metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic19, C19⋊C4, C38.C2, C2.D19, SmallGroup(76,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — Dic19 |
Generators and relations for Dic19
G = < a,b | a38=1, b2=a19, bab-1=a-1 >
Character table of Dic19
| class | 1 | 2 | 4A | 4B | 19A | 19B | 19C | 19D | 19E | 19F | 19G | 19H | 19I | 38A | 38B | 38C | 38D | 38E | 38F | 38G | 38H | 38I | |
| size | 1 | 1 | 19 | 19 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 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 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 0 | 0 | ζ1910+ζ199 | ζ1916+ζ193 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1918+ζ19 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1917+ζ192 | ζ1917+ζ192 | ζ1910+ζ199 | ζ1916+ζ193 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1918+ζ19 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1914+ζ195 | orthogonal lifted from D19 |
| ρ6 | 2 | 2 | 0 | 0 | ζ1913+ζ196 | ζ1917+ζ192 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1912+ζ197 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1914+ζ195 | ζ1914+ζ195 | ζ1913+ζ196 | ζ1917+ζ192 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1912+ζ197 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1916+ζ193 | orthogonal lifted from D19 |
| ρ7 | 2 | 2 | 0 | 0 | ζ1914+ζ195 | ζ1911+ζ198 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1910+ζ199 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1918+ζ19 | ζ1918+ζ19 | ζ1914+ζ195 | ζ1911+ζ198 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1910+ζ199 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1912+ζ197 | orthogonal lifted from D19 |
| ρ8 | 2 | 2 | 0 | 0 | ζ1911+ζ198 | ζ1910+ζ199 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1916+ζ193 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1913+ζ196 | ζ1913+ζ196 | ζ1911+ζ198 | ζ1910+ζ199 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1916+ζ193 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1915+ζ194 | orthogonal lifted from D19 |
| ρ9 | 2 | 2 | 0 | 0 | ζ1912+ζ197 | ζ1915+ζ194 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1914+ζ195 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1910+ζ199 | ζ1910+ζ199 | ζ1912+ζ197 | ζ1915+ζ194 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1914+ζ195 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1913+ζ196 | orthogonal lifted from D19 |
| ρ10 | 2 | 2 | 0 | 0 | ζ1918+ζ19 | ζ1913+ζ196 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1917+ζ192 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1915+ζ194 | ζ1915+ζ194 | ζ1918+ζ19 | ζ1913+ζ196 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1917+ζ192 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1910+ζ199 | orthogonal lifted from D19 |
| ρ11 | 2 | 2 | 0 | 0 | ζ1916+ζ193 | ζ1918+ζ19 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1913+ζ196 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1912+ζ197 | ζ1912+ζ197 | ζ1916+ζ193 | ζ1918+ζ19 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1913+ζ196 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1911+ζ198 | orthogonal lifted from D19 |
| ρ12 | 2 | 2 | 0 | 0 | ζ1915+ζ194 | ζ1914+ζ195 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1911+ζ198 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1916+ζ193 | ζ1916+ζ193 | ζ1915+ζ194 | ζ1914+ζ195 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1911+ζ198 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1917+ζ192 | orthogonal lifted from D19 |
| ρ13 | 2 | 2 | 0 | 0 | ζ1917+ζ192 | ζ1912+ζ197 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1915+ζ194 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1911+ζ198 | ζ1911+ζ198 | ζ1917+ζ192 | ζ1912+ζ197 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1915+ζ194 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1918+ζ19 | orthogonal lifted from D19 |
| ρ14 | 2 | -2 | 0 | 0 | ζ1914+ζ195 | ζ1911+ζ198 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1910+ζ199 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1918+ζ19 | -ζ1918-ζ19 | -ζ1914-ζ195 | -ζ1911-ζ198 | -ζ1917-ζ192 | -ζ1915-ζ194 | -ζ1910-ζ199 | -ζ1916-ζ193 | -ζ1913-ζ196 | -ζ1912-ζ197 | symplectic faithful, Schur index 2 |
| ρ15 | 2 | -2 | 0 | 0 | ζ1913+ζ196 | ζ1917+ζ192 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1912+ζ197 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1914+ζ195 | -ζ1914-ζ195 | -ζ1913-ζ196 | -ζ1917-ζ192 | -ζ1910-ζ199 | -ζ1918-ζ19 | -ζ1912-ζ197 | -ζ1915-ζ194 | -ζ1911-ζ198 | -ζ1916-ζ193 | symplectic faithful, Schur index 2 |
| ρ16 | 2 | -2 | 0 | 0 | ζ1910+ζ199 | ζ1916+ζ193 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1918+ζ19 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1917+ζ192 | -ζ1917-ζ192 | -ζ1910-ζ199 | -ζ1916-ζ193 | -ζ1915-ζ194 | -ζ1911-ζ198 | -ζ1918-ζ19 | -ζ1913-ζ196 | -ζ1912-ζ197 | -ζ1914-ζ195 | symplectic faithful, Schur index 2 |
| ρ17 | 2 | -2 | 0 | 0 | ζ1912+ζ197 | ζ1915+ζ194 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1914+ζ195 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1910+ζ199 | -ζ1910-ζ199 | -ζ1912-ζ197 | -ζ1915-ζ194 | -ζ1918-ζ19 | -ζ1917-ζ192 | -ζ1914-ζ195 | -ζ1911-ζ198 | -ζ1916-ζ193 | -ζ1913-ζ196 | symplectic faithful, Schur index 2 |
| ρ18 | 2 | -2 | 0 | 0 | ζ1911+ζ198 | ζ1910+ζ199 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1916+ζ193 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1913+ζ196 | -ζ1913-ζ196 | -ζ1911-ζ198 | -ζ1910-ζ199 | -ζ1912-ζ197 | -ζ1914-ζ195 | -ζ1916-ζ193 | -ζ1918-ζ19 | -ζ1917-ζ192 | -ζ1915-ζ194 | symplectic faithful, Schur index 2 |
| ρ19 | 2 | -2 | 0 | 0 | ζ1918+ζ19 | ζ1913+ζ196 | ζ1911+ζ198 | ζ1916+ζ193 | ζ1917+ζ192 | ζ1912+ζ197 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1915+ζ194 | -ζ1915-ζ194 | -ζ1918-ζ19 | -ζ1913-ζ196 | -ζ1911-ζ198 | -ζ1916-ζ193 | -ζ1917-ζ192 | -ζ1912-ζ197 | -ζ1914-ζ195 | -ζ1910-ζ199 | symplectic faithful, Schur index 2 |
| ρ20 | 2 | -2 | 0 | 0 | ζ1916+ζ193 | ζ1918+ζ19 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1913+ζ196 | ζ1917+ζ192 | ζ1915+ζ194 | ζ1911+ζ198 | ζ1912+ζ197 | -ζ1912-ζ197 | -ζ1916-ζ193 | -ζ1918-ζ19 | -ζ1914-ζ195 | -ζ1910-ζ199 | -ζ1913-ζ196 | -ζ1917-ζ192 | -ζ1915-ζ194 | -ζ1911-ζ198 | symplectic faithful, Schur index 2 |
| ρ21 | 2 | -2 | 0 | 0 | ζ1915+ζ194 | ζ1914+ζ195 | ζ1913+ζ196 | ζ1912+ζ197 | ζ1911+ζ198 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1917+ζ192 | ζ1916+ζ193 | -ζ1916-ζ193 | -ζ1915-ζ194 | -ζ1914-ζ195 | -ζ1913-ζ196 | -ζ1912-ζ197 | -ζ1911-ζ198 | -ζ1910-ζ199 | -ζ1918-ζ19 | -ζ1917-ζ192 | symplectic faithful, Schur index 2 |
| ρ22 | 2 | -2 | 0 | 0 | ζ1917+ζ192 | ζ1912+ζ197 | ζ1916+ζ193 | ζ1913+ζ196 | ζ1915+ζ194 | ζ1914+ζ195 | ζ1910+ζ199 | ζ1918+ζ19 | ζ1911+ζ198 | -ζ1911-ζ198 | -ζ1917-ζ192 | -ζ1912-ζ197 | -ζ1916-ζ193 | -ζ1913-ζ196 | -ζ1915-ζ194 | -ζ1914-ζ195 | -ζ1910-ζ199 | -ζ1918-ζ19 | symplectic faithful, Schur index 2 |
►Regular action on 76 points
Generators in S76
(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)
(1 57 20 76)(2 56 21 75)(3 55 22 74)(4 54 23 73)(5 53 24 72)(6 52 25 71)(7 51 26 70)(8 50 27 69)(9 49 28 68)(10 48 29 67)(11 47 30 66)(12 46 31 65)(13 45 32 64)(14 44 33 63)(15 43 34 62)(16 42 35 61)(17 41 36 60)(18 40 37 59)(19 39 38 58)
G:=sub<Sym(76)| (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), (1,57,20,76)(2,56,21,75)(3,55,22,74)(4,54,23,73)(5,53,24,72)(6,52,25,71)(7,51,26,70)(8,50,27,69)(9,49,28,68)(10,48,29,67)(11,47,30,66)(12,46,31,65)(13,45,32,64)(14,44,33,63)(15,43,34,62)(16,42,35,61)(17,41,36,60)(18,40,37,59)(19,39,38,58)>;
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), (1,57,20,76)(2,56,21,75)(3,55,22,74)(4,54,23,73)(5,53,24,72)(6,52,25,71)(7,51,26,70)(8,50,27,69)(9,49,28,68)(10,48,29,67)(11,47,30,66)(12,46,31,65)(13,45,32,64)(14,44,33,63)(15,43,34,62)(16,42,35,61)(17,41,36,60)(18,40,37,59)(19,39,38,58) );
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)], [(1,57,20,76),(2,56,21,75),(3,55,22,74),(4,54,23,73),(5,53,24,72),(6,52,25,71),(7,51,26,70),(8,50,27,69),(9,49,28,68),(10,48,29,67),(11,47,30,66),(12,46,31,65),(13,45,32,64),(14,44,33,63),(15,43,34,62),(16,42,35,61),(17,41,36,60),(18,40,37,59),(19,39,38,58)]])
Dic19 is a maximal subgroup of
Dic38 C4×D19 C19⋊D4 C19⋊C12 Dic57 Dic95 C19⋊F5
Dic19 is a maximal quotient of C19⋊C8 Dic57 Dic95 C19⋊F5
Matrix representation of Dic19 ►in GL2(𝔽37) generated by
| 10 | 34 |
| 34 | 1 |
| 6 | 18 |
| 0 | 31 |
G:=sub<GL(2,GF(37))| [10,34,34,1],[6,0,18,31] >;
Dic19 in GAP, Magma, Sage, TeX
{\rm Dic}_{19} % in TeX
G:=Group("Dic19"); // GroupNames label
G:=SmallGroup(76,1);
// by ID
G=gap.SmallGroup(76,1);
# by ID
G:=PCGroup([3,-2,-2,-19,6,650]);
// Polycyclic
G:=Group<a,b|a^38=1,b^2=a^19,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic19 in TeX
Character table of Dic19 in TeX