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## G = C9×D9order 162 = 2·34

### Direct product of C9 and D9

Aliases: C9×D9, C9C2, C93C18, C921C2, C3.1(S3×C9), (C3×C9).6C6, (C3×C9).4S3, C3.4(C3×D9), (C3×D9).2C3, C32.11(C3×S3), SmallGroup(162,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C9×D9
 Chief series C1 — C3 — C9 — C3×C9 — C92 — C9×D9
 Lower central C9 — C9×D9
 Upper central C1 — C9

Generators and relations for C9×D9
G = < a,b,c | a9=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Permutation representations of C9×D9
On 18 points - transitive group 18T74
Generators in S18
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 6 2 7 3 8 4 9 5)(10 14 18 13 17 12 16 11 15)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 10)(9 11)

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

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

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

G:=TransitiveGroup(18,74);

C9×D9 is a maximal subgroup of
C273C18  C9⋊C54  C92⋊S3  C92.S3  C922S3  C927C6  C928C6  C923S3  C924S3  C926S3  C925S3
C9×D9 is a maximal quotient of
C9⋊S3⋊C9  C273C18

54 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 6A 6B 9A ··· 9F 9G ··· 9AM 18A ··· 18F order 1 2 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 1 1 2 2 2 9 9 1 ··· 1 2 ··· 2 9 ··· 9

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + image C1 C2 C3 C6 C9 C18 S3 D9 C3×S3 C3×D9 S3×C9 C9×D9 kernel C9×D9 C92 C3×D9 C3×C9 D9 C9 C3×C9 C9 C32 C3 C3 C1 # reps 1 1 2 2 6 6 1 3 2 6 6 18

Matrix representation of C9×D9 in GL2(𝔽19) generated by

 6 0 0 6
,
 16 0 0 6
,
 0 6 16 0
G:=sub<GL(2,GF(19))| [6,0,0,6],[16,0,0,6],[0,16,6,0] >;

C9×D9 in GAP, Magma, Sage, TeX

C_9\times D_9
% in TeX

G:=Group("C9xD9");
// GroupNames label

G:=SmallGroup(162,3);
// by ID

G=gap.SmallGroup(162,3);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,36,1803,138,2704]);
// Polycyclic

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

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