metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C19⋊C3, SmallGroup(57,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C19⋊C3 |
Generators and relations for C19⋊C3
G = < a,b | a19=b3=1, bab-1=a11 >
Character table of C19⋊C3
| class | 1 | 3A | 3B | 19A | 19B | 19C | 19D | 19E | 19F | |
| size | 1 | 19 | 19 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 3 | 0 | 0 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | complex faithful |
| ρ5 | 3 | 0 | 0 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | complex faithful |
| ρ6 | 3 | 0 | 0 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | complex faithful |
| ρ7 | 3 | 0 | 0 | ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ195 | ζ1918+ζ1912+ζ198 | ζ1915+ζ1913+ζ1910 | ζ1911+ζ197+ζ19 | ζ1914+ζ193+ζ192 | complex faithful |
| ρ8 | 3 | 0 | 0 | ζ1911+ζ197+ζ19 | ζ199+ζ196+ζ194 | ζ1914+ζ193+ζ192 | ζ1918+ζ1912+ζ198 | ζ1917+ζ1916+ζ195 | ζ1915+ζ1913+ζ1910 | complex faithful |
| ρ9 | 3 | 0 | 0 | ζ1917+ζ1916+ζ195 | ζ1911+ζ197+ζ19 | ζ1915+ζ1913+ζ1910 | ζ1914+ζ193+ζ192 | ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ198 | complex faithful |
►On 19 points: primitive - transitive group 19T3
Generators in S19
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)
G:=sub<Sym(19)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16)]])
G:=TransitiveGroup(19,3);
C19⋊C3 is a maximal subgroup of
C19⋊C6 C19⋊C9 C19⋊A4 C133⋊C3 C133⋊4C3
C19⋊C3 is a maximal quotient of C19⋊2C9 C19⋊A4 C133⋊C3 C133⋊4C3
Matrix representation of C19⋊C3 ►in GL3(𝔽7) generated by
| 5 | 0 | 5 |
| 6 | 0 | 3 |
| 2 | 1 | 6 |
| 1 | 3 | 0 |
| 0 | 6 | 1 |
| 0 | 6 | 0 |
G:=sub<GL(3,GF(7))| [5,6,2,0,0,1,5,3,6],[1,0,0,3,6,6,0,1,0] >;
C19⋊C3 in GAP, Magma, Sage, TeX
C_{19}\rtimes C_3 % in TeX
G:=Group("C19:C3"); // GroupNames label
G:=SmallGroup(57,1);
// by ID
G=gap.SmallGroup(57,1);
# by ID
G:=PCGroup([2,-3,-19,85]);
// Polycyclic
G:=Group<a,b|a^19=b^3=1,b*a*b^-1=a^11>;
// generators/relations
Export
Subgroup lattice of C19⋊C3 in TeX
Character table of C19⋊C3 in TeX