direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C9×D9, C9≀C2, C9⋊3C18, C92⋊1C2, 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
| C9 — C9×D9 |
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
C27⋊3C18 C9⋊C54 C92⋊S3 C92.S3 C92⋊2S3 C92⋊7C6 C92⋊8C6 C92⋊3S3 C92⋊4S3 C92⋊6S3 C92⋊5S3
C9×D9 is a maximal quotient of
C9⋊S3⋊C9 C27⋊3C18
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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