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G = C19⋊C6order 114 = 2·3·19

The semidirect product of C19 and C6 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C19⋊C6, D19⋊C3, C19⋊C3⋊C2, SmallGroup(114,1)

Series: Derived Chief Lower central Upper central

C1C19 — C19⋊C6
C1C19C19⋊C3 — C19⋊C6
C19 — C19⋊C6
C1

Generators and relations for C19⋊C6
 G = < a,b | a19=b6=1, bab-1=a12 >

19C2
19C3
19C6

Character table of C19⋊C6

 class 123A3B6A6B19A19B19C
 size 11919191919666
ρ1111111111    trivial
ρ21-111-1-1111    linear of order 2
ρ311ζ32ζ3ζ32ζ3111    linear of order 3
ρ41-1ζ32ζ3ζ6ζ65111    linear of order 6
ρ511ζ3ζ32ζ3ζ32111    linear of order 3
ρ61-1ζ3ζ32ζ65ζ6111    linear of order 6
ρ7600000ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194    orthogonal faithful
ρ8600000ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192    orthogonal faithful
ρ9600000ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719    orthogonal faithful

Permutation representations of C19⋊C6
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)

22810000
22801000
22800100
22800010
22800001
126224232065102
,
2061489229207127
000010
291965175131102
1322001489861
001000
698148200132103

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

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