metacyclic, supersoluble, monomial, Z-group
Aliases: C19⋊C6, D19⋊C3, C19⋊C3⋊C2, SmallGroup(114,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C19⋊C6 |
Generators and relations for C19⋊C6
G = < a,b | a19=b6=1, bab-1=a12 >
Character table of C19⋊C6
| class | 1 | 2 | 3A | 3B | 6A | 6B | 19A | 19B | 19C | |
| size | 1 | 19 | 19 | 19 | 19 | 19 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 6 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | orthogonal faithful |
| ρ8 | 6 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | orthogonal faithful |
| ρ9 | 6 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | orthogonal faithful |
►On 19 points: primitive - transitive group 19T4
Generators in S19
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)
(2 9 8 19 12 13)(3 17 15 18 4 6)(5 14 10 16 7 11)
G:=sub<Sym(19)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,9,8,19,12,13)(3,17,15,18,4,6)(5,14,10,16,7,11)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,9,8,19,12,13)(3,17,15,18,4,6)(5,14,10,16,7,11) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)], [(2,9,8,19,12,13),(3,17,15,18,4,6),(5,14,10,16,7,11)]])
G:=TransitiveGroup(19,4);
C19⋊C6 is a maximal subgroup of
F19 D57⋊C3 D19⋊A4
C19⋊C6 is a maximal quotient of C19⋊C12 C57.C6 D57⋊C3 D19⋊A4
Matrix representation of C19⋊C6 ►in GL6(𝔽229)
| 228 | 1 | 0 | 0 | 0 | 0 |
| 228 | 0 | 1 | 0 | 0 | 0 |
| 228 | 0 | 0 | 1 | 0 | 0 |
| 228 | 0 | 0 | 0 | 1 | 0 |
| 228 | 0 | 0 | 0 | 0 | 1 |
| 126 | 224 | 23 | 206 | 5 | 102 |
| 206 | 148 | 92 | 29 | 207 | 127 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 29 | 196 | 51 | 75 | 131 | 102 |
| 132 | 200 | 148 | 98 | 6 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 6 | 98 | 148 | 200 | 132 | 103 |
G:=sub<GL(6,GF(229))| [228,228,228,228,228,126,1,0,0,0,0,224,0,1,0,0,0,23,0,0,1,0,0,206,0,0,0,1,0,5,0,0,0,0,1,102],[206,0,29,132,0,6,148,0,196,200,0,98,92,0,51,148,1,148,29,0,75,98,0,200,207,1,131,6,0,132,127,0,102,1,0,103] >;
C19⋊C6 in GAP, Magma, Sage, TeX
C_{19}\rtimes C_6 % in TeX
G:=Group("C19:C6"); // GroupNames label
G:=SmallGroup(114,1);
// by ID
G=gap.SmallGroup(114,1);
# by ID
G:=PCGroup([3,-2,-3,-19,974,194]);
// Polycyclic
G:=Group<a,b|a^19=b^6=1,b*a*b^-1=a^12>;
// generators/relations
Export
Subgroup lattice of C19⋊C6 in TeX
Character table of C19⋊C6 in TeX