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G = C13×C3⋊C8order 312 = 23·3·13

Direct product of C13 and C3⋊C8

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

Aliases: C13×C3⋊C8, C3⋊C104, C395C8, C6.C52, C78.5C4, C52.4S3, C156.6C2, C12.2C26, C26.3Dic3, C4.2(S3×C13), C2.(Dic3×C13), SmallGroup(312,3)

Series: Derived Chief Lower central Upper central

C1C3 — C13×C3⋊C8
C1C3C6C12C156 — C13×C3⋊C8
C3 — C13×C3⋊C8
C1C52

Generators and relations for C13×C3⋊C8
 G = < a,b,c | a13=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

3C8
3C104

Smallest permutation representation of C13×C3⋊C8
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)
(1 277 305)(2 278 306)(3 279 307)(4 280 308)(5 281 309)(6 282 310)(7 283 311)(8 284 312)(9 285 300)(10 286 301)(11 274 302)(12 275 303)(13 276 304)(14 97 248)(15 98 249)(16 99 250)(17 100 251)(18 101 252)(19 102 253)(20 103 254)(21 104 255)(22 92 256)(23 93 257)(24 94 258)(25 95 259)(26 96 260)(27 201 293)(28 202 294)(29 203 295)(30 204 296)(31 205 297)(32 206 298)(33 207 299)(34 208 287)(35 196 288)(36 197 289)(37 198 290)(38 199 291)(39 200 292)(40 218 122)(41 219 123)(42 220 124)(43 221 125)(44 209 126)(45 210 127)(46 211 128)(47 212 129)(48 213 130)(49 214 118)(50 215 119)(51 216 120)(52 217 121)(53 111 235)(54 112 236)(55 113 237)(56 114 238)(57 115 239)(58 116 240)(59 117 241)(60 105 242)(61 106 243)(62 107 244)(63 108 245)(64 109 246)(65 110 247)(66 175 268)(67 176 269)(68 177 270)(69 178 271)(70 179 272)(71 180 273)(72 181 261)(73 182 262)(74 170 263)(75 171 264)(76 172 265)(77 173 266)(78 174 267)(79 190 156)(80 191 144)(81 192 145)(82 193 146)(83 194 147)(84 195 148)(85 183 149)(86 184 150)(87 185 151)(88 186 152)(89 187 153)(90 188 154)(91 189 155)(131 159 225)(132 160 226)(133 161 227)(134 162 228)(135 163 229)(136 164 230)(137 165 231)(138 166 232)(139 167 233)(140 168 234)(141 169 222)(142 157 223)(143 158 224)
(1 267 244 163 36 212 97 145)(2 268 245 164 37 213 98 146)(3 269 246 165 38 214 99 147)(4 270 247 166 39 215 100 148)(5 271 235 167 27 216 101 149)(6 272 236 168 28 217 102 150)(7 273 237 169 29 218 103 151)(8 261 238 157 30 219 104 152)(9 262 239 158 31 220 92 153)(10 263 240 159 32 221 93 154)(11 264 241 160 33 209 94 155)(12 265 242 161 34 210 95 156)(13 266 243 162 35 211 96 144)(14 81 305 78 107 229 289 129)(15 82 306 66 108 230 290 130)(16 83 307 67 109 231 291 118)(17 84 308 68 110 232 292 119)(18 85 309 69 111 233 293 120)(19 86 310 70 112 234 294 121)(20 87 311 71 113 222 295 122)(21 88 312 72 114 223 296 123)(22 89 300 73 115 224 297 124)(23 90 301 74 116 225 298 125)(24 91 302 75 117 226 299 126)(25 79 303 76 105 227 287 127)(26 80 304 77 106 228 288 128)(40 254 185 283 180 55 141 203)(41 255 186 284 181 56 142 204)(42 256 187 285 182 57 143 205)(43 257 188 286 170 58 131 206)(44 258 189 274 171 59 132 207)(45 259 190 275 172 60 133 208)(46 260 191 276 173 61 134 196)(47 248 192 277 174 62 135 197)(48 249 193 278 175 63 136 198)(49 250 194 279 176 64 137 199)(50 251 195 280 177 65 138 200)(51 252 183 281 178 53 139 201)(52 253 184 282 179 54 140 202)

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

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

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

156 conjugacy classes

class 1  2  3 4A4B 6 8A8B8C8D12A12B13A···13L26A···26L39A···39L52A···52X78A···78L104A···104AV156A···156X
order1234468888121213···1326···2639···3952···5278···78104···104156···156
size1121123333221···11···12···21···12···23···32···2

156 irreducible representations

dim11111111222222
type+++-
imageC1C2C4C8C13C26C52C104S3Dic3C3⋊C8S3×C13Dic3×C13C13×C3⋊C8
kernelC13×C3⋊C8C156C78C39C3⋊C8C12C6C3C52C26C13C4C2C1
# reps112412122448112121224

Matrix representation of C13×C3⋊C8 in GL3(𝔽313) generated by

100
0440
0044
,
100
00312
01312
,
500
0175271
0133138
G:=sub<GL(3,GF(313))| [1,0,0,0,44,0,0,0,44],[1,0,0,0,0,1,0,312,312],[5,0,0,0,175,133,0,271,138] >;

C13×C3⋊C8 in GAP, Magma, Sage, TeX

C_{13}\times C_3\rtimes C_8
% in TeX

G:=Group("C13xC3:C8");
// GroupNames label

G:=SmallGroup(312,3);
// by ID

G=gap.SmallGroup(312,3);
# by ID

G:=PCGroup([5,-2,-13,-2,-2,-3,130,42,5204]);
// Polycyclic

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

Export

Subgroup lattice of C13×C3⋊C8 in TeX

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