metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic72, C16.D9, C9⋊1Q32, 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
Generators and relations for Dic72
G = < a,b | a144=1, b2=a72, bab-1=a-1 >
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)
(1 199 73 271)(2 198 74 270)(3 197 75 269)(4 196 76 268)(5 195 77 267)(6 194 78 266)(7 193 79 265)(8 192 80 264)(9 191 81 263)(10 190 82 262)(11 189 83 261)(12 188 84 260)(13 187 85 259)(14 186 86 258)(15 185 87 257)(16 184 88 256)(17 183 89 255)(18 182 90 254)(19 181 91 253)(20 180 92 252)(21 179 93 251)(22 178 94 250)(23 177 95 249)(24 176 96 248)(25 175 97 247)(26 174 98 246)(27 173 99 245)(28 172 100 244)(29 171 101 243)(30 170 102 242)(31 169 103 241)(32 168 104 240)(33 167 105 239)(34 166 106 238)(35 165 107 237)(36 164 108 236)(37 163 109 235)(38 162 110 234)(39 161 111 233)(40 160 112 232)(41 159 113 231)(42 158 114 230)(43 157 115 229)(44 156 116 228)(45 155 117 227)(46 154 118 226)(47 153 119 225)(48 152 120 224)(49 151 121 223)(50 150 122 222)(51 149 123 221)(52 148 124 220)(53 147 125 219)(54 146 126 218)(55 145 127 217)(56 288 128 216)(57 287 129 215)(58 286 130 214)(59 285 131 213)(60 284 132 212)(61 283 133 211)(62 282 134 210)(63 281 135 209)(64 280 136 208)(65 279 137 207)(66 278 138 206)(67 277 139 205)(68 276 140 204)(69 275 141 203)(70 274 142 202)(71 273 143 201)(72 272 144 200)
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), (1,199,73,271)(2,198,74,270)(3,197,75,269)(4,196,76,268)(5,195,77,267)(6,194,78,266)(7,193,79,265)(8,192,80,264)(9,191,81,263)(10,190,82,262)(11,189,83,261)(12,188,84,260)(13,187,85,259)(14,186,86,258)(15,185,87,257)(16,184,88,256)(17,183,89,255)(18,182,90,254)(19,181,91,253)(20,180,92,252)(21,179,93,251)(22,178,94,250)(23,177,95,249)(24,176,96,248)(25,175,97,247)(26,174,98,246)(27,173,99,245)(28,172,100,244)(29,171,101,243)(30,170,102,242)(31,169,103,241)(32,168,104,240)(33,167,105,239)(34,166,106,238)(35,165,107,237)(36,164,108,236)(37,163,109,235)(38,162,110,234)(39,161,111,233)(40,160,112,232)(41,159,113,231)(42,158,114,230)(43,157,115,229)(44,156,116,228)(45,155,117,227)(46,154,118,226)(47,153,119,225)(48,152,120,224)(49,151,121,223)(50,150,122,222)(51,149,123,221)(52,148,124,220)(53,147,125,219)(54,146,126,218)(55,145,127,217)(56,288,128,216)(57,287,129,215)(58,286,130,214)(59,285,131,213)(60,284,132,212)(61,283,133,211)(62,282,134,210)(63,281,135,209)(64,280,136,208)(65,279,137,207)(66,278,138,206)(67,277,139,205)(68,276,140,204)(69,275,141,203)(70,274,142,202)(71,273,143,201)(72,272,144,200)>;
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), (1,199,73,271)(2,198,74,270)(3,197,75,269)(4,196,76,268)(5,195,77,267)(6,194,78,266)(7,193,79,265)(8,192,80,264)(9,191,81,263)(10,190,82,262)(11,189,83,261)(12,188,84,260)(13,187,85,259)(14,186,86,258)(15,185,87,257)(16,184,88,256)(17,183,89,255)(18,182,90,254)(19,181,91,253)(20,180,92,252)(21,179,93,251)(22,178,94,250)(23,177,95,249)(24,176,96,248)(25,175,97,247)(26,174,98,246)(27,173,99,245)(28,172,100,244)(29,171,101,243)(30,170,102,242)(31,169,103,241)(32,168,104,240)(33,167,105,239)(34,166,106,238)(35,165,107,237)(36,164,108,236)(37,163,109,235)(38,162,110,234)(39,161,111,233)(40,160,112,232)(41,159,113,231)(42,158,114,230)(43,157,115,229)(44,156,116,228)(45,155,117,227)(46,154,118,226)(47,153,119,225)(48,152,120,224)(49,151,121,223)(50,150,122,222)(51,149,123,221)(52,148,124,220)(53,147,125,219)(54,146,126,218)(55,145,127,217)(56,288,128,216)(57,287,129,215)(58,286,130,214)(59,285,131,213)(60,284,132,212)(61,283,133,211)(62,282,134,210)(63,281,135,209)(64,280,136,208)(65,279,137,207)(66,278,138,206)(67,277,139,205)(68,276,140,204)(69,275,141,203)(70,274,142,202)(71,273,143,201)(72,272,144,200) );
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)], [(1,199,73,271),(2,198,74,270),(3,197,75,269),(4,196,76,268),(5,195,77,267),(6,194,78,266),(7,193,79,265),(8,192,80,264),(9,191,81,263),(10,190,82,262),(11,189,83,261),(12,188,84,260),(13,187,85,259),(14,186,86,258),(15,185,87,257),(16,184,88,256),(17,183,89,255),(18,182,90,254),(19,181,91,253),(20,180,92,252),(21,179,93,251),(22,178,94,250),(23,177,95,249),(24,176,96,248),(25,175,97,247),(26,174,98,246),(27,173,99,245),(28,172,100,244),(29,171,101,243),(30,170,102,242),(31,169,103,241),(32,168,104,240),(33,167,105,239),(34,166,106,238),(35,165,107,237),(36,164,108,236),(37,163,109,235),(38,162,110,234),(39,161,111,233),(40,160,112,232),(41,159,113,231),(42,158,114,230),(43,157,115,229),(44,156,116,228),(45,155,117,227),(46,154,118,226),(47,153,119,225),(48,152,120,224),(49,151,121,223),(50,150,122,222),(51,149,123,221),(52,148,124,220),(53,147,125,219),(54,146,126,218),(55,145,127,217),(56,288,128,216),(57,287,129,215),(58,286,130,214),(59,285,131,213),(60,284,132,212),(61,283,133,211),(62,282,134,210),(63,281,135,209),(64,280,136,208),(65,279,137,207),(66,278,138,206),(67,277,139,205),(68,276,140,204),(69,275,141,203),(70,274,142,202),(71,273,143,201),(72,272,144,200)]])
75 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 16A | 16B | 16C | 16D | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 48A | ··· | 48H | 72A | ··· | 72L | 144A | ··· | 144X |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 16 | 16 | 16 | 16 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 | 144 | ··· | 144 |
| size | 1 | 1 | 2 | 2 | 72 | 72 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D8 | D9 | D12 | Q32 | D18 | D24 | D36 | Dic24 | D72 | Dic72 |
| kernel | Dic72 | C144 | Dic36 | C48 | C36 | C24 | C18 | C16 | C12 | C9 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 3 | 2 | 4 | 3 | 4 | 6 | 8 | 12 | 24 |
Matrix representation of Dic72 ►in GL2(𝔽433) generated by
| 289 | 236 |
| 197 | 53 |
| 62 | 388 |
| 326 | 371 |
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
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