metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C34.C8, C17⋊2C16, Dic17.C4, C2.(C17⋊C8), C17⋊2C8.C2, SmallGroup(272,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C34.C8 |
Generators and relations for C34.C8
G = < a,b | a34=1, b8=a17, bab-1=a19 >
Character table of C34.C8
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 17A | 17B | 34A | 34B | |
| size | 1 | 1 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | i | i | i | i | -i | -i | -i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | -i | -i | -i | -i | i | i | i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 1 | 1 | -1 | -1 | -i | i | i | -i | ζ87 | ζ8 | ζ8 | ζ85 | ζ85 | ζ83 | ζ83 | ζ87 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ6 | 1 | 1 | -1 | -1 | i | -i | -i | i | ζ85 | ζ83 | ζ83 | ζ87 | ζ87 | ζ8 | ζ8 | ζ85 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ7 | 1 | 1 | -1 | -1 | -i | i | i | -i | ζ83 | ζ85 | ζ85 | ζ8 | ζ8 | ζ87 | ζ87 | ζ83 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ8 | 1 | 1 | -1 | -1 | i | -i | -i | i | ζ8 | ζ87 | ζ87 | ζ83 | ζ83 | ζ85 | ζ85 | ζ8 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ9 | 1 | -1 | i | -i | ζ166 | ζ1610 | ζ162 | ζ1614 | ζ167 | ζ169 | ζ16 | ζ1613 | ζ165 | ζ1611 | ζ163 | ζ1615 | 1 | 1 | -1 | -1 | linear of order 16 |
| ρ10 | 1 | -1 | -i | i | ζ162 | ζ1614 | ζ166 | ζ1610 | ζ1613 | ζ163 | ζ1611 | ζ1615 | ζ167 | ζ169 | ζ16 | ζ165 | 1 | 1 | -1 | -1 | linear of order 16 |
| ρ11 | 1 | -1 | i | -i | ζ1614 | ζ162 | ζ1610 | ζ166 | ζ163 | ζ1613 | ζ165 | ζ16 | ζ169 | ζ167 | ζ1615 | ζ1611 | 1 | 1 | -1 | -1 | linear of order 16 |
| ρ12 | 1 | -1 | -i | i | ζ1610 | ζ166 | ζ1614 | ζ162 | ζ169 | ζ167 | ζ1615 | ζ163 | ζ1611 | ζ165 | ζ1613 | ζ16 | 1 | 1 | -1 | -1 | linear of order 16 |
| ρ13 | 1 | -1 | -i | i | ζ162 | ζ1614 | ζ166 | ζ1610 | ζ165 | ζ1611 | ζ163 | ζ167 | ζ1615 | ζ16 | ζ169 | ζ1613 | 1 | 1 | -1 | -1 | linear of order 16 |
| ρ14 | 1 | -1 | -i | i | ζ1610 | ζ166 | ζ1614 | ζ162 | ζ16 | ζ1615 | ζ167 | ζ1611 | ζ163 | ζ1613 | ζ165 | ζ169 | 1 | 1 | -1 | -1 | linear of order 16 |
| ρ15 | 1 | -1 | i | -i | ζ1614 | ζ162 | ζ1610 | ζ166 | ζ1611 | ζ165 | ζ1613 | ζ169 | ζ16 | ζ1615 | ζ167 | ζ163 | 1 | 1 | -1 | -1 | linear of order 16 |
| ρ16 | 1 | -1 | i | -i | ζ166 | ζ1610 | ζ162 | ζ1614 | ζ1615 | ζ16 | ζ169 | ζ165 | ζ1613 | ζ163 | ζ1611 | ζ167 | 1 | 1 | -1 | -1 | linear of order 16 |
| ρ17 | 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√17/2 | -1+√17/2 | -1-√17/2 | -1+√17/2 | orthogonal lifted from C17⋊C8 |
| ρ18 | 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√17/2 | -1-√17/2 | -1+√17/2 | -1-√17/2 | orthogonal lifted from C17⋊C8 |
| ρ19 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√17/2 | -1+√17/2 | 1+√17/2 | 1-√17/2 | symplectic faithful, Schur index 2 |
| ρ20 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√17/2 | -1-√17/2 | 1-√17/2 | 1+√17/2 | symplectic faithful, Schur index 2 |
►Regular action on 272 points
Generators in S272
(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)
(1 268 126 198 39 205 86 160 18 251 109 181 56 222 69 143)(2 243 105 179 38 230 73 145 19 260 122 196 55 213 90 162)(3 252 118 194 37 221 94 164 20 269 135 177 54 238 77 147)(4 261 131 175 36 212 81 149 21 244 114 192 53 229 98 166)(5 270 110 190 35 237 102 168 22 253 127 173 52 220 85 151)(6 245 123 171 68 228 89 153 23 262 106 188 51 211 72 170)(7 254 136 186 67 219 76 138 24 271 119 203 50 236 93 155)(8 263 115 201 66 210 97 157 25 246 132 184 49 227 80 140)(9 272 128 182 65 235 84 142 26 255 111 199 48 218 101 159)(10 247 107 197 64 226 71 161 27 264 124 180 47 209 88 144)(11 256 120 178 63 217 92 146 28 239 103 195 46 234 75 163)(12 265 133 193 62 208 79 165 29 248 116 176 45 225 96 148)(13 240 112 174 61 233 100 150 30 257 129 191 44 216 83 167)(14 249 125 189 60 224 87 169 31 266 108 172 43 207 70 152)(15 258 104 204 59 215 74 154 32 241 121 187 42 232 91 137)(16 267 117 185 58 206 95 139 33 250 134 202 41 223 78 156)(17 242 130 200 57 231 82 158 34 259 113 183 40 214 99 141)
G:=sub<Sym(272)| (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), (1,268,126,198,39,205,86,160,18,251,109,181,56,222,69,143)(2,243,105,179,38,230,73,145,19,260,122,196,55,213,90,162)(3,252,118,194,37,221,94,164,20,269,135,177,54,238,77,147)(4,261,131,175,36,212,81,149,21,244,114,192,53,229,98,166)(5,270,110,190,35,237,102,168,22,253,127,173,52,220,85,151)(6,245,123,171,68,228,89,153,23,262,106,188,51,211,72,170)(7,254,136,186,67,219,76,138,24,271,119,203,50,236,93,155)(8,263,115,201,66,210,97,157,25,246,132,184,49,227,80,140)(9,272,128,182,65,235,84,142,26,255,111,199,48,218,101,159)(10,247,107,197,64,226,71,161,27,264,124,180,47,209,88,144)(11,256,120,178,63,217,92,146,28,239,103,195,46,234,75,163)(12,265,133,193,62,208,79,165,29,248,116,176,45,225,96,148)(13,240,112,174,61,233,100,150,30,257,129,191,44,216,83,167)(14,249,125,189,60,224,87,169,31,266,108,172,43,207,70,152)(15,258,104,204,59,215,74,154,32,241,121,187,42,232,91,137)(16,267,117,185,58,206,95,139,33,250,134,202,41,223,78,156)(17,242,130,200,57,231,82,158,34,259,113,183,40,214,99,141)>;
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), (1,268,126,198,39,205,86,160,18,251,109,181,56,222,69,143)(2,243,105,179,38,230,73,145,19,260,122,196,55,213,90,162)(3,252,118,194,37,221,94,164,20,269,135,177,54,238,77,147)(4,261,131,175,36,212,81,149,21,244,114,192,53,229,98,166)(5,270,110,190,35,237,102,168,22,253,127,173,52,220,85,151)(6,245,123,171,68,228,89,153,23,262,106,188,51,211,72,170)(7,254,136,186,67,219,76,138,24,271,119,203,50,236,93,155)(8,263,115,201,66,210,97,157,25,246,132,184,49,227,80,140)(9,272,128,182,65,235,84,142,26,255,111,199,48,218,101,159)(10,247,107,197,64,226,71,161,27,264,124,180,47,209,88,144)(11,256,120,178,63,217,92,146,28,239,103,195,46,234,75,163)(12,265,133,193,62,208,79,165,29,248,116,176,45,225,96,148)(13,240,112,174,61,233,100,150,30,257,129,191,44,216,83,167)(14,249,125,189,60,224,87,169,31,266,108,172,43,207,70,152)(15,258,104,204,59,215,74,154,32,241,121,187,42,232,91,137)(16,267,117,185,58,206,95,139,33,250,134,202,41,223,78,156)(17,242,130,200,57,231,82,158,34,259,113,183,40,214,99,141) );
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)], [(1,268,126,198,39,205,86,160,18,251,109,181,56,222,69,143),(2,243,105,179,38,230,73,145,19,260,122,196,55,213,90,162),(3,252,118,194,37,221,94,164,20,269,135,177,54,238,77,147),(4,261,131,175,36,212,81,149,21,244,114,192,53,229,98,166),(5,270,110,190,35,237,102,168,22,253,127,173,52,220,85,151),(6,245,123,171,68,228,89,153,23,262,106,188,51,211,72,170),(7,254,136,186,67,219,76,138,24,271,119,203,50,236,93,155),(8,263,115,201,66,210,97,157,25,246,132,184,49,227,80,140),(9,272,128,182,65,235,84,142,26,255,111,199,48,218,101,159),(10,247,107,197,64,226,71,161,27,264,124,180,47,209,88,144),(11,256,120,178,63,217,92,146,28,239,103,195,46,234,75,163),(12,265,133,193,62,208,79,165,29,248,116,176,45,225,96,148),(13,240,112,174,61,233,100,150,30,257,129,191,44,216,83,167),(14,249,125,189,60,224,87,169,31,266,108,172,43,207,70,152),(15,258,104,204,59,215,74,154,32,241,121,187,42,232,91,137),(16,267,117,185,58,206,95,139,33,250,134,202,41,223,78,156),(17,242,130,200,57,231,82,158,34,259,113,183,40,214,99,141)]])
Matrix representation of C34.C8 ►in GL8(𝔽1361)
| 3 | 531 | 528 | 834 | 1359 | 1359 | 302 | 1359 |
| 836 | 530 | 1057 | 306 | 527 | 1359 | 304 | 1358 |
| 305 | 0 | 1055 | 307 | 527 | 528 | 306 | 528 |
| 837 | 531 | 1056 | 307 | 526 | 1358 | 1135 | 1358 |
| 835 | 1 | 1056 | 306 | 528 | 527 | 305 | 1359 |
| 833 | 1 | 527 | 834 | 1360 | 528 | 833 | 0 |
| 531 | 3 | 529 | 0 | 832 | 1358 | 830 | 831 |
| 529 | 2 | 1 | 528 | 833 | 1360 | 1359 | 832 |
| 1349 | 148 | 681 | 1067 | 373 | 408 | 369 | 357 |
| 530 | 543 | 717 | 1062 | 881 | 102 | 356 | 1227 |
| 782 | 1000 | 52 | 573 | 652 | 980 | 444 | 1160 |
| 1204 | 1008 | 315 | 1307 | 1014 | 336 | 1353 | 322 |
| 447 | 1226 | 947 | 33 | 221 | 684 | 142 | 1273 |
| 237 | 1181 | 11 | 761 | 1213 | 211 | 1199 | 880 |
| 173 | 878 | 1097 | 1292 | 37 | 850 | 1142 | 864 |
| 1073 | 422 | 1242 | 287 | 166 | 61 | 861 | 619 |
G:=sub<GL(8,GF(1361))| [3,836,305,837,835,833,531,529,531,530,0,531,1,1,3,2,528,1057,1055,1056,1056,527,529,1,834,306,307,307,306,834,0,528,1359,527,527,526,528,1360,832,833,1359,1359,528,1358,527,528,1358,1360,302,304,306,1135,305,833,830,1359,1359,1358,528,1358,1359,0,831,832],[1349,530,782,1204,447,237,173,1073,148,543,1000,1008,1226,1181,878,422,681,717,52,315,947,11,1097,1242,1067,1062,573,1307,33,761,1292,287,373,881,652,1014,221,1213,37,166,408,102,980,336,684,211,850,61,369,356,444,1353,142,1199,1142,861,357,1227,1160,322,1273,880,864,619] >;
C34.C8 in GAP, Magma, Sage, TeX
C_{34}.C_8 % in TeX
G:=Group("C34.C8"); // GroupNames label
G:=SmallGroup(272,28);
// by ID
G=gap.SmallGroup(272,28);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,10,26,42,3604,2609,1614]);
// Polycyclic
G:=Group<a,b|a^34=1,b^8=a^17,b*a*b^-1=a^19>;
// generators/relations
Export
Subgroup lattice of C34.C8 in TeX
Character table of C34.C8 in TeX