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G = Dic97order 388 = 22·97

Dicyclic group

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

Aliases: Dic97, C972C4, C2.D97, C194.C2, SmallGroup(388,1)

Series: Derived Chief Lower central Upper central

C1C97 — Dic97
C1C97C194 — Dic97
C97 — Dic97
C1C2

Generators and relations for Dic97
 G = < a,b | a194=1, b2=a97, bab-1=a-1 >

97C4

Smallest permutation representation of Dic97
Regular action on 388 points
Generators in S388
(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)
(1 353 98 256)(2 352 99 255)(3 351 100 254)(4 350 101 253)(5 349 102 252)(6 348 103 251)(7 347 104 250)(8 346 105 249)(9 345 106 248)(10 344 107 247)(11 343 108 246)(12 342 109 245)(13 341 110 244)(14 340 111 243)(15 339 112 242)(16 338 113 241)(17 337 114 240)(18 336 115 239)(19 335 116 238)(20 334 117 237)(21 333 118 236)(22 332 119 235)(23 331 120 234)(24 330 121 233)(25 329 122 232)(26 328 123 231)(27 327 124 230)(28 326 125 229)(29 325 126 228)(30 324 127 227)(31 323 128 226)(32 322 129 225)(33 321 130 224)(34 320 131 223)(35 319 132 222)(36 318 133 221)(37 317 134 220)(38 316 135 219)(39 315 136 218)(40 314 137 217)(41 313 138 216)(42 312 139 215)(43 311 140 214)(44 310 141 213)(45 309 142 212)(46 308 143 211)(47 307 144 210)(48 306 145 209)(49 305 146 208)(50 304 147 207)(51 303 148 206)(52 302 149 205)(53 301 150 204)(54 300 151 203)(55 299 152 202)(56 298 153 201)(57 297 154 200)(58 296 155 199)(59 295 156 198)(60 294 157 197)(61 293 158 196)(62 292 159 195)(63 291 160 388)(64 290 161 387)(65 289 162 386)(66 288 163 385)(67 287 164 384)(68 286 165 383)(69 285 166 382)(70 284 167 381)(71 283 168 380)(72 282 169 379)(73 281 170 378)(74 280 171 377)(75 279 172 376)(76 278 173 375)(77 277 174 374)(78 276 175 373)(79 275 176 372)(80 274 177 371)(81 273 178 370)(82 272 179 369)(83 271 180 368)(84 270 181 367)(85 269 182 366)(86 268 183 365)(87 267 184 364)(88 266 185 363)(89 265 186 362)(90 264 187 361)(91 263 188 360)(92 262 189 359)(93 261 190 358)(94 260 191 357)(95 259 192 356)(96 258 193 355)(97 257 194 354)

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

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

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

100 conjugacy classes

class 1  2 4A4B97A···97AV194A···194AV
order124497···97194···194
size1197972···22···2

100 irreducible representations

dim11122
type+++-
imageC1C2C4D97Dic97
kernelDic97C194C97C2C1
# reps1124848

Matrix representation of Dic97 in GL3(𝔽389) generated by

38800
0329388
010
,
27400
0217387
0204172
G:=sub<GL(3,GF(389))| [388,0,0,0,329,1,0,388,0],[274,0,0,0,217,204,0,387,172] >;

Dic97 in GAP, Magma, Sage, TeX

{\rm Dic}_{97}
% in TeX

G:=Group("Dic97");
// GroupNames label

G:=SmallGroup(388,1);
// by ID

G=gap.SmallGroup(388,1);
# by ID

G:=PCGroup([3,-2,-2,-97,6,3458]);
// Polycyclic

G:=Group<a,b|a^194=1,b^2=a^97,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic97 in TeX

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