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G = C19⋊C9  order 171 = 32·19

The semidirect product of C19 and C9 acting faithfully

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

Aliases: C19⋊C9, C19⋊C3.C3, SmallGroup(171,3)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C19⋊C9
 G = < a,b | a19=b9=1, bab-1=a5 >

C1
19C3
C19
19C9
C19⋊C3
C19⋊C9

Character table of C19⋊C9

 class 13A3B9A9B9C9D9E9F19A19B
 size 1191919191919191999
ρ111111111111    trivial
ρ2111ζ3ζ32ζ32ζ3ζ3ζ3211    linear of order 3
ρ3111ζ32ζ3ζ3ζ32ζ32ζ311    linear of order 3
ρ41ζ32ζ3ζ94ζ92ζ98ζ9ζ97ζ9511    linear of order 9
ρ51ζ3ζ32ζ92ζ9ζ94ζ95ζ98ζ9711    linear of order 9
ρ61ζ32ζ3ζ9ζ95ζ92ζ97ζ94ζ9811    linear of order 9
ρ71ζ32ζ3ζ97ζ98ζ95ζ94ζ9ζ9211    linear of order 9
ρ81ζ3ζ32ζ95ζ97ζ9ζ98ζ92ζ9411    linear of order 9
ρ91ζ3ζ32ζ98ζ94ζ97ζ92ζ95ζ911    linear of order 9
ρ10900000000-1+√-19/2-1-√-19/2    complex faithful
ρ11900000000-1-√-19/2-1+√-19/2    complex faithful

Permutation representations of C19⋊C9
►On 19 points: primitive - transitive group 19T5
Generators in S19
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)

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

C19⋊C9 is a maximal subgroup of   F19

Matrix representation of C19⋊C9 ►in GL9(𝔽2053)

205210000000
205201000000
205200100000
205200010000
205200001000
205200000100
205200000010
205200000001
85811971191171653358558611195
,
000010000
85911971191171653358558611195
33785252615363295217334857
100000000
000001000
4333171386415322048171911972051
2050171911981190171543358572
010000000
000000100

G:=sub<GL(9,GF(2053))| [2052,2052,2052,2052,2052,2052,2052,2052,858,1,0,0,0,0,0,0,0,1197,0,1,0,0,0,0,0,0,1191,0,0,1,0,0,0,0,0,1716,0,0,0,1,0,0,0,0,5,0,0,0,0,1,0,0,0,335,0,0,0,0,0,1,0,0,855,0,0,0,0,0,0,1,0,861,0,0,0,0,0,0,0,1,1195],[0,859,337,1,0,4,2050,0,0,0,1197,852,0,0,333,1719,1,0,0,1191,526,0,0,1713,1198,0,0,0,1716,1536,0,0,864,1190,0,0,1,5,329,0,0,1532,1715,0,0,0,335,521,0,1,2048,4,0,0,0,855,7,0,0,1719,335,0,1,0,861,334,0,0,1197,857,0,0,0,1195,857,0,0,2051,2,0,0] >;
 

C19⋊C9 in GAP, Magma, Sage, TeX 

C_{19}\rtimes C_9
 
% in TeX
 
G:=Group("C19:C9");
 
// GroupNames label
 
G:=SmallGroup(171,3);
 
// by ID
 
G=gap.SmallGroup(171,3);
 
# by ID
 
G:=PCGroup([3,-3,-3,-19,9,326,194]);
 
// Polycyclic
 
G:=Group<a,b|a^19=b^9=1,b*a*b^-1=a^5>;
 
// generators/relations
 

Export

Subgroup lattice of C19⋊C9 in TeX
Character table of C19⋊C9 in TeX

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