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G = Dic87order 348 = 22·3·29

Dicyclic group

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

Aliases: Dic87, C873C4, C58.S3, C2.D87, C6.D29, C3⋊Dic29, C292Dic3, C174.1C2, SmallGroup(348,3)

Series: Derived Chief Lower central Upper central

C1C87 — Dic87
C1C29C87C174 — Dic87
C87 — Dic87
C1C2

Generators and relations for Dic87
 G = < a,b | a174=1, b2=a87, bab-1=a-1 >

87C4
29Dic3
3Dic29

Smallest permutation representation of Dic87
Regular action on 348 points
Generators in S348
(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)(175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348)
(1 324 88 237)(2 323 89 236)(3 322 90 235)(4 321 91 234)(5 320 92 233)(6 319 93 232)(7 318 94 231)(8 317 95 230)(9 316 96 229)(10 315 97 228)(11 314 98 227)(12 313 99 226)(13 312 100 225)(14 311 101 224)(15 310 102 223)(16 309 103 222)(17 308 104 221)(18 307 105 220)(19 306 106 219)(20 305 107 218)(21 304 108 217)(22 303 109 216)(23 302 110 215)(24 301 111 214)(25 300 112 213)(26 299 113 212)(27 298 114 211)(28 297 115 210)(29 296 116 209)(30 295 117 208)(31 294 118 207)(32 293 119 206)(33 292 120 205)(34 291 121 204)(35 290 122 203)(36 289 123 202)(37 288 124 201)(38 287 125 200)(39 286 126 199)(40 285 127 198)(41 284 128 197)(42 283 129 196)(43 282 130 195)(44 281 131 194)(45 280 132 193)(46 279 133 192)(47 278 134 191)(48 277 135 190)(49 276 136 189)(50 275 137 188)(51 274 138 187)(52 273 139 186)(53 272 140 185)(54 271 141 184)(55 270 142 183)(56 269 143 182)(57 268 144 181)(58 267 145 180)(59 266 146 179)(60 265 147 178)(61 264 148 177)(62 263 149 176)(63 262 150 175)(64 261 151 348)(65 260 152 347)(66 259 153 346)(67 258 154 345)(68 257 155 344)(69 256 156 343)(70 255 157 342)(71 254 158 341)(72 253 159 340)(73 252 160 339)(74 251 161 338)(75 250 162 337)(76 249 163 336)(77 248 164 335)(78 247 165 334)(79 246 166 333)(80 245 167 332)(81 244 168 331)(82 243 169 330)(83 242 170 329)(84 241 171 328)(85 240 172 327)(86 239 173 326)(87 238 174 325)

G:=sub<Sym(348)| (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)(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348), (1,324,88,237)(2,323,89,236)(3,322,90,235)(4,321,91,234)(5,320,92,233)(6,319,93,232)(7,318,94,231)(8,317,95,230)(9,316,96,229)(10,315,97,228)(11,314,98,227)(12,313,99,226)(13,312,100,225)(14,311,101,224)(15,310,102,223)(16,309,103,222)(17,308,104,221)(18,307,105,220)(19,306,106,219)(20,305,107,218)(21,304,108,217)(22,303,109,216)(23,302,110,215)(24,301,111,214)(25,300,112,213)(26,299,113,212)(27,298,114,211)(28,297,115,210)(29,296,116,209)(30,295,117,208)(31,294,118,207)(32,293,119,206)(33,292,120,205)(34,291,121,204)(35,290,122,203)(36,289,123,202)(37,288,124,201)(38,287,125,200)(39,286,126,199)(40,285,127,198)(41,284,128,197)(42,283,129,196)(43,282,130,195)(44,281,131,194)(45,280,132,193)(46,279,133,192)(47,278,134,191)(48,277,135,190)(49,276,136,189)(50,275,137,188)(51,274,138,187)(52,273,139,186)(53,272,140,185)(54,271,141,184)(55,270,142,183)(56,269,143,182)(57,268,144,181)(58,267,145,180)(59,266,146,179)(60,265,147,178)(61,264,148,177)(62,263,149,176)(63,262,150,175)(64,261,151,348)(65,260,152,347)(66,259,153,346)(67,258,154,345)(68,257,155,344)(69,256,156,343)(70,255,157,342)(71,254,158,341)(72,253,159,340)(73,252,160,339)(74,251,161,338)(75,250,162,337)(76,249,163,336)(77,248,164,335)(78,247,165,334)(79,246,166,333)(80,245,167,332)(81,244,168,331)(82,243,169,330)(83,242,170,329)(84,241,171,328)(85,240,172,327)(86,239,173,326)(87,238,174,325)>;

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)(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348), (1,324,88,237)(2,323,89,236)(3,322,90,235)(4,321,91,234)(5,320,92,233)(6,319,93,232)(7,318,94,231)(8,317,95,230)(9,316,96,229)(10,315,97,228)(11,314,98,227)(12,313,99,226)(13,312,100,225)(14,311,101,224)(15,310,102,223)(16,309,103,222)(17,308,104,221)(18,307,105,220)(19,306,106,219)(20,305,107,218)(21,304,108,217)(22,303,109,216)(23,302,110,215)(24,301,111,214)(25,300,112,213)(26,299,113,212)(27,298,114,211)(28,297,115,210)(29,296,116,209)(30,295,117,208)(31,294,118,207)(32,293,119,206)(33,292,120,205)(34,291,121,204)(35,290,122,203)(36,289,123,202)(37,288,124,201)(38,287,125,200)(39,286,126,199)(40,285,127,198)(41,284,128,197)(42,283,129,196)(43,282,130,195)(44,281,131,194)(45,280,132,193)(46,279,133,192)(47,278,134,191)(48,277,135,190)(49,276,136,189)(50,275,137,188)(51,274,138,187)(52,273,139,186)(53,272,140,185)(54,271,141,184)(55,270,142,183)(56,269,143,182)(57,268,144,181)(58,267,145,180)(59,266,146,179)(60,265,147,178)(61,264,148,177)(62,263,149,176)(63,262,150,175)(64,261,151,348)(65,260,152,347)(66,259,153,346)(67,258,154,345)(68,257,155,344)(69,256,156,343)(70,255,157,342)(71,254,158,341)(72,253,159,340)(73,252,160,339)(74,251,161,338)(75,250,162,337)(76,249,163,336)(77,248,164,335)(78,247,165,334)(79,246,166,333)(80,245,167,332)(81,244,168,331)(82,243,169,330)(83,242,170,329)(84,241,171,328)(85,240,172,327)(86,239,173,326)(87,238,174,325) );

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),(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348)], [(1,324,88,237),(2,323,89,236),(3,322,90,235),(4,321,91,234),(5,320,92,233),(6,319,93,232),(7,318,94,231),(8,317,95,230),(9,316,96,229),(10,315,97,228),(11,314,98,227),(12,313,99,226),(13,312,100,225),(14,311,101,224),(15,310,102,223),(16,309,103,222),(17,308,104,221),(18,307,105,220),(19,306,106,219),(20,305,107,218),(21,304,108,217),(22,303,109,216),(23,302,110,215),(24,301,111,214),(25,300,112,213),(26,299,113,212),(27,298,114,211),(28,297,115,210),(29,296,116,209),(30,295,117,208),(31,294,118,207),(32,293,119,206),(33,292,120,205),(34,291,121,204),(35,290,122,203),(36,289,123,202),(37,288,124,201),(38,287,125,200),(39,286,126,199),(40,285,127,198),(41,284,128,197),(42,283,129,196),(43,282,130,195),(44,281,131,194),(45,280,132,193),(46,279,133,192),(47,278,134,191),(48,277,135,190),(49,276,136,189),(50,275,137,188),(51,274,138,187),(52,273,139,186),(53,272,140,185),(54,271,141,184),(55,270,142,183),(56,269,143,182),(57,268,144,181),(58,267,145,180),(59,266,146,179),(60,265,147,178),(61,264,148,177),(62,263,149,176),(63,262,150,175),(64,261,151,348),(65,260,152,347),(66,259,153,346),(67,258,154,345),(68,257,155,344),(69,256,156,343),(70,255,157,342),(71,254,158,341),(72,253,159,340),(73,252,160,339),(74,251,161,338),(75,250,162,337),(76,249,163,336),(77,248,164,335),(78,247,165,334),(79,246,166,333),(80,245,167,332),(81,244,168,331),(82,243,169,330),(83,242,170,329),(84,241,171,328),(85,240,172,327),(86,239,173,326),(87,238,174,325)])

90 conjugacy classes

class 1  2  3 4A4B 6 29A···29N58A···58N87A···87AB174A···174AB
order12344629···2958···5887···87174···174
size112878722···22···22···22···2

90 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4S3Dic3D29Dic29D87Dic87
kernelDic87C174C87C58C29C6C3C2C1
# reps1121114142828

Matrix representation of Dic87 in GL4(𝔽349) generated by

259100
348000
0077118
00317164
,
28522200
466400
007543
0064274
G:=sub<GL(4,GF(349))| [259,348,0,0,1,0,0,0,0,0,77,317,0,0,118,164],[285,46,0,0,222,64,0,0,0,0,75,64,0,0,43,274] >;

Dic87 in GAP, Magma, Sage, TeX

{\rm Dic}_{87}
% in TeX

G:=Group("Dic87");
// GroupNames label

G:=SmallGroup(348,3);
// by ID

G=gap.SmallGroup(348,3);
# by ID

G:=PCGroup([4,-2,-2,-3,-29,8,98,5379]);
// Polycyclic

G:=Group<a,b|a^174=1,b^2=a^87,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic87 in TeX

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