metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic84, C8.D21, C21⋊4Q16, C56.1S3, C24.1D7, C2.5D84, C6.3D28, C3⋊1Dic28, C7⋊1Dic12, C168.1C2, C14.3D12, C4.10D42, C42.21D4, C28.45D6, C12.45D14, C84.52C22, Dic42.1C2, SmallGroup(336,94)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic84
G = < a,b | a168=1, b2=a84, bab-1=a-1 >
Smallest permutation representation of Dic84
►Regular action on 336 points
Generators in S336
(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)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336)
(1 327 85 243)(2 326 86 242)(3 325 87 241)(4 324 88 240)(5 323 89 239)(6 322 90 238)(7 321 91 237)(8 320 92 236)(9 319 93 235)(10 318 94 234)(11 317 95 233)(12 316 96 232)(13 315 97 231)(14 314 98 230)(15 313 99 229)(16 312 100 228)(17 311 101 227)(18 310 102 226)(19 309 103 225)(20 308 104 224)(21 307 105 223)(22 306 106 222)(23 305 107 221)(24 304 108 220)(25 303 109 219)(26 302 110 218)(27 301 111 217)(28 300 112 216)(29 299 113 215)(30 298 114 214)(31 297 115 213)(32 296 116 212)(33 295 117 211)(34 294 118 210)(35 293 119 209)(36 292 120 208)(37 291 121 207)(38 290 122 206)(39 289 123 205)(40 288 124 204)(41 287 125 203)(42 286 126 202)(43 285 127 201)(44 284 128 200)(45 283 129 199)(46 282 130 198)(47 281 131 197)(48 280 132 196)(49 279 133 195)(50 278 134 194)(51 277 135 193)(52 276 136 192)(53 275 137 191)(54 274 138 190)(55 273 139 189)(56 272 140 188)(57 271 141 187)(58 270 142 186)(59 269 143 185)(60 268 144 184)(61 267 145 183)(62 266 146 182)(63 265 147 181)(64 264 148 180)(65 263 149 179)(66 262 150 178)(67 261 151 177)(68 260 152 176)(69 259 153 175)(70 258 154 174)(71 257 155 173)(72 256 156 172)(73 255 157 171)(74 254 158 170)(75 253 159 169)(76 252 160 336)(77 251 161 335)(78 250 162 334)(79 249 163 333)(80 248 164 332)(81 247 165 331)(82 246 166 330)(83 245 167 329)(84 244 168 328)
G:=sub<Sym(336)| (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336), (1,327,85,243)(2,326,86,242)(3,325,87,241)(4,324,88,240)(5,323,89,239)(6,322,90,238)(7,321,91,237)(8,320,92,236)(9,319,93,235)(10,318,94,234)(11,317,95,233)(12,316,96,232)(13,315,97,231)(14,314,98,230)(15,313,99,229)(16,312,100,228)(17,311,101,227)(18,310,102,226)(19,309,103,225)(20,308,104,224)(21,307,105,223)(22,306,106,222)(23,305,107,221)(24,304,108,220)(25,303,109,219)(26,302,110,218)(27,301,111,217)(28,300,112,216)(29,299,113,215)(30,298,114,214)(31,297,115,213)(32,296,116,212)(33,295,117,211)(34,294,118,210)(35,293,119,209)(36,292,120,208)(37,291,121,207)(38,290,122,206)(39,289,123,205)(40,288,124,204)(41,287,125,203)(42,286,126,202)(43,285,127,201)(44,284,128,200)(45,283,129,199)(46,282,130,198)(47,281,131,197)(48,280,132,196)(49,279,133,195)(50,278,134,194)(51,277,135,193)(52,276,136,192)(53,275,137,191)(54,274,138,190)(55,273,139,189)(56,272,140,188)(57,271,141,187)(58,270,142,186)(59,269,143,185)(60,268,144,184)(61,267,145,183)(62,266,146,182)(63,265,147,181)(64,264,148,180)(65,263,149,179)(66,262,150,178)(67,261,151,177)(68,260,152,176)(69,259,153,175)(70,258,154,174)(71,257,155,173)(72,256,156,172)(73,255,157,171)(74,254,158,170)(75,253,159,169)(76,252,160,336)(77,251,161,335)(78,250,162,334)(79,249,163,333)(80,248,164,332)(81,247,165,331)(82,246,166,330)(83,245,167,329)(84,244,168,328)>;
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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336), (1,327,85,243)(2,326,86,242)(3,325,87,241)(4,324,88,240)(5,323,89,239)(6,322,90,238)(7,321,91,237)(8,320,92,236)(9,319,93,235)(10,318,94,234)(11,317,95,233)(12,316,96,232)(13,315,97,231)(14,314,98,230)(15,313,99,229)(16,312,100,228)(17,311,101,227)(18,310,102,226)(19,309,103,225)(20,308,104,224)(21,307,105,223)(22,306,106,222)(23,305,107,221)(24,304,108,220)(25,303,109,219)(26,302,110,218)(27,301,111,217)(28,300,112,216)(29,299,113,215)(30,298,114,214)(31,297,115,213)(32,296,116,212)(33,295,117,211)(34,294,118,210)(35,293,119,209)(36,292,120,208)(37,291,121,207)(38,290,122,206)(39,289,123,205)(40,288,124,204)(41,287,125,203)(42,286,126,202)(43,285,127,201)(44,284,128,200)(45,283,129,199)(46,282,130,198)(47,281,131,197)(48,280,132,196)(49,279,133,195)(50,278,134,194)(51,277,135,193)(52,276,136,192)(53,275,137,191)(54,274,138,190)(55,273,139,189)(56,272,140,188)(57,271,141,187)(58,270,142,186)(59,269,143,185)(60,268,144,184)(61,267,145,183)(62,266,146,182)(63,265,147,181)(64,264,148,180)(65,263,149,179)(66,262,150,178)(67,261,151,177)(68,260,152,176)(69,259,153,175)(70,258,154,174)(71,257,155,173)(72,256,156,172)(73,255,157,171)(74,254,158,170)(75,253,159,169)(76,252,160,336)(77,251,161,335)(78,250,162,334)(79,249,163,333)(80,248,164,332)(81,247,165,331)(82,246,166,330)(83,245,167,329)(84,244,168,328) );
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),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336)], [(1,327,85,243),(2,326,86,242),(3,325,87,241),(4,324,88,240),(5,323,89,239),(6,322,90,238),(7,321,91,237),(8,320,92,236),(9,319,93,235),(10,318,94,234),(11,317,95,233),(12,316,96,232),(13,315,97,231),(14,314,98,230),(15,313,99,229),(16,312,100,228),(17,311,101,227),(18,310,102,226),(19,309,103,225),(20,308,104,224),(21,307,105,223),(22,306,106,222),(23,305,107,221),(24,304,108,220),(25,303,109,219),(26,302,110,218),(27,301,111,217),(28,300,112,216),(29,299,113,215),(30,298,114,214),(31,297,115,213),(32,296,116,212),(33,295,117,211),(34,294,118,210),(35,293,119,209),(36,292,120,208),(37,291,121,207),(38,290,122,206),(39,289,123,205),(40,288,124,204),(41,287,125,203),(42,286,126,202),(43,285,127,201),(44,284,128,200),(45,283,129,199),(46,282,130,198),(47,281,131,197),(48,280,132,196),(49,279,133,195),(50,278,134,194),(51,277,135,193),(52,276,136,192),(53,275,137,191),(54,274,138,190),(55,273,139,189),(56,272,140,188),(57,271,141,187),(58,270,142,186),(59,269,143,185),(60,268,144,184),(61,267,145,183),(62,266,146,182),(63,265,147,181),(64,264,148,180),(65,263,149,179),(66,262,150,178),(67,261,151,177),(68,260,152,176),(69,259,153,175),(70,258,154,174),(71,257,155,173),(72,256,156,172),(73,255,157,171),(74,254,158,170),(75,253,159,169),(76,252,160,336),(77,251,161,335),(78,250,162,334),(79,249,163,333),(80,248,164,332),(81,247,165,331),(82,246,166,330),(83,245,167,329),(84,244,168,328)]])
87 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 7A | 7B | 7C | 8A | 8B | 12A | 12B | 14A | 14B | 14C | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 42A | ··· | 42F | 56A | ··· | 56L | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 2 | 2 | 84 | 84 | 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 |
87 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D7 | Q16 | D12 | D14 | D21 | Dic12 | D28 | D42 | Dic28 | D84 | Dic84 |
| kernel | Dic84 | C168 | Dic42 | C56 | C42 | C28 | C24 | C21 | C14 | C12 | C8 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 4 | 6 | 6 | 12 | 12 | 24 |
Matrix representation of Dic84 ►in GL2(𝔽337) generated by
| 184 | 299 |
| 38 | 320 |
| 120 | 209 |
| 152 | 217 |
G:=sub<GL(2,GF(337))| [184,38,299,320],[120,152,209,217] >;
Dic84 in GAP, Magma, Sage, TeX
{\rm Dic}_{84} % in TeX
G:=Group("Dic84"); // GroupNames label
G:=SmallGroup(336,94);
// by ID
G=gap.SmallGroup(336,94);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,73,79,218,50,964,10373]);
// Polycyclic
G:=Group<a,b|a^168=1,b^2=a^84,b*a*b^-1=a^-1>;
// generators/relations
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