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G = C3×C132C8order 312 = 23·3·13

Direct product of C3 and C132C8

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

Aliases: C3×C132C8, C394C8, C135C24, C78.4C4, C52.6C6, C156.4C2, C26.5C12, C12.4D13, C6.2Dic13, C4.2(C3×D13), C2.(C3×Dic13), SmallGroup(312,4)

Series: Derived Chief Lower central Upper central

C1C13 — C3×C132C8
C1C13C26C52C156 — C3×C132C8
C13 — C3×C132C8
C1C12

Generators and relations for C3×C132C8
 G = < a,b,c | a3=b13=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

13C8
13C24

Smallest permutation representation of C3×C132C8
Regular action on 312 points
Generators in S312
(1 105 53)(2 106 54)(3 107 55)(4 108 56)(5 109 57)(6 110 58)(7 111 59)(8 112 60)(9 113 61)(10 114 62)(11 115 63)(12 116 64)(13 117 65)(14 118 66)(15 119 67)(16 120 68)(17 121 69)(18 122 70)(19 123 71)(20 124 72)(21 125 73)(22 126 74)(23 127 75)(24 128 76)(25 129 77)(26 130 78)(27 131 79)(28 132 80)(29 133 81)(30 134 82)(31 135 83)(32 136 84)(33 137 85)(34 138 86)(35 139 87)(36 140 88)(37 141 89)(38 142 90)(39 143 91)(40 144 92)(41 145 93)(42 146 94)(43 147 95)(44 148 96)(45 149 97)(46 150 98)(47 151 99)(48 152 100)(49 153 101)(50 154 102)(51 155 103)(52 156 104)(157 267 211)(158 268 212)(159 269 213)(160 270 214)(161 271 215)(162 272 216)(163 273 217)(164 261 218)(165 262 219)(166 263 220)(167 264 221)(168 265 209)(169 266 210)(170 275 224)(171 276 225)(172 277 226)(173 278 227)(174 279 228)(175 280 229)(176 281 230)(177 282 231)(178 283 232)(179 284 233)(180 285 234)(181 286 222)(182 274 223)(183 292 241)(184 293 242)(185 294 243)(186 295 244)(187 296 245)(188 297 246)(189 298 247)(190 299 235)(191 287 236)(192 288 237)(193 289 238)(194 290 239)(195 291 240)(196 300 251)(197 301 252)(198 302 253)(199 303 254)(200 304 255)(201 305 256)(202 306 257)(203 307 258)(204 308 259)(205 309 260)(206 310 248)(207 311 249)(208 312 250)
(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)
(1 197 40 176 14 186 27 166)(2 196 41 175 15 185 28 165)(3 208 42 174 16 184 29 164)(4 207 43 173 17 183 30 163)(5 206 44 172 18 195 31 162)(6 205 45 171 19 194 32 161)(7 204 46 170 20 193 33 160)(8 203 47 182 21 192 34 159)(9 202 48 181 22 191 35 158)(10 201 49 180 23 190 36 157)(11 200 50 179 24 189 37 169)(12 199 51 178 25 188 38 168)(13 198 52 177 26 187 39 167)(53 252 92 230 66 244 79 220)(54 251 93 229 67 243 80 219)(55 250 94 228 68 242 81 218)(56 249 95 227 69 241 82 217)(57 248 96 226 70 240 83 216)(58 260 97 225 71 239 84 215)(59 259 98 224 72 238 85 214)(60 258 99 223 73 237 86 213)(61 257 100 222 74 236 87 212)(62 256 101 234 75 235 88 211)(63 255 102 233 76 247 89 210)(64 254 103 232 77 246 90 209)(65 253 104 231 78 245 91 221)(105 301 144 281 118 295 131 263)(106 300 145 280 119 294 132 262)(107 312 146 279 120 293 133 261)(108 311 147 278 121 292 134 273)(109 310 148 277 122 291 135 272)(110 309 149 276 123 290 136 271)(111 308 150 275 124 289 137 270)(112 307 151 274 125 288 138 269)(113 306 152 286 126 287 139 268)(114 305 153 285 127 299 140 267)(115 304 154 284 128 298 141 266)(116 303 155 283 129 297 142 265)(117 302 156 282 130 296 143 264)

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

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

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

96 conjugacy classes

class 1  2 3A3B4A4B6A6B8A8B8C8D12A12B12C12D13A···13F24A···24H26A···26F39A···39L52A···52L78A···78L156A···156X
order1233446688881212121213···1324···2426···2639···3952···5278···78156···156
size111111111313131311112···213···132···22···22···22···22···2

96 irreducible representations

dim11111111222222
type+++-
imageC1C2C3C4C6C8C12C24D13Dic13C3×D13C132C8C3×Dic13C3×C132C8
kernelC3×C132C8C156C132C8C78C52C39C26C13C12C6C4C3C2C1
# reps112224486612121224

Matrix representation of C3×C132C8 in GL4(𝔽5) generated by

2424
1403
0104
1012
,
1323
0144
2013
3221
,
3143
0223
4102
4220
G:=sub<GL(4,GF(5))| [2,1,0,1,4,4,1,0,2,0,0,1,4,3,4,2],[1,0,2,3,3,1,0,2,2,4,1,2,3,4,3,1],[3,0,4,4,1,2,1,2,4,2,0,2,3,3,2,0] >;

C3×C132C8 in GAP, Magma, Sage, TeX

C_3\times C_{13}\rtimes_2C_8
% in TeX

G:=Group("C3xC13:2C8");
// GroupNames label

G:=SmallGroup(312,4);
// by ID

G=gap.SmallGroup(312,4);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-13,30,42,7204]);
// Polycyclic

G:=Group<a,b,c|a^3=b^13=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×C132C8 in TeX

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