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G = F19  order 342 = 2·32·19

Frobenius group

metacyclic, supersoluble, monomial, Z-group

Aliases: F19, AGL1(𝔽19), C19⋊C18, D19⋊C9, C19⋊C9⋊C2, C19⋊C3.C6, C19⋊C6.C3, Aut(D19), Hol(C19), SmallGroup(342,7)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C19 — F19
Chief series
C1 — C19 — C19⋊C3 — C19⋊C9 — F19
Lower central
C19 — F19
Upper central
C1

Generators and relations for F19
 G = < a,b | a19=b18=1, bab-1=a3 >

C1
19C2
19C3
C19
19C6
19C9
D19
C19⋊C3
19C18
C19⋊C6
C19⋊C9
F19

Character table of F19

 class 123A3B6A6B9A9B9C9D9E9F18A18B18C18D18E18F19
 size 1191919191919191919191919191919191918
ρ11111111111111111111    trivial
ρ21-111-1-1111111-1-1-1-1-1-11    linear of order 2
ρ31-111-1-1ζ3ζ3ζ32ζ32ζ32ζ3ζ65ζ6ζ6ζ6ζ65ζ651    linear of order 6
ρ4111111ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ31    linear of order 3
ρ51-111-1-1ζ32ζ32ζ3ζ3ζ3ζ32ζ6ζ65ζ65ζ65ζ6ζ61    linear of order 6
ρ6111111ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ321    linear of order 3
ρ711ζ3ζ32ζ3ζ32ζ94ζ97ζ95ζ92ζ98ζ9ζ94ζ95ζ92ζ98ζ9ζ971    linear of order 9
ρ81-1ζ3ζ32ζ65ζ6ζ94ζ97ζ95ζ92ζ98ζ9-ζ94-ζ95-ζ92-ζ98-ζ9-ζ971    linear of order 18
ρ91-1ζ32ζ3ζ6ζ65ζ95ζ92ζ94ζ97ζ9ζ98-ζ95-ζ94-ζ97-ζ9-ζ98-ζ921    linear of order 18
ρ1011ζ32ζ3ζ32ζ3ζ92ζ98ζ97ζ9ζ94ζ95ζ92ζ97ζ9ζ94ζ95ζ981    linear of order 9
ρ1111ζ32ζ3ζ32ζ3ζ98ζ95ζ9ζ94ζ97ζ92ζ98ζ9ζ94ζ97ζ92ζ951    linear of order 9
ρ121-1ζ3ζ32ζ65ζ6ζ97ζ9ζ92ζ98ζ95ζ94-ζ97-ζ92-ζ98-ζ95-ζ94-ζ91    linear of order 18
ρ1311ζ3ζ32ζ3ζ32ζ9ζ94ζ98ζ95ζ92ζ97ζ9ζ98ζ95ζ92ζ97ζ941    linear of order 9
ρ1411ζ3ζ32ζ3ζ32ζ97ζ9ζ92ζ98ζ95ζ94ζ97ζ92ζ98ζ95ζ94ζ91    linear of order 9
ρ151-1ζ3ζ32ζ65ζ6ζ9ζ94ζ98ζ95ζ92ζ97-ζ9-ζ98-ζ95-ζ92-ζ97-ζ941    linear of order 18
ρ1611ζ32ζ3ζ32ζ3ζ95ζ92ζ94ζ97ζ9ζ98ζ95ζ94ζ97ζ9ζ98ζ921    linear of order 9
ρ171-1ζ32ζ3ζ6ζ65ζ92ζ98ζ97ζ9ζ94ζ95-ζ92-ζ97-ζ9-ζ94-ζ95-ζ981    linear of order 18
ρ181-1ζ32ζ3ζ6ζ65ζ98ζ95ζ9ζ94ζ97ζ92-ζ98-ζ9-ζ94-ζ97-ζ92-ζ951    linear of order 18
ρ191800000000000000000-1    orthogonal faithful

Permutation representations of F19
►On 19 points: primitive, sharply doubly transitive - transitive group 19T6
Generators in S19
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)

(2 14 18 13 5 15 12 11 17 19 7 3 8 16 6 9 10 4)
 
G:=sub<Sym(19)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,14,18,13,5,15,12,11,17,19,7,3,8,16,6,9,10,4)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,14,18,13,5,15,12,11,17,19,7,3,8,16,6,9,10,4) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)], [(2,14,18,13,5,15,12,11,17,19,7,3,8,16,6,9,10,4)]])
 
G:=TransitiveGroup(19,6);
 

Matrix representation of F19 ►in GL18(ℤ)

010000000000000000
001000000000000000
000100000000000000
000010000000000000
000001000000000000
000000100000000000
000000010000000000
000000001000000000
000000000100000000
000000000010000000
000000000001000000
000000000000100000
000000000000010000
000000000000001000
000000000000000100
000000000000000010
000000000000000001
-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1
,
100000000000000000
000100000000000000
000000100000000000
000000000100000000
000000000000100000
000000000000000100
-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1
001000000000000000
000001000000000000
000000001000000000
000000000001000000
000000000000001000
000000000000000001
010000000000000000
000010000000000000
000000010000000000
000000000010000000
000000000000010000

G:=sub<GL(18,Integers())| [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0] >;
 

F19 in GAP, Magma, Sage, TeX 

F_{19}
 
% in TeX
 
G:=Group("F19");
 
// GroupNames label
 
G:=SmallGroup(342,7);
 
// by ID
 
G=gap.SmallGroup(342,7);
 
# by ID
 
G:=PCGroup([4,-2,-3,-3,-19,29,5187,2455,539]);
 
// Polycyclic
 
G:=Group<a,b|a^19=b^18=1,b*a*b^-1=a^3>;
 
// generators/relations
 

Export

Subgroup lattice of F19 in TeX
Character table of F19 in TeX

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