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G = C19⋊C3order 57 = 3·19

The semidirect product of C19 and C3 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C19⋊C3, SmallGroup(57,1)

Series: Derived Chief Lower central Upper central

C1C19 — C19⋊C3
C1C19 — C19⋊C3
C19 — C19⋊C3
C1

Generators and relations for C19⋊C3
 G = < a,b | a19=b3=1, bab-1=a11 >

19C3

Character table of C19⋊C3

 class 13A3B19A19B19C19D19E19F
 size 11919333333
ρ1111111111    trivial
ρ21ζ3ζ32111111    linear of order 3
ρ31ζ32ζ3111111    linear of order 3
ρ4300ζ19181912198ζ191519131910ζ19171916195ζ191119719ζ1914193192ζ199196194    complex faithful
ρ5300ζ191519131910ζ1914193192ζ191119719ζ199196194ζ19181912198ζ19171916195    complex faithful
ρ6300ζ1914193192ζ19181912198ζ199196194ζ19171916195ζ191519131910ζ191119719    complex faithful
ρ7300ζ199196194ζ19171916195ζ19181912198ζ191519131910ζ191119719ζ1914193192    complex faithful
ρ8300ζ191119719ζ199196194ζ1914193192ζ19181912198ζ19171916195ζ191519131910    complex faithful
ρ9300ζ19171916195ζ191119719ζ191519131910ζ1914193192ζ199196194ζ19181912198    complex faithful

Permutation representations of C19⋊C3
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);

Matrix representation of C19⋊C3 in GL3(𝔽7) generated by

505
603
216
,
130
061
060
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

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