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G = Dic98order 392 = 23·72

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic98, C49⋊Q8, C4.D49, C2.3D98, C28.1D7, C7.Dic14, C196.1C2, C14.6D14, Dic49.C2, C98.1C22, SmallGroup(392,3)

Series: Derived Chief Lower central Upper central

C1C98 — Dic98
C1C7C49C98Dic49 — Dic98
C49C98 — Dic98
C1C2C4

Generators and relations for Dic98
 G = < a,b | a196=1, b2=a98, bab-1=a-1 >

49C4
49C4
49Q8
7Dic7
7Dic7
7Dic14

Smallest permutation representation of Dic98
Regular action on 392 points
Generators in S392
(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)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392)
(1 212 99 310)(2 211 100 309)(3 210 101 308)(4 209 102 307)(5 208 103 306)(6 207 104 305)(7 206 105 304)(8 205 106 303)(9 204 107 302)(10 203 108 301)(11 202 109 300)(12 201 110 299)(13 200 111 298)(14 199 112 297)(15 198 113 296)(16 197 114 295)(17 392 115 294)(18 391 116 293)(19 390 117 292)(20 389 118 291)(21 388 119 290)(22 387 120 289)(23 386 121 288)(24 385 122 287)(25 384 123 286)(26 383 124 285)(27 382 125 284)(28 381 126 283)(29 380 127 282)(30 379 128 281)(31 378 129 280)(32 377 130 279)(33 376 131 278)(34 375 132 277)(35 374 133 276)(36 373 134 275)(37 372 135 274)(38 371 136 273)(39 370 137 272)(40 369 138 271)(41 368 139 270)(42 367 140 269)(43 366 141 268)(44 365 142 267)(45 364 143 266)(46 363 144 265)(47 362 145 264)(48 361 146 263)(49 360 147 262)(50 359 148 261)(51 358 149 260)(52 357 150 259)(53 356 151 258)(54 355 152 257)(55 354 153 256)(56 353 154 255)(57 352 155 254)(58 351 156 253)(59 350 157 252)(60 349 158 251)(61 348 159 250)(62 347 160 249)(63 346 161 248)(64 345 162 247)(65 344 163 246)(66 343 164 245)(67 342 165 244)(68 341 166 243)(69 340 167 242)(70 339 168 241)(71 338 169 240)(72 337 170 239)(73 336 171 238)(74 335 172 237)(75 334 173 236)(76 333 174 235)(77 332 175 234)(78 331 176 233)(79 330 177 232)(80 329 178 231)(81 328 179 230)(82 327 180 229)(83 326 181 228)(84 325 182 227)(85 324 183 226)(86 323 184 225)(87 322 185 224)(88 321 186 223)(89 320 187 222)(90 319 188 221)(91 318 189 220)(92 317 190 219)(93 316 191 218)(94 315 192 217)(95 314 193 216)(96 313 194 215)(97 312 195 214)(98 311 196 213)

G:=sub<Sym(392)| (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)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392), (1,212,99,310)(2,211,100,309)(3,210,101,308)(4,209,102,307)(5,208,103,306)(6,207,104,305)(7,206,105,304)(8,205,106,303)(9,204,107,302)(10,203,108,301)(11,202,109,300)(12,201,110,299)(13,200,111,298)(14,199,112,297)(15,198,113,296)(16,197,114,295)(17,392,115,294)(18,391,116,293)(19,390,117,292)(20,389,118,291)(21,388,119,290)(22,387,120,289)(23,386,121,288)(24,385,122,287)(25,384,123,286)(26,383,124,285)(27,382,125,284)(28,381,126,283)(29,380,127,282)(30,379,128,281)(31,378,129,280)(32,377,130,279)(33,376,131,278)(34,375,132,277)(35,374,133,276)(36,373,134,275)(37,372,135,274)(38,371,136,273)(39,370,137,272)(40,369,138,271)(41,368,139,270)(42,367,140,269)(43,366,141,268)(44,365,142,267)(45,364,143,266)(46,363,144,265)(47,362,145,264)(48,361,146,263)(49,360,147,262)(50,359,148,261)(51,358,149,260)(52,357,150,259)(53,356,151,258)(54,355,152,257)(55,354,153,256)(56,353,154,255)(57,352,155,254)(58,351,156,253)(59,350,157,252)(60,349,158,251)(61,348,159,250)(62,347,160,249)(63,346,161,248)(64,345,162,247)(65,344,163,246)(66,343,164,245)(67,342,165,244)(68,341,166,243)(69,340,167,242)(70,339,168,241)(71,338,169,240)(72,337,170,239)(73,336,171,238)(74,335,172,237)(75,334,173,236)(76,333,174,235)(77,332,175,234)(78,331,176,233)(79,330,177,232)(80,329,178,231)(81,328,179,230)(82,327,180,229)(83,326,181,228)(84,325,182,227)(85,324,183,226)(86,323,184,225)(87,322,185,224)(88,321,186,223)(89,320,187,222)(90,319,188,221)(91,318,189,220)(92,317,190,219)(93,316,191,218)(94,315,192,217)(95,314,193,216)(96,313,194,215)(97,312,195,214)(98,311,196,213)>;

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)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392), (1,212,99,310)(2,211,100,309)(3,210,101,308)(4,209,102,307)(5,208,103,306)(6,207,104,305)(7,206,105,304)(8,205,106,303)(9,204,107,302)(10,203,108,301)(11,202,109,300)(12,201,110,299)(13,200,111,298)(14,199,112,297)(15,198,113,296)(16,197,114,295)(17,392,115,294)(18,391,116,293)(19,390,117,292)(20,389,118,291)(21,388,119,290)(22,387,120,289)(23,386,121,288)(24,385,122,287)(25,384,123,286)(26,383,124,285)(27,382,125,284)(28,381,126,283)(29,380,127,282)(30,379,128,281)(31,378,129,280)(32,377,130,279)(33,376,131,278)(34,375,132,277)(35,374,133,276)(36,373,134,275)(37,372,135,274)(38,371,136,273)(39,370,137,272)(40,369,138,271)(41,368,139,270)(42,367,140,269)(43,366,141,268)(44,365,142,267)(45,364,143,266)(46,363,144,265)(47,362,145,264)(48,361,146,263)(49,360,147,262)(50,359,148,261)(51,358,149,260)(52,357,150,259)(53,356,151,258)(54,355,152,257)(55,354,153,256)(56,353,154,255)(57,352,155,254)(58,351,156,253)(59,350,157,252)(60,349,158,251)(61,348,159,250)(62,347,160,249)(63,346,161,248)(64,345,162,247)(65,344,163,246)(66,343,164,245)(67,342,165,244)(68,341,166,243)(69,340,167,242)(70,339,168,241)(71,338,169,240)(72,337,170,239)(73,336,171,238)(74,335,172,237)(75,334,173,236)(76,333,174,235)(77,332,175,234)(78,331,176,233)(79,330,177,232)(80,329,178,231)(81,328,179,230)(82,327,180,229)(83,326,181,228)(84,325,182,227)(85,324,183,226)(86,323,184,225)(87,322,185,224)(88,321,186,223)(89,320,187,222)(90,319,188,221)(91,318,189,220)(92,317,190,219)(93,316,191,218)(94,315,192,217)(95,314,193,216)(96,313,194,215)(97,312,195,214)(98,311,196,213) );

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),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392)], [(1,212,99,310),(2,211,100,309),(3,210,101,308),(4,209,102,307),(5,208,103,306),(6,207,104,305),(7,206,105,304),(8,205,106,303),(9,204,107,302),(10,203,108,301),(11,202,109,300),(12,201,110,299),(13,200,111,298),(14,199,112,297),(15,198,113,296),(16,197,114,295),(17,392,115,294),(18,391,116,293),(19,390,117,292),(20,389,118,291),(21,388,119,290),(22,387,120,289),(23,386,121,288),(24,385,122,287),(25,384,123,286),(26,383,124,285),(27,382,125,284),(28,381,126,283),(29,380,127,282),(30,379,128,281),(31,378,129,280),(32,377,130,279),(33,376,131,278),(34,375,132,277),(35,374,133,276),(36,373,134,275),(37,372,135,274),(38,371,136,273),(39,370,137,272),(40,369,138,271),(41,368,139,270),(42,367,140,269),(43,366,141,268),(44,365,142,267),(45,364,143,266),(46,363,144,265),(47,362,145,264),(48,361,146,263),(49,360,147,262),(50,359,148,261),(51,358,149,260),(52,357,150,259),(53,356,151,258),(54,355,152,257),(55,354,153,256),(56,353,154,255),(57,352,155,254),(58,351,156,253),(59,350,157,252),(60,349,158,251),(61,348,159,250),(62,347,160,249),(63,346,161,248),(64,345,162,247),(65,344,163,246),(66,343,164,245),(67,342,165,244),(68,341,166,243),(69,340,167,242),(70,339,168,241),(71,338,169,240),(72,337,170,239),(73,336,171,238),(74,335,172,237),(75,334,173,236),(76,333,174,235),(77,332,175,234),(78,331,176,233),(79,330,177,232),(80,329,178,231),(81,328,179,230),(82,327,180,229),(83,326,181,228),(84,325,182,227),(85,324,183,226),(86,323,184,225),(87,322,185,224),(88,321,186,223),(89,320,187,222),(90,319,188,221),(91,318,189,220),(92,317,190,219),(93,316,191,218),(94,315,192,217),(95,314,193,216),(96,313,194,215),(97,312,195,214),(98,311,196,213)])

101 conjugacy classes

class 1  2 4A4B4C7A7B7C14A14B14C28A···28F49A···49U98A···98U196A···196AP
order1244477714141428···2849···4998···98196···196
size11298982222222···22···22···22···2

101 irreducible representations

dim1112222222
type+++-++-++-
imageC1C2C2Q8D7D14Dic14D49D98Dic98
kernelDic98Dic49C196C49C28C14C7C4C2C1
# reps1211336212142

Matrix representation of Dic98 in GL2(𝔽197) generated by

11414
183159
,
12187
113185
G:=sub<GL(2,GF(197))| [114,183,14,159],[12,113,187,185] >;

Dic98 in GAP, Magma, Sage, TeX

{\rm Dic}_{98}
% in TeX

G:=Group("Dic98");
// GroupNames label

G:=SmallGroup(392,3);
// by ID

G=gap.SmallGroup(392,3);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-7,20,61,26,2083,858,8404]);
// Polycyclic

G:=Group<a,b|a^196=1,b^2=a^98,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic98 in TeX

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