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G = Dic84order 336 = 24·3·7

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic84, C8.D21, C214Q16, C56.1S3, C24.1D7, C2.5D84, C6.3D28, C31Dic28, C71Dic12, 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

C1C84 — Dic84
C1C7C21C42C84Dic42 — Dic84
C21C42C84 — Dic84
C1C2C4C8

Generators and relations for Dic84
 G = < a,b | a168=1, b2=a84, bab-1=a-1 >

42C4
42C4
21Q8
21Q8
14Dic3
14Dic3
6Dic7
6Dic7
21Q16
7Dic6
7Dic6
3Dic14
3Dic14
2Dic21
2Dic21
7Dic12
3Dic28

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)
(1 296 85 212)(2 295 86 211)(3 294 87 210)(4 293 88 209)(5 292 89 208)(6 291 90 207)(7 290 91 206)(8 289 92 205)(9 288 93 204)(10 287 94 203)(11 286 95 202)(12 285 96 201)(13 284 97 200)(14 283 98 199)(15 282 99 198)(16 281 100 197)(17 280 101 196)(18 279 102 195)(19 278 103 194)(20 277 104 193)(21 276 105 192)(22 275 106 191)(23 274 107 190)(24 273 108 189)(25 272 109 188)(26 271 110 187)(27 270 111 186)(28 269 112 185)(29 268 113 184)(30 267 114 183)(31 266 115 182)(32 265 116 181)(33 264 117 180)(34 263 118 179)(35 262 119 178)(36 261 120 177)(37 260 121 176)(38 259 122 175)(39 258 123 174)(40 257 124 173)(41 256 125 172)(42 255 126 171)(43 254 127 170)(44 253 128 169)(45 252 129 336)(46 251 130 335)(47 250 131 334)(48 249 132 333)(49 248 133 332)(50 247 134 331)(51 246 135 330)(52 245 136 329)(53 244 137 328)(54 243 138 327)(55 242 139 326)(56 241 140 325)(57 240 141 324)(58 239 142 323)(59 238 143 322)(60 237 144 321)(61 236 145 320)(62 235 146 319)(63 234 147 318)(64 233 148 317)(65 232 149 316)(66 231 150 315)(67 230 151 314)(68 229 152 313)(69 228 153 312)(70 227 154 311)(71 226 155 310)(72 225 156 309)(73 224 157 308)(74 223 158 307)(75 222 159 306)(76 221 160 305)(77 220 161 304)(78 219 162 303)(79 218 163 302)(80 217 164 301)(81 216 165 300)(82 215 166 299)(83 214 167 298)(84 213 168 297)

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), (1,296,85,212)(2,295,86,211)(3,294,87,210)(4,293,88,209)(5,292,89,208)(6,291,90,207)(7,290,91,206)(8,289,92,205)(9,288,93,204)(10,287,94,203)(11,286,95,202)(12,285,96,201)(13,284,97,200)(14,283,98,199)(15,282,99,198)(16,281,100,197)(17,280,101,196)(18,279,102,195)(19,278,103,194)(20,277,104,193)(21,276,105,192)(22,275,106,191)(23,274,107,190)(24,273,108,189)(25,272,109,188)(26,271,110,187)(27,270,111,186)(28,269,112,185)(29,268,113,184)(30,267,114,183)(31,266,115,182)(32,265,116,181)(33,264,117,180)(34,263,118,179)(35,262,119,178)(36,261,120,177)(37,260,121,176)(38,259,122,175)(39,258,123,174)(40,257,124,173)(41,256,125,172)(42,255,126,171)(43,254,127,170)(44,253,128,169)(45,252,129,336)(46,251,130,335)(47,250,131,334)(48,249,132,333)(49,248,133,332)(50,247,134,331)(51,246,135,330)(52,245,136,329)(53,244,137,328)(54,243,138,327)(55,242,139,326)(56,241,140,325)(57,240,141,324)(58,239,142,323)(59,238,143,322)(60,237,144,321)(61,236,145,320)(62,235,146,319)(63,234,147,318)(64,233,148,317)(65,232,149,316)(66,231,150,315)(67,230,151,314)(68,229,152,313)(69,228,153,312)(70,227,154,311)(71,226,155,310)(72,225,156,309)(73,224,157,308)(74,223,158,307)(75,222,159,306)(76,221,160,305)(77,220,161,304)(78,219,162,303)(79,218,163,302)(80,217,164,301)(81,216,165,300)(82,215,166,299)(83,214,167,298)(84,213,168,297)>;

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), (1,296,85,212)(2,295,86,211)(3,294,87,210)(4,293,88,209)(5,292,89,208)(6,291,90,207)(7,290,91,206)(8,289,92,205)(9,288,93,204)(10,287,94,203)(11,286,95,202)(12,285,96,201)(13,284,97,200)(14,283,98,199)(15,282,99,198)(16,281,100,197)(17,280,101,196)(18,279,102,195)(19,278,103,194)(20,277,104,193)(21,276,105,192)(22,275,106,191)(23,274,107,190)(24,273,108,189)(25,272,109,188)(26,271,110,187)(27,270,111,186)(28,269,112,185)(29,268,113,184)(30,267,114,183)(31,266,115,182)(32,265,116,181)(33,264,117,180)(34,263,118,179)(35,262,119,178)(36,261,120,177)(37,260,121,176)(38,259,122,175)(39,258,123,174)(40,257,124,173)(41,256,125,172)(42,255,126,171)(43,254,127,170)(44,253,128,169)(45,252,129,336)(46,251,130,335)(47,250,131,334)(48,249,132,333)(49,248,133,332)(50,247,134,331)(51,246,135,330)(52,245,136,329)(53,244,137,328)(54,243,138,327)(55,242,139,326)(56,241,140,325)(57,240,141,324)(58,239,142,323)(59,238,143,322)(60,237,144,321)(61,236,145,320)(62,235,146,319)(63,234,147,318)(64,233,148,317)(65,232,149,316)(66,231,150,315)(67,230,151,314)(68,229,152,313)(69,228,153,312)(70,227,154,311)(71,226,155,310)(72,225,156,309)(73,224,157,308)(74,223,158,307)(75,222,159,306)(76,221,160,305)(77,220,161,304)(78,219,162,303)(79,218,163,302)(80,217,164,301)(81,216,165,300)(82,215,166,299)(83,214,167,298)(84,213,168,297) );

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)], [(1,296,85,212),(2,295,86,211),(3,294,87,210),(4,293,88,209),(5,292,89,208),(6,291,90,207),(7,290,91,206),(8,289,92,205),(9,288,93,204),(10,287,94,203),(11,286,95,202),(12,285,96,201),(13,284,97,200),(14,283,98,199),(15,282,99,198),(16,281,100,197),(17,280,101,196),(18,279,102,195),(19,278,103,194),(20,277,104,193),(21,276,105,192),(22,275,106,191),(23,274,107,190),(24,273,108,189),(25,272,109,188),(26,271,110,187),(27,270,111,186),(28,269,112,185),(29,268,113,184),(30,267,114,183),(31,266,115,182),(32,265,116,181),(33,264,117,180),(34,263,118,179),(35,262,119,178),(36,261,120,177),(37,260,121,176),(38,259,122,175),(39,258,123,174),(40,257,124,173),(41,256,125,172),(42,255,126,171),(43,254,127,170),(44,253,128,169),(45,252,129,336),(46,251,130,335),(47,250,131,334),(48,249,132,333),(49,248,133,332),(50,247,134,331),(51,246,135,330),(52,245,136,329),(53,244,137,328),(54,243,138,327),(55,242,139,326),(56,241,140,325),(57,240,141,324),(58,239,142,323),(59,238,143,322),(60,237,144,321),(61,236,145,320),(62,235,146,319),(63,234,147,318),(64,233,148,317),(65,232,149,316),(66,231,150,315),(67,230,151,314),(68,229,152,313),(69,228,153,312),(70,227,154,311),(71,226,155,310),(72,225,156,309),(73,224,157,308),(74,223,158,307),(75,222,159,306),(76,221,160,305),(77,220,161,304),(78,219,162,303),(79,218,163,302),(80,217,164,301),(81,216,165,300),(82,215,166,299),(83,214,167,298),(84,213,168,297)])

87 conjugacy classes

class 1  2  3 4A4B4C 6 7A7B7C8A8B12A12B14A14B14C21A···21F24A24B24C24D28A···28F42A···42F56A···56L84A···84L168A···168X
order123444677788121214141421···212424242428···2842···4256···5684···84168···168
size11228484222222222222···222222···22···22···22···22···2

87 irreducible representations

dim11122222222222222
type+++++++-+++-++-+-
imageC1C2C2S3D4D6D7Q16D12D14D21Dic12D28D42Dic28D84Dic84
kernelDic84C168Dic42C56C42C28C24C21C14C12C8C7C6C4C3C2C1
# reps11211132236466121224

Matrix representation of Dic84 in GL2(𝔽337) generated by

184299
38320
,
120209
152217
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

Export

Subgroup lattice of Dic84 in TeX

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