c,d | a3=b6=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 168 in 96 conjugacy classes, 78 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C2×C4, C32, C10, C10, C12, C2×C6, C15, C3×C6, C3×C6, Dic5, C2×C10, C2×C12, C30, C3×C12, C62, C2×Dic5, C3×C15, C3×Dic5, C2×C30, C6×C12, C3×C30, C3×C30, C6×Dic5, C32×Dic5, C6×C30, C3×C6×Dic5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, D5, C12, C2×C6, C3×C6, Dic5, D10, C2×C12, C3×D5, C3×C12, C62, C2×Dic5, C3×Dic5, C6×D5, C6×C12, C32×D5, C6×Dic5, C32×Dic5, D5×C3×C6, C3×C6×Dic5

Smallest permutation representation of C3×C6×Dic5
Regular action on 360 points
Generators in S360
(1 69 39)(2 70 40)(3 61 31)(4 62 32)(5 63 33)(6 64 34)(7 65 35)(8 66 36)(9 67 37)(10 68 38)(11 71 41)(12 72 42)(13 73 43)(14 74 44)(15 75 45)(16 76 46)(17 77 47)(18 78 48)(19 79 49)(20 80 50)(21 81 51)(22 82 52)(23 83 53)(24 84 54)(25 85 55)(26 86 56)(27 87 57)(28 88 58)(29 89 59)(30 90 60)(91 151 121)(92 152 122)(93 153 123)(94 154 124)(95 155 125)(96 156 126)(97 157 127)(98 158 128)(99 159 129)(100 160 130)(101 161 131)(102 162 132)(103 163 133)(104 164 134)(105 165 135)(106 166 136)(107 167 137)(108 168 138)(109 169 139)(110 170 140)(111 171 141)(112 172 142)(113 173 143)(114 174 144)(115 175 145)(116 176 146)(117 177 147)(118 178 148)(119 179 149)(120 180 150)(181 241 211)(182 242 212)(183 243 213)(184 244 214)(185 245 215)(186 246 216)(187 247 217)(188 248 218)(189 249 219)(190 250 220)(191 251 221)(192 252 222)(193 253 223)(194 254 224)(195 255 225)(196 256 226)(197 257 227)(198 258 228)(199 259 229)(200 260 230)(201 261 231)(202 262 232)(203 263 233)(204 264 234)(205 265 235)(206 266 236)(207 267 237)(208 268 238)(209 269 239)(210 270 240)(271 331 301)(272 332 302)(273 333 303)(274 334 304)(275 335 305)(276 336 306)(277 337 307)(278 338 308)(279 339 309)(280 340 310)(281 341 311)(282 342 312)(283 343 313)(284 344 314)(285 345 315)(286 346 316)(287 347 317)(288 348 318)(289 349 319)(290 350 320)(291 351 321)(292 352 322)(293 353 323)(294 354 324)(295 355 325)(296 356 326)(297 357 327)(298 358 328)(299 359 329)(300 360 330)
(1 109 29 99 19 119)(2 110 30 100 20 120)(3 101 21 91 11 111)(4 102 22 92 12 112)(5 103 23 93 13 113)(6 104 24 94 14 114)(7 105 25 95 15 115)(8 106 26 96 16 116)(9 107 27 97 17 117)(10 108 28 98 18 118)(31 131 51 121 41 141)(32 132 52 122 42 142)(33 133 53 123 43 143)(34 134 54 124 44 144)(35 135 55 125 45 145)(36 136 56 126 46 146)(37 137 57 127 47 147)(38 138 58 128 48 148)(39 139 59 129 49 149)(40 140 60 130 50 150)(61 161 81 151 71 171)(62 162 82 152 72 172)(63 163 83 153 73 173)(64 164 84 154 74 174)(65 165 85 155 75 175)(66 166 86 156 76 176)(67 167 87 157 77 177)(68 168 88 158 78 178)(69 169 89 159 79 179)(70 170 90 160 80 180)(181 281 201 271 191 291)(182 282 202 272 192 292)(183 283 203 273 193 293)(184 284 204 274 194 294)(185 285 205 275 195 295)(186 286 206 276 196 296)(187 287 207 277 197 297)(188 288 208 278 198 298)(189 289 209 279 199 299)(190 290 210 280 200 300)(211 311 231 301 221 321)(212 312 232 302 222 322)(213 313 233 303 223 323)(214 314 234 304 224 324)(215 315 235 305 225 325)(216 316 236 306 226 326)(217 317 237 307 227 327)(218 318 238 308 228 328)(219 319 239 309 229 329)(220 320 240 310 230 330)(241 341 261 331 251 351)(242 342 262 332 252 352)(243 343 263 333 253 353)(244 344 264 334 254 354)(245 345 265 335 255 355)(246 346 266 336 256 356)(247 347 267 337 257 357)(248 348 268 338 258 358)(249 349 269 339 259 359)(250 350 270 340 260 360)
(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)
(1 186 6 181)(2 185 7 190)(3 184 8 189)(4 183 9 188)(5 182 10 187)(11 194 16 199)(12 193 17 198)(13 192 18 197)(14 191 19 196)(15 200 20 195)(21 204 26 209)(22 203 27 208)(23 202 28 207)(24 201 29 206)(25 210 30 205)(31 214 36 219)(32 213 37 218)(33 212 38 217)(34 211 39 216)(35 220 40 215)(41 224 46 229)(42 223 47 228)(43 222 48 227)(44 221 49 226)(45 230 50 225)(51 234 56 239)(52 233 57 238)(53 232 58 237)(54 231 59 236)(55 240 60 235)(61 244 66 249)(62 243 67 248)(63 242 68 247)(64 241 69 246)(65 250 70 245)(71 254 76 259)(72 253 77 258)(73 252 78 257)(74 251 79 256)(75 260 80 255)(81 264 86 269)(82 263 87 268)(83 262 88 267)(84 261 89 266)(85 270 90 265)(91 274 96 279)(92 273 97 278)(93 272 98 277)(94 271 99 276)(95 280 100 275)(101 284 106 289)(102 283 107 288)(103 282 108 287)(104 281 109 286)(105 290 110 285)(111 294 116 299)(112 293 117 298)(113 292 118 297)(114 291 119 296)(115 300 120 295)(121 304 126 309)(122 303 127 308)(123 302 128 307)(124 301 129 306)(125 310 130 305)(131 314 136 319)(132 313 137 318)(133 312 138 317)(134 311 139 316)(135 320 140 315)(141 324 146 329)(142 323 147 328)(143 322 148 327)(144 321 149 326)(145 330 150 325)(151 334 156 339)(152 333 157 338)(153 332 158 337)(154 331 159 336)(155 340 160 335)(161 344 166 349)(162 343 167 348)(163 342 168 347)(164 341 169 346)(165 350 170 345)(171 354 176 359)(172 353 177 358)(173 352 178 357)(174 351 179 356)(175 360 180 355)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A 4B 4C 4D 5A 5B 6A ··· 6X 10A ··· 10F 12A ··· 12AF 15A ··· 15P 30A ··· 30AV order 1 2 2 2 3 ··· 3 4 4 4 4 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 ··· 15 30 ··· 30 size 1 1 1 1 1 ··· 1 5 5 5 5 2 2 1 ··· 1 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 C2 C3 C4 C6 C6 C12 D5 Dic5 D10 C3×D5 C3×Dic5 C6×D5 kernel C3×C6×Dic5 C32×Dic5 C6×C30 C6×Dic5 C3×C30 C3×Dic5 C2×C30 C30 C62 C3×C6 C3×C6 C2×C6 C6 C6 # reps 1 2 1 8 4 16 8 32 2 4 2 16 32 16

Matrix representation of C3×C6×Dic5 in GL4(𝔽61) generated by

 47 0 0 0 0 1 0 0 0 0 13 0 0 0 0 13
,
 47 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 1 0 0 0 0 1 60 0 0 45 17
,
 1 0 0 0 0 60 0 0 0 0 10 29 0 0 47 51
G:=sub<GL(4,GF(61))| [47,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[47,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,1,45,0,0,60,17],[1,0,0,0,0,60,0,0,0,0,10,47,0,0,29,51] >;

C3×C6×Dic5 in GAP, Magma, Sage, TeX

C_3\times C_6\times {\rm Dic}_5
% in TeX

G:=Group("C3xC6xDic5");
// GroupNames label

G:=SmallGroup(360,93);
// by ID

G=gap.SmallGroup(360,93);
# by ID

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

G:=Group<a,b,c,d|a^3=b^6=c^10=1,d^2=c^5,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

׿
×
𝔽