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G = C13×Dic6order 312 = 23·3·13

Direct product of C13 and Dic6

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

Aliases: C13×Dic6, C394Q8, C52.3S3, C156.5C2, C12.1C26, C26.13D6, Dic3.C26, C78.18C22, C3⋊(Q8×C13), C4.(S3×C13), C2.3(S3×C26), C6.1(C2×C26), (Dic3×C13).2C2, SmallGroup(312,32)

Series: Derived Chief Lower central Upper central

C1C6 — C13×Dic6
C1C3C6C78Dic3×C13 — C13×Dic6
C3C6 — C13×Dic6
C1C26C52

Generators and relations for C13×Dic6
 G = < a,b,c | a13=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C4
3Q8
3C52
3C52
3Q8×C13

Smallest permutation representation of C13×Dic6
Regular action on 312 points
Generators in S312
(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)
(1 307 227 112 167 274 88 212 297 173 266 68)(2 308 228 113 168 275 89 213 298 174 267 69)(3 309 229 114 169 276 90 214 299 175 268 70)(4 310 230 115 157 277 91 215 287 176 269 71)(5 311 231 116 158 278 79 216 288 177 270 72)(6 312 232 117 159 279 80 217 289 178 271 73)(7 300 233 105 160 280 81 218 290 179 272 74)(8 301 234 106 161 281 82 219 291 180 273 75)(9 302 222 107 162 282 83 220 292 181 261 76)(10 303 223 108 163 283 84 221 293 182 262 77)(11 304 224 109 164 284 85 209 294 170 263 78)(12 305 225 110 165 285 86 210 295 171 264 66)(13 306 226 111 166 286 87 211 296 172 265 67)(14 60 35 45 191 152 126 208 98 256 240 131)(15 61 36 46 192 153 127 196 99 257 241 132)(16 62 37 47 193 154 128 197 100 258 242 133)(17 63 38 48 194 155 129 198 101 259 243 134)(18 64 39 49 195 156 130 199 102 260 244 135)(19 65 27 50 183 144 118 200 103 248 245 136)(20 53 28 51 184 145 119 201 104 249 246 137)(21 54 29 52 185 146 120 202 92 250 247 138)(22 55 30 40 186 147 121 203 93 251 235 139)(23 56 31 41 187 148 122 204 94 252 236 140)(24 57 32 42 188 149 123 205 95 253 237 141)(25 58 33 43 189 150 124 206 96 254 238 142)(26 59 34 44 190 151 125 207 97 255 239 143)
(1 92 88 29)(2 93 89 30)(3 94 90 31)(4 95 91 32)(5 96 79 33)(6 97 80 34)(7 98 81 35)(8 99 82 36)(9 100 83 37)(10 101 84 38)(11 102 85 39)(12 103 86 27)(13 104 87 28)(14 233 126 290)(15 234 127 291)(16 222 128 292)(17 223 129 293)(18 224 130 294)(19 225 118 295)(20 226 119 296)(21 227 120 297)(22 228 121 298)(23 229 122 299)(24 230 123 287)(25 231 124 288)(26 232 125 289)(40 69 251 275)(41 70 252 276)(42 71 253 277)(43 72 254 278)(44 73 255 279)(45 74 256 280)(46 75 257 281)(47 76 258 282)(48 77 259 283)(49 78 260 284)(50 66 248 285)(51 67 249 286)(52 68 250 274)(53 306 201 211)(54 307 202 212)(55 308 203 213)(56 309 204 214)(57 310 205 215)(58 311 206 216)(59 312 207 217)(60 300 208 218)(61 301 196 219)(62 302 197 220)(63 303 198 221)(64 304 199 209)(65 305 200 210)(105 152 179 131)(106 153 180 132)(107 154 181 133)(108 155 182 134)(109 156 170 135)(110 144 171 136)(111 145 172 137)(112 146 173 138)(113 147 174 139)(114 148 175 140)(115 149 176 141)(116 150 177 142)(117 151 178 143)(157 188 269 237)(158 189 270 238)(159 190 271 239)(160 191 272 240)(161 192 273 241)(162 193 261 242)(163 194 262 243)(164 195 263 244)(165 183 264 245)(166 184 265 246)(167 185 266 247)(168 186 267 235)(169 187 268 236)

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

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

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

117 conjugacy classes

class 1  2  3 4A4B4C 6 12A12B13A···13L26A···26L39A···39L52A···52L52M···52AJ78A···78L156A···156X
order1234446121213···1326···2639···3952···5252···5278···78156···156
size1122662221···11···12···22···26···62···22···2

117 irreducible representations

dim11111122222222
type++++-+-
imageC1C2C2C13C26C26S3Q8D6Dic6S3×C13Q8×C13S3×C26C13×Dic6
kernelC13×Dic6Dic3×C13C156Dic6Dic3C12C52C39C26C13C4C3C2C1
# reps121122412111212121224

Matrix representation of C13×Dic6 in GL2(𝔽157) generated by

390
039
,
4824
13324
,
12785
11530
G:=sub<GL(2,GF(157))| [39,0,0,39],[48,133,24,24],[127,115,85,30] >;

C13×Dic6 in GAP, Magma, Sage, TeX

C_{13}\times {\rm Dic}_6
% in TeX

G:=Group("C13xDic6");
// GroupNames label

G:=SmallGroup(312,32);
// by ID

G=gap.SmallGroup(312,32);
# by ID

G:=PCGroup([5,-2,-2,-13,-2,-3,260,541,266,5204]);
// Polycyclic

G:=Group<a,b,c|a^13=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C13×Dic6 in TeX

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