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G = C3×D9order 54 = 2·33

Direct product of C3 and D9

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C3×D9, C93C6, C32.2S3, (C3×C9)⋊2C2, C3.1(C3×S3), SmallGroup(54,3)

Series: Derived Chief Lower central Upper central

C1C9 — C3×D9
C1C3C9C3×C9 — C3×D9
C9 — C3×D9
C1C3

Generators and relations for C3×D9
 G = < a,b,c | a3=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
2C3
3S3
9C6
2C9
3C3×S3

Character table of C3×D9

 class 123A3B3C3D3E6A6B9A9B9C9D9E9F9G9H9I
 size 191122299222222222
ρ1111111111111111111    trivial
ρ21-111111-1-1111111111    linear of order 2
ρ31-1ζ3ζ32ζ3ζ321ζ6ζ65ζ311ζ32ζ32ζ321ζ3ζ3    linear of order 6
ρ411ζ3ζ32ζ3ζ321ζ32ζ3ζ311ζ32ζ32ζ321ζ3ζ3    linear of order 3
ρ511ζ32ζ3ζ32ζ31ζ3ζ32ζ3211ζ3ζ3ζ31ζ32ζ32    linear of order 3
ρ61-1ζ32ζ3ζ32ζ31ζ65ζ6ζ3211ζ3ζ3ζ31ζ32ζ32    linear of order 6
ρ7202222200-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ82022-1-1-100ζ989ζ989ζ9792ζ9792ζ9594ζ989ζ9594ζ9792ζ9594    orthogonal lifted from D9
ρ92022-1-1-100ζ9594ζ9594ζ989ζ989ζ9792ζ9594ζ9792ζ989ζ9792    orthogonal lifted from D9
ρ102022-1-1-100ζ9792ζ9792ζ9594ζ9594ζ989ζ9792ζ989ζ9594ζ989    orthogonal lifted from D9
ρ1120-1+-3-1--3-1+-3-1--3200ζ65-1-1ζ6ζ6ζ6-1ζ65ζ65    complex lifted from C3×S3
ρ1220-1--3-1+-3-1--3-1+-3200ζ6-1-1ζ65ζ65ζ65-1ζ6ζ6    complex lifted from C3×S3
ρ1320-1+-3-1--3ζ65ζ6-100ζ9897ζ9594ζ989ζ9795ζ9894ζ929ζ9792ζ9492ζ959    complex faithful
ρ1420-1--3-1+-3ζ6ζ65-100ζ9894ζ9792ζ9594ζ9897ζ9492ζ959ζ989ζ929ζ9795    complex faithful
ρ1520-1+-3-1--3ζ65ζ6-100ζ959ζ9792ζ9594ζ929ζ9795ζ9894ζ989ζ9897ζ9492    complex faithful
ρ1620-1--3-1+-3ζ6ζ65-100ζ9795ζ989ζ9792ζ959ζ9897ζ9492ζ9594ζ9894ζ929    complex faithful
ρ1720-1+-3-1--3ζ65ζ6-100ζ9492ζ989ζ9792ζ9894ζ929ζ9795ζ9594ζ959ζ9897    complex faithful
ρ1820-1--3-1+-3ζ6ζ65-100ζ929ζ9594ζ989ζ9492ζ959ζ9897ζ9792ζ9795ζ9894    complex faithful

Permutation representations of C3×D9
On 18 points - transitive group 18T19
Generators in S18
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 12)(2 11)(3 10)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)

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

G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,12)(2,11)(3,10)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13) );

G=PermutationGroup([(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,12),(2,11),(3,10),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13)])

G:=TransitiveGroup(18,19);

On 27 points - transitive group 27T9
Generators in S27
(1 23 14)(2 24 15)(3 25 16)(4 26 17)(5 27 18)(6 19 10)(7 20 11)(8 21 12)(9 22 13)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)
(1 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)

G:=sub<Sym(27)| (1,23,14)(2,24,15)(3,25,16)(4,26,17)(5,27,18)(6,19,10)(7,20,11)(8,21,12)(9,22,13), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)>;

G:=Group( (1,23,14)(2,24,15)(3,25,16)(4,26,17)(5,27,18)(6,19,10)(7,20,11)(8,21,12)(9,22,13), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23) );

G=PermutationGroup([(1,23,14),(2,24,15),(3,25,16),(4,26,17),(5,27,18),(6,19,10),(7,20,11),(8,21,12),(9,22,13)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,26),(20,25),(21,24),(22,23)])

G:=TransitiveGroup(27,9);

Matrix representation of C3×D9 in GL2(𝔽19) generated by

110
011
,
50
04
,
04
50
G:=sub<GL(2,GF(19))| [11,0,0,11],[5,0,0,4],[0,5,4,0] >;

C3×D9 in GAP, Magma, Sage, TeX

C_3\times D_9
% in TeX

G:=Group("C3xD9");
// GroupNames label

G:=SmallGroup(54,3);
// by ID

G=gap.SmallGroup(54,3);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,362,82,579]);
// Polycyclic

G:=Group<a,b,c|a^3=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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