direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic3×C25, C3⋊C100, C75⋊5C4, C6.C50, C15.C20, C50.2S3, C150.3C2, C30.3C10, C2.(S3×C25), C5.(C5×Dic3), C10.2(C5×S3), (C5×Dic3).C5, SmallGroup(300,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C25 |
Generators and relations for Dic3×C25
G = < a,b,c | a25=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C25
►Regular action on 300 points
Generators in S300
(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)
(1 300 233 172 89 213)(2 276 234 173 90 214)(3 277 235 174 91 215)(4 278 236 175 92 216)(5 279 237 151 93 217)(6 280 238 152 94 218)(7 281 239 153 95 219)(8 282 240 154 96 220)(9 283 241 155 97 221)(10 284 242 156 98 222)(11 285 243 157 99 223)(12 286 244 158 100 224)(13 287 245 159 76 225)(14 288 246 160 77 201)(15 289 247 161 78 202)(16 290 248 162 79 203)(17 291 249 163 80 204)(18 292 250 164 81 205)(19 293 226 165 82 206)(20 294 227 166 83 207)(21 295 228 167 84 208)(22 296 229 168 85 209)(23 297 230 169 86 210)(24 298 231 170 87 211)(25 299 232 171 88 212)(26 144 54 107 259 183)(27 145 55 108 260 184)(28 146 56 109 261 185)(29 147 57 110 262 186)(30 148 58 111 263 187)(31 149 59 112 264 188)(32 150 60 113 265 189)(33 126 61 114 266 190)(34 127 62 115 267 191)(35 128 63 116 268 192)(36 129 64 117 269 193)(37 130 65 118 270 194)(38 131 66 119 271 195)(39 132 67 120 272 196)(40 133 68 121 273 197)(41 134 69 122 274 198)(42 135 70 123 275 199)(43 136 71 124 251 200)(44 137 72 125 252 176)(45 138 73 101 253 177)(46 139 74 102 254 178)(47 140 75 103 255 179)(48 141 51 104 256 180)(49 142 52 105 257 181)(50 143 53 106 258 182)
(1 46 172 102)(2 47 173 103)(3 48 174 104)(4 49 175 105)(5 50 151 106)(6 26 152 107)(7 27 153 108)(8 28 154 109)(9 29 155 110)(10 30 156 111)(11 31 157 112)(12 32 158 113)(13 33 159 114)(14 34 160 115)(15 35 161 116)(16 36 162 117)(17 37 163 118)(18 38 164 119)(19 39 165 120)(20 40 166 121)(21 41 167 122)(22 42 168 123)(23 43 169 124)(24 44 170 125)(25 45 171 101)(51 277 180 91)(52 278 181 92)(53 279 182 93)(54 280 183 94)(55 281 184 95)(56 282 185 96)(57 283 186 97)(58 284 187 98)(59 285 188 99)(60 286 189 100)(61 287 190 76)(62 288 191 77)(63 289 192 78)(64 290 193 79)(65 291 194 80)(66 292 195 81)(67 293 196 82)(68 294 197 83)(69 295 198 84)(70 296 199 85)(71 297 200 86)(72 298 176 87)(73 299 177 88)(74 300 178 89)(75 276 179 90)(126 245 266 225)(127 246 267 201)(128 247 268 202)(129 248 269 203)(130 249 270 204)(131 250 271 205)(132 226 272 206)(133 227 273 207)(134 228 274 208)(135 229 275 209)(136 230 251 210)(137 231 252 211)(138 232 253 212)(139 233 254 213)(140 234 255 214)(141 235 256 215)(142 236 257 216)(143 237 258 217)(144 238 259 218)(145 239 260 219)(146 240 261 220)(147 241 262 221)(148 242 263 222)(149 243 264 223)(150 244 265 224)
G:=sub<Sym(300)| (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), (1,300,233,172,89,213)(2,276,234,173,90,214)(3,277,235,174,91,215)(4,278,236,175,92,216)(5,279,237,151,93,217)(6,280,238,152,94,218)(7,281,239,153,95,219)(8,282,240,154,96,220)(9,283,241,155,97,221)(10,284,242,156,98,222)(11,285,243,157,99,223)(12,286,244,158,100,224)(13,287,245,159,76,225)(14,288,246,160,77,201)(15,289,247,161,78,202)(16,290,248,162,79,203)(17,291,249,163,80,204)(18,292,250,164,81,205)(19,293,226,165,82,206)(20,294,227,166,83,207)(21,295,228,167,84,208)(22,296,229,168,85,209)(23,297,230,169,86,210)(24,298,231,170,87,211)(25,299,232,171,88,212)(26,144,54,107,259,183)(27,145,55,108,260,184)(28,146,56,109,261,185)(29,147,57,110,262,186)(30,148,58,111,263,187)(31,149,59,112,264,188)(32,150,60,113,265,189)(33,126,61,114,266,190)(34,127,62,115,267,191)(35,128,63,116,268,192)(36,129,64,117,269,193)(37,130,65,118,270,194)(38,131,66,119,271,195)(39,132,67,120,272,196)(40,133,68,121,273,197)(41,134,69,122,274,198)(42,135,70,123,275,199)(43,136,71,124,251,200)(44,137,72,125,252,176)(45,138,73,101,253,177)(46,139,74,102,254,178)(47,140,75,103,255,179)(48,141,51,104,256,180)(49,142,52,105,257,181)(50,143,53,106,258,182), (1,46,172,102)(2,47,173,103)(3,48,174,104)(4,49,175,105)(5,50,151,106)(6,26,152,107)(7,27,153,108)(8,28,154,109)(9,29,155,110)(10,30,156,111)(11,31,157,112)(12,32,158,113)(13,33,159,114)(14,34,160,115)(15,35,161,116)(16,36,162,117)(17,37,163,118)(18,38,164,119)(19,39,165,120)(20,40,166,121)(21,41,167,122)(22,42,168,123)(23,43,169,124)(24,44,170,125)(25,45,171,101)(51,277,180,91)(52,278,181,92)(53,279,182,93)(54,280,183,94)(55,281,184,95)(56,282,185,96)(57,283,186,97)(58,284,187,98)(59,285,188,99)(60,286,189,100)(61,287,190,76)(62,288,191,77)(63,289,192,78)(64,290,193,79)(65,291,194,80)(66,292,195,81)(67,293,196,82)(68,294,197,83)(69,295,198,84)(70,296,199,85)(71,297,200,86)(72,298,176,87)(73,299,177,88)(74,300,178,89)(75,276,179,90)(126,245,266,225)(127,246,267,201)(128,247,268,202)(129,248,269,203)(130,249,270,204)(131,250,271,205)(132,226,272,206)(133,227,273,207)(134,228,274,208)(135,229,275,209)(136,230,251,210)(137,231,252,211)(138,232,253,212)(139,233,254,213)(140,234,255,214)(141,235,256,215)(142,236,257,216)(143,237,258,217)(144,238,259,218)(145,239,260,219)(146,240,261,220)(147,241,262,221)(148,242,263,222)(149,243,264,223)(150,244,265,224)>;
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), (1,300,233,172,89,213)(2,276,234,173,90,214)(3,277,235,174,91,215)(4,278,236,175,92,216)(5,279,237,151,93,217)(6,280,238,152,94,218)(7,281,239,153,95,219)(8,282,240,154,96,220)(9,283,241,155,97,221)(10,284,242,156,98,222)(11,285,243,157,99,223)(12,286,244,158,100,224)(13,287,245,159,76,225)(14,288,246,160,77,201)(15,289,247,161,78,202)(16,290,248,162,79,203)(17,291,249,163,80,204)(18,292,250,164,81,205)(19,293,226,165,82,206)(20,294,227,166,83,207)(21,295,228,167,84,208)(22,296,229,168,85,209)(23,297,230,169,86,210)(24,298,231,170,87,211)(25,299,232,171,88,212)(26,144,54,107,259,183)(27,145,55,108,260,184)(28,146,56,109,261,185)(29,147,57,110,262,186)(30,148,58,111,263,187)(31,149,59,112,264,188)(32,150,60,113,265,189)(33,126,61,114,266,190)(34,127,62,115,267,191)(35,128,63,116,268,192)(36,129,64,117,269,193)(37,130,65,118,270,194)(38,131,66,119,271,195)(39,132,67,120,272,196)(40,133,68,121,273,197)(41,134,69,122,274,198)(42,135,70,123,275,199)(43,136,71,124,251,200)(44,137,72,125,252,176)(45,138,73,101,253,177)(46,139,74,102,254,178)(47,140,75,103,255,179)(48,141,51,104,256,180)(49,142,52,105,257,181)(50,143,53,106,258,182), (1,46,172,102)(2,47,173,103)(3,48,174,104)(4,49,175,105)(5,50,151,106)(6,26,152,107)(7,27,153,108)(8,28,154,109)(9,29,155,110)(10,30,156,111)(11,31,157,112)(12,32,158,113)(13,33,159,114)(14,34,160,115)(15,35,161,116)(16,36,162,117)(17,37,163,118)(18,38,164,119)(19,39,165,120)(20,40,166,121)(21,41,167,122)(22,42,168,123)(23,43,169,124)(24,44,170,125)(25,45,171,101)(51,277,180,91)(52,278,181,92)(53,279,182,93)(54,280,183,94)(55,281,184,95)(56,282,185,96)(57,283,186,97)(58,284,187,98)(59,285,188,99)(60,286,189,100)(61,287,190,76)(62,288,191,77)(63,289,192,78)(64,290,193,79)(65,291,194,80)(66,292,195,81)(67,293,196,82)(68,294,197,83)(69,295,198,84)(70,296,199,85)(71,297,200,86)(72,298,176,87)(73,299,177,88)(74,300,178,89)(75,276,179,90)(126,245,266,225)(127,246,267,201)(128,247,268,202)(129,248,269,203)(130,249,270,204)(131,250,271,205)(132,226,272,206)(133,227,273,207)(134,228,274,208)(135,229,275,209)(136,230,251,210)(137,231,252,211)(138,232,253,212)(139,233,254,213)(140,234,255,214)(141,235,256,215)(142,236,257,216)(143,237,258,217)(144,238,259,218)(145,239,260,219)(146,240,261,220)(147,241,262,221)(148,242,263,222)(149,243,264,223)(150,244,265,224) );
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)], [(1,300,233,172,89,213),(2,276,234,173,90,214),(3,277,235,174,91,215),(4,278,236,175,92,216),(5,279,237,151,93,217),(6,280,238,152,94,218),(7,281,239,153,95,219),(8,282,240,154,96,220),(9,283,241,155,97,221),(10,284,242,156,98,222),(11,285,243,157,99,223),(12,286,244,158,100,224),(13,287,245,159,76,225),(14,288,246,160,77,201),(15,289,247,161,78,202),(16,290,248,162,79,203),(17,291,249,163,80,204),(18,292,250,164,81,205),(19,293,226,165,82,206),(20,294,227,166,83,207),(21,295,228,167,84,208),(22,296,229,168,85,209),(23,297,230,169,86,210),(24,298,231,170,87,211),(25,299,232,171,88,212),(26,144,54,107,259,183),(27,145,55,108,260,184),(28,146,56,109,261,185),(29,147,57,110,262,186),(30,148,58,111,263,187),(31,149,59,112,264,188),(32,150,60,113,265,189),(33,126,61,114,266,190),(34,127,62,115,267,191),(35,128,63,116,268,192),(36,129,64,117,269,193),(37,130,65,118,270,194),(38,131,66,119,271,195),(39,132,67,120,272,196),(40,133,68,121,273,197),(41,134,69,122,274,198),(42,135,70,123,275,199),(43,136,71,124,251,200),(44,137,72,125,252,176),(45,138,73,101,253,177),(46,139,74,102,254,178),(47,140,75,103,255,179),(48,141,51,104,256,180),(49,142,52,105,257,181),(50,143,53,106,258,182)], [(1,46,172,102),(2,47,173,103),(3,48,174,104),(4,49,175,105),(5,50,151,106),(6,26,152,107),(7,27,153,108),(8,28,154,109),(9,29,155,110),(10,30,156,111),(11,31,157,112),(12,32,158,113),(13,33,159,114),(14,34,160,115),(15,35,161,116),(16,36,162,117),(17,37,163,118),(18,38,164,119),(19,39,165,120),(20,40,166,121),(21,41,167,122),(22,42,168,123),(23,43,169,124),(24,44,170,125),(25,45,171,101),(51,277,180,91),(52,278,181,92),(53,279,182,93),(54,280,183,94),(55,281,184,95),(56,282,185,96),(57,283,186,97),(58,284,187,98),(59,285,188,99),(60,286,189,100),(61,287,190,76),(62,288,191,77),(63,289,192,78),(64,290,193,79),(65,291,194,80),(66,292,195,81),(67,293,196,82),(68,294,197,83),(69,295,198,84),(70,296,199,85),(71,297,200,86),(72,298,176,87),(73,299,177,88),(74,300,178,89),(75,276,179,90),(126,245,266,225),(127,246,267,201),(128,247,268,202),(129,248,269,203),(130,249,270,204),(131,250,271,205),(132,226,272,206),(133,227,273,207),(134,228,274,208),(135,229,275,209),(136,230,251,210),(137,231,252,211),(138,232,253,212),(139,233,254,213),(140,234,255,214),(141,235,256,215),(142,236,257,216),(143,237,258,217),(144,238,259,218),(145,239,260,219),(146,240,261,220),(147,241,262,221),(148,242,263,222),(149,243,264,223),(150,244,265,224)]])
150 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 25A | ··· | 25T | 30A | 30B | 30C | 30D | 50A | ··· | 50T | 75A | ··· | 75T | 100A | ··· | 100AN | 150A | ··· | 150T |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 25 | ··· | 25 | 30 | 30 | 30 | 30 | 50 | ··· | 50 | 75 | ··· | 75 | 100 | ··· | 100 | 150 | ··· | 150 |
| size | 1 | 1 | 2 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | C25 | C50 | C100 | S3 | Dic3 | C5×S3 | C5×Dic3 | S3×C25 | Dic3×C25 |
| kernel | Dic3×C25 | C150 | C75 | C5×Dic3 | C30 | C15 | Dic3 | C6 | C3 | C50 | C25 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 20 | 20 | 40 | 1 | 1 | 4 | 4 | 20 | 20 |
Matrix representation of Dic3×C25 ►in GL2(𝔽601) generated by
| 245 | 0 |
| 0 | 245 |
| 0 | 600 |
| 1 | 1 |
| 542 | 438 |
| 497 | 59 |
G:=sub<GL(2,GF(601))| [245,0,0,245],[0,1,600,1],[542,497,438,59] >;
Dic3×C25 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{25} % in TeX
G:=Group("Dic3xC25"); // GroupNames label
G:=SmallGroup(300,1);
// by ID
G=gap.SmallGroup(300,1);
# by ID
G:=PCGroup([5,-2,-5,-2,-5,-3,50,106,5004]);
// Polycyclic
G:=Group<a,b,c|a^25=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export