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G = C3×Dic26order 312 = 23·3·13

Direct product of C3 and Dic26

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C3×Dic26, C393Q8, C52.5C6, C156.3C2, C6.13D26, C12.3D13, C78.13C22, Dic13.3C6, C4.(C3×D13), C133(C3×Q8), C26.9(C2×C6), C2.3(C6×D13), (C3×Dic13).2C2, SmallGroup(312,27)

Series: Derived Chief Lower central Upper central

C1C26 — C3×Dic26
C1C13C26C78C3×Dic13 — C3×Dic26
C13C26 — C3×Dic26
C1C6C12

Generators and relations for C3×Dic26
 G = < a,b,c | a3=b52=1, c2=b26, ab=ba, ac=ca, cbc-1=b-1 >

13C4
13C4
13Q8
13C12
13C12
13C3×Q8

Smallest permutation representation of C3×Dic26
Regular action on 312 points
Generators in S312
(1 264 186)(2 265 187)(3 266 188)(4 267 189)(5 268 190)(6 269 191)(7 270 192)(8 271 193)(9 272 194)(10 273 195)(11 274 196)(12 275 197)(13 276 198)(14 277 199)(15 278 200)(16 279 201)(17 280 202)(18 281 203)(19 282 204)(20 283 205)(21 284 206)(22 285 207)(23 286 208)(24 287 157)(25 288 158)(26 289 159)(27 290 160)(28 291 161)(29 292 162)(30 293 163)(31 294 164)(32 295 165)(33 296 166)(34 297 167)(35 298 168)(36 299 169)(37 300 170)(38 301 171)(39 302 172)(40 303 173)(41 304 174)(42 305 175)(43 306 176)(44 307 177)(45 308 178)(46 309 179)(47 310 180)(48 311 181)(49 312 182)(50 261 183)(51 262 184)(52 263 185)(53 214 120)(54 215 121)(55 216 122)(56 217 123)(57 218 124)(58 219 125)(59 220 126)(60 221 127)(61 222 128)(62 223 129)(63 224 130)(64 225 131)(65 226 132)(66 227 133)(67 228 134)(68 229 135)(69 230 136)(70 231 137)(71 232 138)(72 233 139)(73 234 140)(74 235 141)(75 236 142)(76 237 143)(77 238 144)(78 239 145)(79 240 146)(80 241 147)(81 242 148)(82 243 149)(83 244 150)(84 245 151)(85 246 152)(86 247 153)(87 248 154)(88 249 155)(89 250 156)(90 251 105)(91 252 106)(92 253 107)(93 254 108)(94 255 109)(95 256 110)(96 257 111)(97 258 112)(98 259 113)(99 260 114)(100 209 115)(101 210 116)(102 211 117)(103 212 118)(104 213 119)
(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)
(1 259 27 233)(2 258 28 232)(3 257 29 231)(4 256 30 230)(5 255 31 229)(6 254 32 228)(7 253 33 227)(8 252 34 226)(9 251 35 225)(10 250 36 224)(11 249 37 223)(12 248 38 222)(13 247 39 221)(14 246 40 220)(15 245 41 219)(16 244 42 218)(17 243 43 217)(18 242 44 216)(19 241 45 215)(20 240 46 214)(21 239 47 213)(22 238 48 212)(23 237 49 211)(24 236 50 210)(25 235 51 209)(26 234 52 260)(53 205 79 179)(54 204 80 178)(55 203 81 177)(56 202 82 176)(57 201 83 175)(58 200 84 174)(59 199 85 173)(60 198 86 172)(61 197 87 171)(62 196 88 170)(63 195 89 169)(64 194 90 168)(65 193 91 167)(66 192 92 166)(67 191 93 165)(68 190 94 164)(69 189 95 163)(70 188 96 162)(71 187 97 161)(72 186 98 160)(73 185 99 159)(74 184 100 158)(75 183 101 157)(76 182 102 208)(77 181 103 207)(78 180 104 206)(105 298 131 272)(106 297 132 271)(107 296 133 270)(108 295 134 269)(109 294 135 268)(110 293 136 267)(111 292 137 266)(112 291 138 265)(113 290 139 264)(114 289 140 263)(115 288 141 262)(116 287 142 261)(117 286 143 312)(118 285 144 311)(119 284 145 310)(120 283 146 309)(121 282 147 308)(122 281 148 307)(123 280 149 306)(124 279 150 305)(125 278 151 304)(126 277 152 303)(127 276 153 302)(128 275 154 301)(129 274 155 300)(130 273 156 299)

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

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

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

87 conjugacy classes

class 1  2 3A3B4A4B4C6A6B12A12B12C12D12E12F13A···13F26A···26F39A···39L52A···52L78A···78L156A···156X
order12334446612121212121213···1326···2639···3952···5278···78156···156
size1111226261122262626262···22···22···22···22···22···2

87 irreducible representations

dim11111122222222
type+++-++-
imageC1C2C2C3C6C6Q8C3×Q8D13D26C3×D13Dic26C6×D13C3×Dic26
kernelC3×Dic26C3×Dic13C156Dic26Dic13C52C39C13C12C6C4C3C2C1
# reps121242126612121224

Matrix representation of C3×Dic26 in GL2(𝔽157) generated by

1440
0144
,
6106
6458
,
13641
3921
G:=sub<GL(2,GF(157))| [144,0,0,144],[6,64,106,58],[136,39,41,21] >;

C3×Dic26 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{26}
% in TeX

G:=Group("C3xDic26");
// GroupNames label

G:=SmallGroup(312,27);
// by ID

G=gap.SmallGroup(312,27);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,60,141,66,7204]);
// Polycyclic

G:=Group<a,b,c|a^3=b^52=1,c^2=b^26,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic26 in TeX

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