direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×Dic25, C75⋊4C4, C50.C6, C25⋊2C12, C30.4D5, C6.2D25, C150.2C2, C15.2Dic5, C2.(C3×D25), C5.(C3×Dic5), C10.1(C3×D5), SmallGroup(300,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C3×Dic25 |
Generators and relations for C3×Dic25
G = < a,b,c | a3=b50=1, c2=b25, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic25
►Regular action on 300 points
Generators in S300
(1 148 85)(2 149 86)(3 150 87)(4 101 88)(5 102 89)(6 103 90)(7 104 91)(8 105 92)(9 106 93)(10 107 94)(11 108 95)(12 109 96)(13 110 97)(14 111 98)(15 112 99)(16 113 100)(17 114 51)(18 115 52)(19 116 53)(20 117 54)(21 118 55)(22 119 56)(23 120 57)(24 121 58)(25 122 59)(26 123 60)(27 124 61)(28 125 62)(29 126 63)(30 127 64)(31 128 65)(32 129 66)(33 130 67)(34 131 68)(35 132 69)(36 133 70)(37 134 71)(38 135 72)(39 136 73)(40 137 74)(41 138 75)(42 139 76)(43 140 77)(44 141 78)(45 142 79)(46 143 80)(47 144 81)(48 145 82)(49 146 83)(50 147 84)(151 278 228)(152 279 229)(153 280 230)(154 281 231)(155 282 232)(156 283 233)(157 284 234)(158 285 235)(159 286 236)(160 287 237)(161 288 238)(162 289 239)(163 290 240)(164 291 241)(165 292 242)(166 293 243)(167 294 244)(168 295 245)(169 296 246)(170 297 247)(171 298 248)(172 299 249)(173 300 250)(174 251 201)(175 252 202)(176 253 203)(177 254 204)(178 255 205)(179 256 206)(180 257 207)(181 258 208)(182 259 209)(183 260 210)(184 261 211)(185 262 212)(186 263 213)(187 264 214)(188 265 215)(189 266 216)(190 267 217)(191 268 218)(192 269 219)(193 270 220)(194 271 221)(195 272 222)(196 273 223)(197 274 224)(198 275 225)(199 276 226)(200 277 227)
(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)
(1 177 26 152)(2 176 27 151)(3 175 28 200)(4 174 29 199)(5 173 30 198)(6 172 31 197)(7 171 32 196)(8 170 33 195)(9 169 34 194)(10 168 35 193)(11 167 36 192)(12 166 37 191)(13 165 38 190)(14 164 39 189)(15 163 40 188)(16 162 41 187)(17 161 42 186)(18 160 43 185)(19 159 44 184)(20 158 45 183)(21 157 46 182)(22 156 47 181)(23 155 48 180)(24 154 49 179)(25 153 50 178)(51 238 76 213)(52 237 77 212)(53 236 78 211)(54 235 79 210)(55 234 80 209)(56 233 81 208)(57 232 82 207)(58 231 83 206)(59 230 84 205)(60 229 85 204)(61 228 86 203)(62 227 87 202)(63 226 88 201)(64 225 89 250)(65 224 90 249)(66 223 91 248)(67 222 92 247)(68 221 93 246)(69 220 94 245)(70 219 95 244)(71 218 96 243)(72 217 97 242)(73 216 98 241)(74 215 99 240)(75 214 100 239)(101 251 126 276)(102 300 127 275)(103 299 128 274)(104 298 129 273)(105 297 130 272)(106 296 131 271)(107 295 132 270)(108 294 133 269)(109 293 134 268)(110 292 135 267)(111 291 136 266)(112 290 137 265)(113 289 138 264)(114 288 139 263)(115 287 140 262)(116 286 141 261)(117 285 142 260)(118 284 143 259)(119 283 144 258)(120 282 145 257)(121 281 146 256)(122 280 147 255)(123 279 148 254)(124 278 149 253)(125 277 150 252)
G:=sub<Sym(300)| (1,148,85)(2,149,86)(3,150,87)(4,101,88)(5,102,89)(6,103,90)(7,104,91)(8,105,92)(9,106,93)(10,107,94)(11,108,95)(12,109,96)(13,110,97)(14,111,98)(15,112,99)(16,113,100)(17,114,51)(18,115,52)(19,116,53)(20,117,54)(21,118,55)(22,119,56)(23,120,57)(24,121,58)(25,122,59)(26,123,60)(27,124,61)(28,125,62)(29,126,63)(30,127,64)(31,128,65)(32,129,66)(33,130,67)(34,131,68)(35,132,69)(36,133,70)(37,134,71)(38,135,72)(39,136,73)(40,137,74)(41,138,75)(42,139,76)(43,140,77)(44,141,78)(45,142,79)(46,143,80)(47,144,81)(48,145,82)(49,146,83)(50,147,84)(151,278,228)(152,279,229)(153,280,230)(154,281,231)(155,282,232)(156,283,233)(157,284,234)(158,285,235)(159,286,236)(160,287,237)(161,288,238)(162,289,239)(163,290,240)(164,291,241)(165,292,242)(166,293,243)(167,294,244)(168,295,245)(169,296,246)(170,297,247)(171,298,248)(172,299,249)(173,300,250)(174,251,201)(175,252,202)(176,253,203)(177,254,204)(178,255,205)(179,256,206)(180,257,207)(181,258,208)(182,259,209)(183,260,210)(184,261,211)(185,262,212)(186,263,213)(187,264,214)(188,265,215)(189,266,216)(190,267,217)(191,268,218)(192,269,219)(193,270,220)(194,271,221)(195,272,222)(196,273,223)(197,274,224)(198,275,225)(199,276,226)(200,277,227), (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), (1,177,26,152)(2,176,27,151)(3,175,28,200)(4,174,29,199)(5,173,30,198)(6,172,31,197)(7,171,32,196)(8,170,33,195)(9,169,34,194)(10,168,35,193)(11,167,36,192)(12,166,37,191)(13,165,38,190)(14,164,39,189)(15,163,40,188)(16,162,41,187)(17,161,42,186)(18,160,43,185)(19,159,44,184)(20,158,45,183)(21,157,46,182)(22,156,47,181)(23,155,48,180)(24,154,49,179)(25,153,50,178)(51,238,76,213)(52,237,77,212)(53,236,78,211)(54,235,79,210)(55,234,80,209)(56,233,81,208)(57,232,82,207)(58,231,83,206)(59,230,84,205)(60,229,85,204)(61,228,86,203)(62,227,87,202)(63,226,88,201)(64,225,89,250)(65,224,90,249)(66,223,91,248)(67,222,92,247)(68,221,93,246)(69,220,94,245)(70,219,95,244)(71,218,96,243)(72,217,97,242)(73,216,98,241)(74,215,99,240)(75,214,100,239)(101,251,126,276)(102,300,127,275)(103,299,128,274)(104,298,129,273)(105,297,130,272)(106,296,131,271)(107,295,132,270)(108,294,133,269)(109,293,134,268)(110,292,135,267)(111,291,136,266)(112,290,137,265)(113,289,138,264)(114,288,139,263)(115,287,140,262)(116,286,141,261)(117,285,142,260)(118,284,143,259)(119,283,144,258)(120,282,145,257)(121,281,146,256)(122,280,147,255)(123,279,148,254)(124,278,149,253)(125,277,150,252)>;
G:=Group( (1,148,85)(2,149,86)(3,150,87)(4,101,88)(5,102,89)(6,103,90)(7,104,91)(8,105,92)(9,106,93)(10,107,94)(11,108,95)(12,109,96)(13,110,97)(14,111,98)(15,112,99)(16,113,100)(17,114,51)(18,115,52)(19,116,53)(20,117,54)(21,118,55)(22,119,56)(23,120,57)(24,121,58)(25,122,59)(26,123,60)(27,124,61)(28,125,62)(29,126,63)(30,127,64)(31,128,65)(32,129,66)(33,130,67)(34,131,68)(35,132,69)(36,133,70)(37,134,71)(38,135,72)(39,136,73)(40,137,74)(41,138,75)(42,139,76)(43,140,77)(44,141,78)(45,142,79)(46,143,80)(47,144,81)(48,145,82)(49,146,83)(50,147,84)(151,278,228)(152,279,229)(153,280,230)(154,281,231)(155,282,232)(156,283,233)(157,284,234)(158,285,235)(159,286,236)(160,287,237)(161,288,238)(162,289,239)(163,290,240)(164,291,241)(165,292,242)(166,293,243)(167,294,244)(168,295,245)(169,296,246)(170,297,247)(171,298,248)(172,299,249)(173,300,250)(174,251,201)(175,252,202)(176,253,203)(177,254,204)(178,255,205)(179,256,206)(180,257,207)(181,258,208)(182,259,209)(183,260,210)(184,261,211)(185,262,212)(186,263,213)(187,264,214)(188,265,215)(189,266,216)(190,267,217)(191,268,218)(192,269,219)(193,270,220)(194,271,221)(195,272,222)(196,273,223)(197,274,224)(198,275,225)(199,276,226)(200,277,227), (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), (1,177,26,152)(2,176,27,151)(3,175,28,200)(4,174,29,199)(5,173,30,198)(6,172,31,197)(7,171,32,196)(8,170,33,195)(9,169,34,194)(10,168,35,193)(11,167,36,192)(12,166,37,191)(13,165,38,190)(14,164,39,189)(15,163,40,188)(16,162,41,187)(17,161,42,186)(18,160,43,185)(19,159,44,184)(20,158,45,183)(21,157,46,182)(22,156,47,181)(23,155,48,180)(24,154,49,179)(25,153,50,178)(51,238,76,213)(52,237,77,212)(53,236,78,211)(54,235,79,210)(55,234,80,209)(56,233,81,208)(57,232,82,207)(58,231,83,206)(59,230,84,205)(60,229,85,204)(61,228,86,203)(62,227,87,202)(63,226,88,201)(64,225,89,250)(65,224,90,249)(66,223,91,248)(67,222,92,247)(68,221,93,246)(69,220,94,245)(70,219,95,244)(71,218,96,243)(72,217,97,242)(73,216,98,241)(74,215,99,240)(75,214,100,239)(101,251,126,276)(102,300,127,275)(103,299,128,274)(104,298,129,273)(105,297,130,272)(106,296,131,271)(107,295,132,270)(108,294,133,269)(109,293,134,268)(110,292,135,267)(111,291,136,266)(112,290,137,265)(113,289,138,264)(114,288,139,263)(115,287,140,262)(116,286,141,261)(117,285,142,260)(118,284,143,259)(119,283,144,258)(120,282,145,257)(121,281,146,256)(122,280,147,255)(123,279,148,254)(124,278,149,253)(125,277,150,252) );
G=PermutationGroup([[(1,148,85),(2,149,86),(3,150,87),(4,101,88),(5,102,89),(6,103,90),(7,104,91),(8,105,92),(9,106,93),(10,107,94),(11,108,95),(12,109,96),(13,110,97),(14,111,98),(15,112,99),(16,113,100),(17,114,51),(18,115,52),(19,116,53),(20,117,54),(21,118,55),(22,119,56),(23,120,57),(24,121,58),(25,122,59),(26,123,60),(27,124,61),(28,125,62),(29,126,63),(30,127,64),(31,128,65),(32,129,66),(33,130,67),(34,131,68),(35,132,69),(36,133,70),(37,134,71),(38,135,72),(39,136,73),(40,137,74),(41,138,75),(42,139,76),(43,140,77),(44,141,78),(45,142,79),(46,143,80),(47,144,81),(48,145,82),(49,146,83),(50,147,84),(151,278,228),(152,279,229),(153,280,230),(154,281,231),(155,282,232),(156,283,233),(157,284,234),(158,285,235),(159,286,236),(160,287,237),(161,288,238),(162,289,239),(163,290,240),(164,291,241),(165,292,242),(166,293,243),(167,294,244),(168,295,245),(169,296,246),(170,297,247),(171,298,248),(172,299,249),(173,300,250),(174,251,201),(175,252,202),(176,253,203),(177,254,204),(178,255,205),(179,256,206),(180,257,207),(181,258,208),(182,259,209),(183,260,210),(184,261,211),(185,262,212),(186,263,213),(187,264,214),(188,265,215),(189,266,216),(190,267,217),(191,268,218),(192,269,219),(193,270,220),(194,271,221),(195,272,222),(196,273,223),(197,274,224),(198,275,225),(199,276,226),(200,277,227)], [(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)], [(1,177,26,152),(2,176,27,151),(3,175,28,200),(4,174,29,199),(5,173,30,198),(6,172,31,197),(7,171,32,196),(8,170,33,195),(9,169,34,194),(10,168,35,193),(11,167,36,192),(12,166,37,191),(13,165,38,190),(14,164,39,189),(15,163,40,188),(16,162,41,187),(17,161,42,186),(18,160,43,185),(19,159,44,184),(20,158,45,183),(21,157,46,182),(22,156,47,181),(23,155,48,180),(24,154,49,179),(25,153,50,178),(51,238,76,213),(52,237,77,212),(53,236,78,211),(54,235,79,210),(55,234,80,209),(56,233,81,208),(57,232,82,207),(58,231,83,206),(59,230,84,205),(60,229,85,204),(61,228,86,203),(62,227,87,202),(63,226,88,201),(64,225,89,250),(65,224,90,249),(66,223,91,248),(67,222,92,247),(68,221,93,246),(69,220,94,245),(70,219,95,244),(71,218,96,243),(72,217,97,242),(73,216,98,241),(74,215,99,240),(75,214,100,239),(101,251,126,276),(102,300,127,275),(103,299,128,274),(104,298,129,273),(105,297,130,272),(106,296,131,271),(107,295,132,270),(108,294,133,269),(109,293,134,268),(110,292,135,267),(111,291,136,266),(112,290,137,265),(113,289,138,264),(114,288,139,263),(115,287,140,262),(116,286,141,261),(117,285,142,260),(118,284,143,259),(119,283,144,258),(120,282,145,257),(121,281,146,256),(122,280,147,255),(123,279,148,254),(124,278,149,253),(125,277,150,252)]])
84 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 10A | 10B | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 25A | ··· | 25J | 30A | 30B | 30C | 30D | 50A | ··· | 50J | 75A | ··· | 75T | 150A | ··· | 150T |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 25 | ··· | 25 | 30 | 30 | 30 | 30 | 50 | ··· | 50 | 75 | ··· | 75 | 150 | ··· | 150 |
| size | 1 | 1 | 1 | 1 | 25 | 25 | 2 | 2 | 1 | 1 | 2 | 2 | 25 | 25 | 25 | 25 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D5 | Dic5 | C3×D5 | D25 | C3×Dic5 | Dic25 | C3×D25 | C3×Dic25 |
| kernel | C3×Dic25 | C150 | Dic25 | C75 | C50 | C25 | C30 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 10 | 4 | 10 | 20 | 20 |
Matrix representation of C3×Dic25 ►in GL2(𝔽601) generated by
| 24 | 0 |
| 0 | 24 |
| 469 | 373 |
| 228 | 453 |
| 21 | 77 |
| 205 | 580 |
G:=sub<GL(2,GF(601))| [24,0,0,24],[469,228,373,453],[21,205,77,580] >;
C3×Dic25 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{25} % in TeX
G:=Group("C3xDic25"); // GroupNames label
G:=SmallGroup(300,2);
// by ID
G=gap.SmallGroup(300,2);
# by ID
G:=PCGroup([5,-2,-3,-2,-5,-5,30,2163,418,6004]);
// Polycyclic
G:=Group<a,b,c|a^3=b^50=1,c^2=b^25,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export