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G = Dic93order 372 = 22·3·31

Dicyclic group

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

Aliases: Dic93, C931C4, C62.S3, C6.D31, C2.D93, C3⋊Dic31, C31⋊Dic3, C186.1C2, SmallGroup(372,5)

Series: Derived Chief Lower central Upper central

C1C93 — Dic93
C1C31C93C186 — Dic93
C93 — Dic93
C1C2

Generators and relations for Dic93
 G = < a,b | a186=1, b2=a93, bab-1=a-1 >

93C4
31Dic3
3Dic31

Smallest permutation representation of Dic93
Regular action on 372 points
Generators in S372
(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)
(1 254 94 347)(2 253 95 346)(3 252 96 345)(4 251 97 344)(5 250 98 343)(6 249 99 342)(7 248 100 341)(8 247 101 340)(9 246 102 339)(10 245 103 338)(11 244 104 337)(12 243 105 336)(13 242 106 335)(14 241 107 334)(15 240 108 333)(16 239 109 332)(17 238 110 331)(18 237 111 330)(19 236 112 329)(20 235 113 328)(21 234 114 327)(22 233 115 326)(23 232 116 325)(24 231 117 324)(25 230 118 323)(26 229 119 322)(27 228 120 321)(28 227 121 320)(29 226 122 319)(30 225 123 318)(31 224 124 317)(32 223 125 316)(33 222 126 315)(34 221 127 314)(35 220 128 313)(36 219 129 312)(37 218 130 311)(38 217 131 310)(39 216 132 309)(40 215 133 308)(41 214 134 307)(42 213 135 306)(43 212 136 305)(44 211 137 304)(45 210 138 303)(46 209 139 302)(47 208 140 301)(48 207 141 300)(49 206 142 299)(50 205 143 298)(51 204 144 297)(52 203 145 296)(53 202 146 295)(54 201 147 294)(55 200 148 293)(56 199 149 292)(57 198 150 291)(58 197 151 290)(59 196 152 289)(60 195 153 288)(61 194 154 287)(62 193 155 286)(63 192 156 285)(64 191 157 284)(65 190 158 283)(66 189 159 282)(67 188 160 281)(68 187 161 280)(69 372 162 279)(70 371 163 278)(71 370 164 277)(72 369 165 276)(73 368 166 275)(74 367 167 274)(75 366 168 273)(76 365 169 272)(77 364 170 271)(78 363 171 270)(79 362 172 269)(80 361 173 268)(81 360 174 267)(82 359 175 266)(83 358 176 265)(84 357 177 264)(85 356 178 263)(86 355 179 262)(87 354 180 261)(88 353 181 260)(89 352 182 259)(90 351 183 258)(91 350 184 257)(92 349 185 256)(93 348 186 255)

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

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

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

96 conjugacy classes

class 1  2  3 4A4B 6 31A···31O62A···62O93A···93AD186A···186AD
order12344631···3162···6293···93186···186
size112939322···22···22···22···2

96 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4S3Dic3D31Dic31D93Dic93
kernelDic93C186C93C62C31C6C3C2C1
# reps1121115153030

Matrix representation of Dic93 in GL2(𝔽373) generated by

83175
19831
,
81147
270292
G:=sub<GL(2,GF(373))| [83,198,175,31],[81,270,147,292] >;

Dic93 in GAP, Magma, Sage, TeX

{\rm Dic}_{93}
% in TeX

G:=Group("Dic93");
// GroupNames label

G:=SmallGroup(372,5);
// by ID

G=gap.SmallGroup(372,5);
# by ID

G:=PCGroup([4,-2,-2,-3,-31,8,98,5763]);
// Polycyclic

G:=Group<a,b|a^186=1,b^2=a^93,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic93 in TeX

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