Copied to
clipboard

G = Dic72order 288 = 25·32

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic72, C16.D9, C91Q32, C48.2S3, C6.3D24, C18.3D8, C2.5D72, C4.3D36, C3.Dic24, C144.1C2, C24.69D6, C36.26D4, C8.15D18, C12.36D12, C72.16C22, Dic36.1C2, SmallGroup(288,8)

Series: Derived Chief Lower central Upper central

C1C72 — Dic72
C1C3C9C18C36C72Dic36 — Dic72
C9C18C36C72 — Dic72
C1C2C4C8C16

Generators and relations for Dic72
 G = < a,b | a144=1, b2=a72, bab-1=a-1 >

36C4
36C4
18Q8
18Q8
12Dic3
12Dic3
9Q16
9Q16
6Dic6
6Dic6
4Dic9
4Dic9
9Q32
3Dic12
3Dic12
2Dic18
2Dic18
3Dic24

Smallest permutation representation of Dic72
Regular action on 288 points
Generators in S288
(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)
(1 244 73 172)(2 243 74 171)(3 242 75 170)(4 241 76 169)(5 240 77 168)(6 239 78 167)(7 238 79 166)(8 237 80 165)(9 236 81 164)(10 235 82 163)(11 234 83 162)(12 233 84 161)(13 232 85 160)(14 231 86 159)(15 230 87 158)(16 229 88 157)(17 228 89 156)(18 227 90 155)(19 226 91 154)(20 225 92 153)(21 224 93 152)(22 223 94 151)(23 222 95 150)(24 221 96 149)(25 220 97 148)(26 219 98 147)(27 218 99 146)(28 217 100 145)(29 216 101 288)(30 215 102 287)(31 214 103 286)(32 213 104 285)(33 212 105 284)(34 211 106 283)(35 210 107 282)(36 209 108 281)(37 208 109 280)(38 207 110 279)(39 206 111 278)(40 205 112 277)(41 204 113 276)(42 203 114 275)(43 202 115 274)(44 201 116 273)(45 200 117 272)(46 199 118 271)(47 198 119 270)(48 197 120 269)(49 196 121 268)(50 195 122 267)(51 194 123 266)(52 193 124 265)(53 192 125 264)(54 191 126 263)(55 190 127 262)(56 189 128 261)(57 188 129 260)(58 187 130 259)(59 186 131 258)(60 185 132 257)(61 184 133 256)(62 183 134 255)(63 182 135 254)(64 181 136 253)(65 180 137 252)(66 179 138 251)(67 178 139 250)(68 177 140 249)(69 176 141 248)(70 175 142 247)(71 174 143 246)(72 173 144 245)

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

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

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

75 conjugacy classes

class 1  2  3 4A4B4C 6 8A8B9A9B9C12A12B16A16B16C16D18A18B18C24A24B24C24D36A···36F48A···48H72A···72L144A···144X
order1234446889991212161616161818182424242436···3648···4872···72144···144
size1122727222222222222222222222···22···22···22···2

75 irreducible representations

dim1112222222222222
type+++++++++-+++-+-
imageC1C2C2S3D4D6D8D9D12Q32D18D24D36Dic24D72Dic72
kernelDic72C144Dic36C48C36C24C18C16C12C9C8C6C4C3C2C1
# reps112111232434681224

Matrix representation of Dic72 in GL2(𝔽433) generated by

289236
19753
,
62388
326371
G:=sub<GL(2,GF(433))| [289,197,236,53],[62,326,388,371] >;

Dic72 in GAP, Magma, Sage, TeX

{\rm Dic}_{72}
% in TeX

G:=Group("Dic72");
// GroupNames label

G:=SmallGroup(288,8);
// by ID

G=gap.SmallGroup(288,8);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,92,254,142,675,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b|a^144=1,b^2=a^72,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic72 in TeX

׿
×
𝔽