metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic75, C75⋊3C4, C50.S3, C2.D75, C6.D25, C3⋊Dic25, C30.1D5, C25⋊2Dic3, C5.Dic15, C150.1C2, C10.1D15, C15.1Dic5, SmallGroup(300,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C75 — Dic75 |
Generators and relations for Dic75
G = < a,b | a150=1, b2=a75, bab-1=a-1 >
Smallest permutation representation of Dic75
►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)
(1 187 76 262)(2 186 77 261)(3 185 78 260)(4 184 79 259)(5 183 80 258)(6 182 81 257)(7 181 82 256)(8 180 83 255)(9 179 84 254)(10 178 85 253)(11 177 86 252)(12 176 87 251)(13 175 88 250)(14 174 89 249)(15 173 90 248)(16 172 91 247)(17 171 92 246)(18 170 93 245)(19 169 94 244)(20 168 95 243)(21 167 96 242)(22 166 97 241)(23 165 98 240)(24 164 99 239)(25 163 100 238)(26 162 101 237)(27 161 102 236)(28 160 103 235)(29 159 104 234)(30 158 105 233)(31 157 106 232)(32 156 107 231)(33 155 108 230)(34 154 109 229)(35 153 110 228)(36 152 111 227)(37 151 112 226)(38 300 113 225)(39 299 114 224)(40 298 115 223)(41 297 116 222)(42 296 117 221)(43 295 118 220)(44 294 119 219)(45 293 120 218)(46 292 121 217)(47 291 122 216)(48 290 123 215)(49 289 124 214)(50 288 125 213)(51 287 126 212)(52 286 127 211)(53 285 128 210)(54 284 129 209)(55 283 130 208)(56 282 131 207)(57 281 132 206)(58 280 133 205)(59 279 134 204)(60 278 135 203)(61 277 136 202)(62 276 137 201)(63 275 138 200)(64 274 139 199)(65 273 140 198)(66 272 141 197)(67 271 142 196)(68 270 143 195)(69 269 144 194)(70 268 145 193)(71 267 146 192)(72 266 147 191)(73 265 148 190)(74 264 149 189)(75 263 150 188)
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), (1,187,76,262)(2,186,77,261)(3,185,78,260)(4,184,79,259)(5,183,80,258)(6,182,81,257)(7,181,82,256)(8,180,83,255)(9,179,84,254)(10,178,85,253)(11,177,86,252)(12,176,87,251)(13,175,88,250)(14,174,89,249)(15,173,90,248)(16,172,91,247)(17,171,92,246)(18,170,93,245)(19,169,94,244)(20,168,95,243)(21,167,96,242)(22,166,97,241)(23,165,98,240)(24,164,99,239)(25,163,100,238)(26,162,101,237)(27,161,102,236)(28,160,103,235)(29,159,104,234)(30,158,105,233)(31,157,106,232)(32,156,107,231)(33,155,108,230)(34,154,109,229)(35,153,110,228)(36,152,111,227)(37,151,112,226)(38,300,113,225)(39,299,114,224)(40,298,115,223)(41,297,116,222)(42,296,117,221)(43,295,118,220)(44,294,119,219)(45,293,120,218)(46,292,121,217)(47,291,122,216)(48,290,123,215)(49,289,124,214)(50,288,125,213)(51,287,126,212)(52,286,127,211)(53,285,128,210)(54,284,129,209)(55,283,130,208)(56,282,131,207)(57,281,132,206)(58,280,133,205)(59,279,134,204)(60,278,135,203)(61,277,136,202)(62,276,137,201)(63,275,138,200)(64,274,139,199)(65,273,140,198)(66,272,141,197)(67,271,142,196)(68,270,143,195)(69,269,144,194)(70,268,145,193)(71,267,146,192)(72,266,147,191)(73,265,148,190)(74,264,149,189)(75,263,150,188)>;
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), (1,187,76,262)(2,186,77,261)(3,185,78,260)(4,184,79,259)(5,183,80,258)(6,182,81,257)(7,181,82,256)(8,180,83,255)(9,179,84,254)(10,178,85,253)(11,177,86,252)(12,176,87,251)(13,175,88,250)(14,174,89,249)(15,173,90,248)(16,172,91,247)(17,171,92,246)(18,170,93,245)(19,169,94,244)(20,168,95,243)(21,167,96,242)(22,166,97,241)(23,165,98,240)(24,164,99,239)(25,163,100,238)(26,162,101,237)(27,161,102,236)(28,160,103,235)(29,159,104,234)(30,158,105,233)(31,157,106,232)(32,156,107,231)(33,155,108,230)(34,154,109,229)(35,153,110,228)(36,152,111,227)(37,151,112,226)(38,300,113,225)(39,299,114,224)(40,298,115,223)(41,297,116,222)(42,296,117,221)(43,295,118,220)(44,294,119,219)(45,293,120,218)(46,292,121,217)(47,291,122,216)(48,290,123,215)(49,289,124,214)(50,288,125,213)(51,287,126,212)(52,286,127,211)(53,285,128,210)(54,284,129,209)(55,283,130,208)(56,282,131,207)(57,281,132,206)(58,280,133,205)(59,279,134,204)(60,278,135,203)(61,277,136,202)(62,276,137,201)(63,275,138,200)(64,274,139,199)(65,273,140,198)(66,272,141,197)(67,271,142,196)(68,270,143,195)(69,269,144,194)(70,268,145,193)(71,267,146,192)(72,266,147,191)(73,265,148,190)(74,264,149,189)(75,263,150,188) );
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)], [(1,187,76,262),(2,186,77,261),(3,185,78,260),(4,184,79,259),(5,183,80,258),(6,182,81,257),(7,181,82,256),(8,180,83,255),(9,179,84,254),(10,178,85,253),(11,177,86,252),(12,176,87,251),(13,175,88,250),(14,174,89,249),(15,173,90,248),(16,172,91,247),(17,171,92,246),(18,170,93,245),(19,169,94,244),(20,168,95,243),(21,167,96,242),(22,166,97,241),(23,165,98,240),(24,164,99,239),(25,163,100,238),(26,162,101,237),(27,161,102,236),(28,160,103,235),(29,159,104,234),(30,158,105,233),(31,157,106,232),(32,156,107,231),(33,155,108,230),(34,154,109,229),(35,153,110,228),(36,152,111,227),(37,151,112,226),(38,300,113,225),(39,299,114,224),(40,298,115,223),(41,297,116,222),(42,296,117,221),(43,295,118,220),(44,294,119,219),(45,293,120,218),(46,292,121,217),(47,291,122,216),(48,290,123,215),(49,289,124,214),(50,288,125,213),(51,287,126,212),(52,286,127,211),(53,285,128,210),(54,284,129,209),(55,283,130,208),(56,282,131,207),(57,281,132,206),(58,280,133,205),(59,279,134,204),(60,278,135,203),(61,277,136,202),(62,276,137,201),(63,275,138,200),(64,274,139,199),(65,273,140,198),(66,272,141,197),(67,271,142,196),(68,270,143,195),(69,269,144,194),(70,268,145,193),(71,267,146,192),(72,266,147,191),(73,265,148,190),(74,264,149,189),(75,263,150,188)]])
78 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 6 | 10A | 10B | 15A | 15B | 15C | 15D | 25A | ··· | 25J | 30A | 30B | 30C | 30D | 50A | ··· | 50J | 75A | ··· | 75T | 150A | ··· | 150T |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 10 | 10 | 15 | 15 | 15 | 15 | 25 | ··· | 25 | 30 | 30 | 30 | 30 | 50 | ··· | 50 | 75 | ··· | 75 | 150 | ··· | 150 |
| size | 1 | 1 | 2 | 75 | 75 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | + | - | - | + | - | |
| image | C1 | C2 | C4 | S3 | D5 | Dic3 | Dic5 | D15 | D25 | Dic15 | Dic25 | D75 | Dic75 |
| kernel | Dic75 | C150 | C75 | C50 | C30 | C25 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 10 | 4 | 10 | 20 | 20 |
Matrix representation of Dic75 ►in GL2(𝔽601) generated by
| 486 | 513 |
| 88 | 522 |
| 382 | 247 |
| 200 | 219 |
G:=sub<GL(2,GF(601))| [486,88,513,522],[382,200,247,219] >;
Dic75 in GAP, Magma, Sage, TeX
{\rm Dic}_{75} % in TeX
G:=Group("Dic75"); // GroupNames label
G:=SmallGroup(300,3);
// by ID
G=gap.SmallGroup(300,3);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5,10,122,2163,418,6004]);
// Polycyclic
G:=Group<a,b|a^150=1,b^2=a^75,b*a*b^-1=a^-1>;
// generators/relations
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