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G = Dic91order 364 = 22·7·13

Dicyclic group

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

Aliases: Dic91, C913C4, C26.D7, C2.D91, C7⋊Dic13, C14.D13, C132Dic7, C182.1C2, SmallGroup(364,3)

Series: Derived Chief Lower central Upper central

C1C91 — Dic91
C1C13C91C182 — Dic91
C91 — Dic91
C1C2

Generators and relations for Dic91
 G = < a,b | a182=1, b2=a91, bab-1=a-1 >

91C4
13Dic7
7Dic13

Smallest permutation representation of Dic91
Regular action on 364 points
Generators in S364
(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)
(1 197 92 288)(2 196 93 287)(3 195 94 286)(4 194 95 285)(5 193 96 284)(6 192 97 283)(7 191 98 282)(8 190 99 281)(9 189 100 280)(10 188 101 279)(11 187 102 278)(12 186 103 277)(13 185 104 276)(14 184 105 275)(15 183 106 274)(16 364 107 273)(17 363 108 272)(18 362 109 271)(19 361 110 270)(20 360 111 269)(21 359 112 268)(22 358 113 267)(23 357 114 266)(24 356 115 265)(25 355 116 264)(26 354 117 263)(27 353 118 262)(28 352 119 261)(29 351 120 260)(30 350 121 259)(31 349 122 258)(32 348 123 257)(33 347 124 256)(34 346 125 255)(35 345 126 254)(36 344 127 253)(37 343 128 252)(38 342 129 251)(39 341 130 250)(40 340 131 249)(41 339 132 248)(42 338 133 247)(43 337 134 246)(44 336 135 245)(45 335 136 244)(46 334 137 243)(47 333 138 242)(48 332 139 241)(49 331 140 240)(50 330 141 239)(51 329 142 238)(52 328 143 237)(53 327 144 236)(54 326 145 235)(55 325 146 234)(56 324 147 233)(57 323 148 232)(58 322 149 231)(59 321 150 230)(60 320 151 229)(61 319 152 228)(62 318 153 227)(63 317 154 226)(64 316 155 225)(65 315 156 224)(66 314 157 223)(67 313 158 222)(68 312 159 221)(69 311 160 220)(70 310 161 219)(71 309 162 218)(72 308 163 217)(73 307 164 216)(74 306 165 215)(75 305 166 214)(76 304 167 213)(77 303 168 212)(78 302 169 211)(79 301 170 210)(80 300 171 209)(81 299 172 208)(82 298 173 207)(83 297 174 206)(84 296 175 205)(85 295 176 204)(86 294 177 203)(87 293 178 202)(88 292 179 201)(89 291 180 200)(90 290 181 199)(91 289 182 198)

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

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

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

94 conjugacy classes

class 1  2 4A4B7A7B7C13A···13F14A14B14C26A···26F91A···91AJ182A···182AJ
order124477713···1314141426···2691···91182···182
size1191912222···22222···22···22···2

94 irreducible representations

dim111222222
type++++--+-
imageC1C2C4D7D13Dic7Dic13D91Dic91
kernelDic91C182C91C26C14C13C7C2C1
# reps11236363636

Matrix representation of Dic91 in GL3(𝔽1093) generated by

109200
0512922
0171404
,
53000
0219703
0151874
G:=sub<GL(3,GF(1093))| [1092,0,0,0,512,171,0,922,404],[530,0,0,0,219,151,0,703,874] >;

Dic91 in GAP, Magma, Sage, TeX

{\rm Dic}_{91}
% in TeX

G:=Group("Dic91");
// GroupNames label

G:=SmallGroup(364,3);
// by ID

G=gap.SmallGroup(364,3);
# by ID

G:=PCGroup([4,-2,-2,-7,-13,8,290,5379]);
// Polycyclic

G:=Group<a,b|a^182=1,b^2=a^91,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic91 in TeX

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