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G = Dic75order 300 = 22·3·52

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic75, C753C4, C50.S3, C2.D75, C6.D25, C3⋊Dic25, C30.1D5, C252Dic3, C5.Dic15, C150.1C2, C10.1D15, C15.1Dic5, SmallGroup(300,3)

Series: Derived Chief Lower central Upper central

C1C75 — Dic75
C1C5C25C75C150 — Dic75
C75 — Dic75
C1C2

Generators and relations for Dic75
 G = < a,b | a150=1, b2=a75, bab-1=a-1 >

75C4
25Dic3
15Dic5
5Dic15
3Dic25

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

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

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

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

78 conjugacy classes

class 1  2  3 4A4B5A5B 6 10A10B15A15B15C15D25A···25J30A30B30C30D50A···50J75A···75T150A···150T
order1234455610101515151525···253030303050···5075···75150···150
size11275752222222222···222222···22···22···2

78 irreducible representations

dim1112222222222
type++++--++--+-
imageC1C2C4S3D5Dic3Dic5D15D25Dic15Dic25D75Dic75
kernelDic75C150C75C50C30C25C15C10C6C5C3C2C1
# reps11212124104102020

Matrix representation of Dic75 in GL2(𝔽601) generated by

486513
88522
,
382247
200219
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

Export

Subgroup lattice of Dic75 in TeX

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