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## G = Dic3×C52order 300 = 22·3·52

### Direct product of C52 and Dic3

Aliases: Dic3×C52, C155C20, C30.5C10, C3⋊(C5×C20), C6.(C5×C10), (C5×C15)⋊13C4, C2.(S3×C52), C10.4(C5×S3), (C5×C30).5C2, (C5×C10).4S3, SmallGroup(300,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C52
 Chief series C1 — C3 — C6 — C30 — C5×C30 — Dic3×C52
 Lower central C3 — Dic3×C52
 Upper central C1 — C5×C10

Generators and relations for Dic3×C52
G = < a,b,c,d | a5=b5=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of Dic3×C52
Regular action on 300 points
Generators in S300
(1 149 113 77 41)(2 150 114 78 42)(3 145 109 73 37)(4 146 110 74 38)(5 147 111 75 39)(6 148 112 76 40)(7 121 115 79 43)(8 122 116 80 44)(9 123 117 81 45)(10 124 118 82 46)(11 125 119 83 47)(12 126 120 84 48)(13 127 91 85 49)(14 128 92 86 50)(15 129 93 87 51)(16 130 94 88 52)(17 131 95 89 53)(18 132 96 90 54)(19 133 97 61 55)(20 134 98 62 56)(21 135 99 63 57)(22 136 100 64 58)(23 137 101 65 59)(24 138 102 66 60)(25 139 103 67 31)(26 140 104 68 32)(27 141 105 69 33)(28 142 106 70 34)(29 143 107 71 35)(30 144 108 72 36)(151 295 259 223 187)(152 296 260 224 188)(153 297 261 225 189)(154 298 262 226 190)(155 299 263 227 191)(156 300 264 228 192)(157 271 265 229 193)(158 272 266 230 194)(159 273 267 231 195)(160 274 268 232 196)(161 275 269 233 197)(162 276 270 234 198)(163 277 241 235 199)(164 278 242 236 200)(165 279 243 237 201)(166 280 244 238 202)(167 281 245 239 203)(168 282 246 240 204)(169 283 247 211 205)(170 284 248 212 206)(171 285 249 213 207)(172 286 250 214 208)(173 287 251 215 209)(174 288 252 216 210)(175 289 253 217 181)(176 290 254 218 182)(177 291 255 219 183)(178 292 256 220 184)(179 293 257 221 185)(180 294 258 222 186)
(1 29 23 17 11)(2 30 24 18 12)(3 25 19 13 7)(4 26 20 14 8)(5 27 21 15 9)(6 28 22 16 10)(31 55 49 43 37)(32 56 50 44 38)(33 57 51 45 39)(34 58 52 46 40)(35 59 53 47 41)(36 60 54 48 42)(61 85 79 73 67)(62 86 80 74 68)(63 87 81 75 69)(64 88 82 76 70)(65 89 83 77 71)(66 90 84 78 72)(91 115 109 103 97)(92 116 110 104 98)(93 117 111 105 99)(94 118 112 106 100)(95 119 113 107 101)(96 120 114 108 102)(121 145 139 133 127)(122 146 140 134 128)(123 147 141 135 129)(124 148 142 136 130)(125 149 143 137 131)(126 150 144 138 132)(151 175 169 163 157)(152 176 170 164 158)(153 177 171 165 159)(154 178 172 166 160)(155 179 173 167 161)(156 180 174 168 162)(181 205 199 193 187)(182 206 200 194 188)(183 207 201 195 189)(184 208 202 196 190)(185 209 203 197 191)(186 210 204 198 192)(211 235 229 223 217)(212 236 230 224 218)(213 237 231 225 219)(214 238 232 226 220)(215 239 233 227 221)(216 240 234 228 222)(241 265 259 253 247)(242 266 260 254 248)(243 267 261 255 249)(244 268 262 256 250)(245 269 263 257 251)(246 270 264 258 252)(271 295 289 283 277)(272 296 290 284 278)(273 297 291 285 279)(274 298 292 286 280)(275 299 293 287 281)(276 300 294 288 282)
(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)
(1 154 4 151)(2 153 5 156)(3 152 6 155)(7 158 10 161)(8 157 11 160)(9 162 12 159)(13 164 16 167)(14 163 17 166)(15 168 18 165)(19 170 22 173)(20 169 23 172)(21 174 24 171)(25 176 28 179)(26 175 29 178)(27 180 30 177)(31 182 34 185)(32 181 35 184)(33 186 36 183)(37 188 40 191)(38 187 41 190)(39 192 42 189)(43 194 46 197)(44 193 47 196)(45 198 48 195)(49 200 52 203)(50 199 53 202)(51 204 54 201)(55 206 58 209)(56 205 59 208)(57 210 60 207)(61 212 64 215)(62 211 65 214)(63 216 66 213)(67 218 70 221)(68 217 71 220)(69 222 72 219)(73 224 76 227)(74 223 77 226)(75 228 78 225)(79 230 82 233)(80 229 83 232)(81 234 84 231)(85 236 88 239)(86 235 89 238)(87 240 90 237)(91 242 94 245)(92 241 95 244)(93 246 96 243)(97 248 100 251)(98 247 101 250)(99 252 102 249)(103 254 106 257)(104 253 107 256)(105 258 108 255)(109 260 112 263)(110 259 113 262)(111 264 114 261)(115 266 118 269)(116 265 119 268)(117 270 120 267)(121 272 124 275)(122 271 125 274)(123 276 126 273)(127 278 130 281)(128 277 131 280)(129 282 132 279)(133 284 136 287)(134 283 137 286)(135 288 138 285)(139 290 142 293)(140 289 143 292)(141 294 144 291)(145 296 148 299)(146 295 149 298)(147 300 150 297)

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

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

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

150 conjugacy classes

 class 1 2 3 4A 4B 5A ··· 5X 6 10A ··· 10X 15A ··· 15X 20A ··· 20AV 30A ··· 30X order 1 2 3 4 4 5 ··· 5 6 10 ··· 10 15 ··· 15 20 ··· 20 30 ··· 30 size 1 1 2 3 3 1 ··· 1 2 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C5 C10 C20 S3 Dic3 C5×S3 C5×Dic3 kernel Dic3×C52 C5×C30 C5×C15 C5×Dic3 C30 C15 C5×C10 C52 C10 C5 # reps 1 1 2 24 24 48 1 1 24 24

Matrix representation of Dic3×C52 in GL3(𝔽61) generated by

 58 0 0 0 34 0 0 0 34
,
 9 0 0 0 34 0 0 0 34
,
 1 0 0 0 1 60 0 1 0
,
 1 0 0 0 34 57 0 30 27
G:=sub<GL(3,GF(61))| [58,0,0,0,34,0,0,0,34],[9,0,0,0,34,0,0,0,34],[1,0,0,0,1,1,0,60,0],[1,0,0,0,34,30,0,57,27] >;

Dic3×C52 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_5^2
% in TeX

G:=Group("Dic3xC5^2");
// GroupNames label

G:=SmallGroup(300,18);
// by ID

G=gap.SmallGroup(300,18);
# by ID

G:=PCGroup([5,-2,-5,-5,-2,-3,250,5004]);
// Polycyclic

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

Export

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