metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C39⋊3C8, C78.3C4, C52.2S3, C4.2D39, C6.Dic13, C2.Dic39, C156.2C2, C12.2D13, C26.2Dic3, C3⋊(C13⋊2C8), C13⋊2(C3⋊C8), SmallGroup(312,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — C39⋊3C8 |
Generators and relations for C39⋊3C8
G = < a,b | a39=b8=1, bab-1=a-1 >
Smallest permutation representation of C39⋊3C8
►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)
(1 289 141 221 45 264 106 180)(2 288 142 220 46 263 107 179)(3 287 143 219 47 262 108 178)(4 286 144 218 48 261 109 177)(5 285 145 217 49 260 110 176)(6 284 146 216 50 259 111 175)(7 283 147 215 51 258 112 174)(8 282 148 214 52 257 113 173)(9 281 149 213 53 256 114 172)(10 280 150 212 54 255 115 171)(11 279 151 211 55 254 116 170)(12 278 152 210 56 253 117 169)(13 277 153 209 57 252 79 168)(14 276 154 208 58 251 80 167)(15 275 155 207 59 250 81 166)(16 274 156 206 60 249 82 165)(17 312 118 205 61 248 83 164)(18 311 119 204 62 247 84 163)(19 310 120 203 63 246 85 162)(20 309 121 202 64 245 86 161)(21 308 122 201 65 244 87 160)(22 307 123 200 66 243 88 159)(23 306 124 199 67 242 89 158)(24 305 125 198 68 241 90 157)(25 304 126 197 69 240 91 195)(26 303 127 196 70 239 92 194)(27 302 128 234 71 238 93 193)(28 301 129 233 72 237 94 192)(29 300 130 232 73 236 95 191)(30 299 131 231 74 235 96 190)(31 298 132 230 75 273 97 189)(32 297 133 229 76 272 98 188)(33 296 134 228 77 271 99 187)(34 295 135 227 78 270 100 186)(35 294 136 226 40 269 101 185)(36 293 137 225 41 268 102 184)(37 292 138 224 42 267 103 183)(38 291 139 223 43 266 104 182)(39 290 140 222 44 265 105 181)
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), (1,289,141,221,45,264,106,180)(2,288,142,220,46,263,107,179)(3,287,143,219,47,262,108,178)(4,286,144,218,48,261,109,177)(5,285,145,217,49,260,110,176)(6,284,146,216,50,259,111,175)(7,283,147,215,51,258,112,174)(8,282,148,214,52,257,113,173)(9,281,149,213,53,256,114,172)(10,280,150,212,54,255,115,171)(11,279,151,211,55,254,116,170)(12,278,152,210,56,253,117,169)(13,277,153,209,57,252,79,168)(14,276,154,208,58,251,80,167)(15,275,155,207,59,250,81,166)(16,274,156,206,60,249,82,165)(17,312,118,205,61,248,83,164)(18,311,119,204,62,247,84,163)(19,310,120,203,63,246,85,162)(20,309,121,202,64,245,86,161)(21,308,122,201,65,244,87,160)(22,307,123,200,66,243,88,159)(23,306,124,199,67,242,89,158)(24,305,125,198,68,241,90,157)(25,304,126,197,69,240,91,195)(26,303,127,196,70,239,92,194)(27,302,128,234,71,238,93,193)(28,301,129,233,72,237,94,192)(29,300,130,232,73,236,95,191)(30,299,131,231,74,235,96,190)(31,298,132,230,75,273,97,189)(32,297,133,229,76,272,98,188)(33,296,134,228,77,271,99,187)(34,295,135,227,78,270,100,186)(35,294,136,226,40,269,101,185)(36,293,137,225,41,268,102,184)(37,292,138,224,42,267,103,183)(38,291,139,223,43,266,104,182)(39,290,140,222,44,265,105,181)>;
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), (1,289,141,221,45,264,106,180)(2,288,142,220,46,263,107,179)(3,287,143,219,47,262,108,178)(4,286,144,218,48,261,109,177)(5,285,145,217,49,260,110,176)(6,284,146,216,50,259,111,175)(7,283,147,215,51,258,112,174)(8,282,148,214,52,257,113,173)(9,281,149,213,53,256,114,172)(10,280,150,212,54,255,115,171)(11,279,151,211,55,254,116,170)(12,278,152,210,56,253,117,169)(13,277,153,209,57,252,79,168)(14,276,154,208,58,251,80,167)(15,275,155,207,59,250,81,166)(16,274,156,206,60,249,82,165)(17,312,118,205,61,248,83,164)(18,311,119,204,62,247,84,163)(19,310,120,203,63,246,85,162)(20,309,121,202,64,245,86,161)(21,308,122,201,65,244,87,160)(22,307,123,200,66,243,88,159)(23,306,124,199,67,242,89,158)(24,305,125,198,68,241,90,157)(25,304,126,197,69,240,91,195)(26,303,127,196,70,239,92,194)(27,302,128,234,71,238,93,193)(28,301,129,233,72,237,94,192)(29,300,130,232,73,236,95,191)(30,299,131,231,74,235,96,190)(31,298,132,230,75,273,97,189)(32,297,133,229,76,272,98,188)(33,296,134,228,77,271,99,187)(34,295,135,227,78,270,100,186)(35,294,136,226,40,269,101,185)(36,293,137,225,41,268,102,184)(37,292,138,224,42,267,103,183)(38,291,139,223,43,266,104,182)(39,290,140,222,44,265,105,181) );
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)], [(1,289,141,221,45,264,106,180),(2,288,142,220,46,263,107,179),(3,287,143,219,47,262,108,178),(4,286,144,218,48,261,109,177),(5,285,145,217,49,260,110,176),(6,284,146,216,50,259,111,175),(7,283,147,215,51,258,112,174),(8,282,148,214,52,257,113,173),(9,281,149,213,53,256,114,172),(10,280,150,212,54,255,115,171),(11,279,151,211,55,254,116,170),(12,278,152,210,56,253,117,169),(13,277,153,209,57,252,79,168),(14,276,154,208,58,251,80,167),(15,275,155,207,59,250,81,166),(16,274,156,206,60,249,82,165),(17,312,118,205,61,248,83,164),(18,311,119,204,62,247,84,163),(19,310,120,203,63,246,85,162),(20,309,121,202,64,245,86,161),(21,308,122,201,65,244,87,160),(22,307,123,200,66,243,88,159),(23,306,124,199,67,242,89,158),(24,305,125,198,68,241,90,157),(25,304,126,197,69,240,91,195),(26,303,127,196,70,239,92,194),(27,302,128,234,71,238,93,193),(28,301,129,233,72,237,94,192),(29,300,130,232,73,236,95,191),(30,299,131,231,74,235,96,190),(31,298,132,230,75,273,97,189),(32,297,133,229,76,272,98,188),(33,296,134,228,77,271,99,187),(34,295,135,227,78,270,100,186),(35,294,136,226,40,269,101,185),(36,293,137,225,41,268,102,184),(37,292,138,224,42,267,103,183),(38,291,139,223,43,266,104,182),(39,290,140,222,44,265,105,181)]])
84 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39L | 52A | ··· | 52L | 78A | ··· | 78L | 156A | ··· | 156X |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 39 | 39 | 39 | 39 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | S3 | Dic3 | C3⋊C8 | D13 | Dic13 | D39 | C13⋊2C8 | Dic39 | C39⋊3C8 |
| kernel | C39⋊3C8 | C156 | C78 | C39 | C52 | C26 | C13 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 6 | 6 | 12 | 12 | 12 | 24 |
Matrix representation of C39⋊3C8 ►in GL2(𝔽313) generated by
| 228 | 61 |
| 252 | 99 |
| 210 | 215 |
| 166 | 103 |
G:=sub<GL(2,GF(313))| [228,252,61,99],[210,166,215,103] >;
C39⋊3C8 in GAP, Magma, Sage, TeX
C_{39}\rtimes_3C_8 % in TeX
G:=Group("C39:3C8"); // GroupNames label
G:=SmallGroup(312,5);
// by ID
G=gap.SmallGroup(312,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,10,26,323,7204]);
// Polycyclic
G:=Group<a,b|a^39=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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