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## G = C3×D9order 54 = 2·33

### Direct product of C3 and D9

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C3×D9
 Chief series C1 — C3 — C9 — C3×C9 — C3×D9
 Lower central C9 — C3×D9
 Upper central C1 — C3

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

Character table of C3×D9

 class 1 2 3A 3B 3C 3D 3E 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I size 1 9 1 1 2 2 2 9 9 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 ζ3 ζ32 ζ3 ζ32 1 ζ6 ζ65 ζ3 1 1 ζ32 ζ32 ζ32 1 ζ3 ζ3 linear of order 6 ρ4 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ3 1 1 ζ32 ζ32 ζ32 1 ζ3 ζ3 linear of order 3 ρ5 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ32 1 1 ζ3 ζ3 ζ3 1 ζ32 ζ32 linear of order 3 ρ6 1 -1 ζ32 ζ3 ζ32 ζ3 1 ζ65 ζ6 ζ32 1 1 ζ3 ζ3 ζ3 1 ζ32 ζ32 linear of order 6 ρ7 2 0 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 0 2 2 -1 -1 -1 0 0 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ9 2 0 2 2 -1 -1 -1 0 0 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ10 2 0 2 2 -1 -1 -1 0 0 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ11 2 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 2 0 0 ζ65 -1 -1 ζ6 ζ6 ζ6 -1 ζ65 ζ65 complex lifted from C3×S3 ρ12 2 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 2 0 0 ζ6 -1 -1 ζ65 ζ65 ζ65 -1 ζ6 ζ6 complex lifted from C3×S3 ρ13 2 0 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 ζ98+ζ97 ζ95+ζ94 ζ98+ζ9 ζ97+ζ95 ζ98+ζ94 ζ92+ζ9 ζ97+ζ92 ζ94+ζ92 ζ95+ζ9 complex faithful ρ14 2 0 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 ζ98+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ97 ζ94+ζ92 ζ95+ζ9 ζ98+ζ9 ζ92+ζ9 ζ97+ζ95 complex faithful ρ15 2 0 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 ζ95+ζ9 ζ97+ζ92 ζ95+ζ94 ζ92+ζ9 ζ97+ζ95 ζ98+ζ94 ζ98+ζ9 ζ98+ζ97 ζ94+ζ92 complex faithful ρ16 2 0 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 ζ97+ζ95 ζ98+ζ9 ζ97+ζ92 ζ95+ζ9 ζ98+ζ97 ζ94+ζ92 ζ95+ζ94 ζ98+ζ94 ζ92+ζ9 complex faithful ρ17 2 0 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 ζ94+ζ92 ζ98+ζ9 ζ97+ζ92 ζ98+ζ94 ζ92+ζ9 ζ97+ζ95 ζ95+ζ94 ζ95+ζ9 ζ98+ζ97 complex faithful ρ18 2 0 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 ζ92+ζ9 ζ95+ζ94 ζ98+ζ9 ζ94+ζ92 ζ95+ζ9 ζ98+ζ97 ζ97+ζ92 ζ97+ζ95 ζ98+ζ94 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);

C3×D9 is a maximal subgroup of
C9⋊C18  C27⋊C6  C322D9  He3.3S3  He3⋊S3  3- 1+2.S3  He3.4S3  C95F7
C3×D9 is a maximal quotient of
C32⋊D9  C27⋊C6  C95F7

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

 11 0 0 11
,
 5 0 0 4
,
 0 4 5 0
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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