direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C9×Dic9, C9⋊3C36, C92⋊2C4, C18.6D9, C18.5C18, C2.(C9×D9), C6.1(S3×C9), C6.8(C3×D9), (C3×C9).6C12, (C9×C18).1C2, (C3×C18).15S3, (C3×C18).24C6, C3.4(C3×Dic9), C3.1(C9×Dic3), (C3×C9).4Dic3, (C3×Dic9).2C3, C32.11(C3×Dic3), (C3×C6).25(C3×S3), SmallGroup(324,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C9×Dic9 |
Generators and relations for C9×Dic9
G = < a,b,c | a9=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C9×Dic9
►On 36 points
Generators in S36
(1 3 5 7 9 11 13 15 17)(2 4 6 8 10 12 14 16 18)(19 35 33 31 29 27 25 23 21)(20 36 34 32 30 28 26 24 22)
(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 28 29 30 31 32 33 34 35 36)
(1 36 10 27)(2 35 11 26)(3 34 12 25)(4 33 13 24)(5 32 14 23)(6 31 15 22)(7 30 16 21)(8 29 17 20)(9 28 18 19)
G:=sub<Sym(36)| (1,3,5,7,9,11,13,15,17)(2,4,6,8,10,12,14,16,18)(19,35,33,31,29,27,25,23,21)(20,36,34,32,30,28,26,24,22), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36), (1,36,10,27)(2,35,11,26)(3,34,12,25)(4,33,13,24)(5,32,14,23)(6,31,15,22)(7,30,16,21)(8,29,17,20)(9,28,18,19)>;
G:=Group( (1,3,5,7,9,11,13,15,17)(2,4,6,8,10,12,14,16,18)(19,35,33,31,29,27,25,23,21)(20,36,34,32,30,28,26,24,22), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36), (1,36,10,27)(2,35,11,26)(3,34,12,25)(4,33,13,24)(5,32,14,23)(6,31,15,22)(7,30,16,21)(8,29,17,20)(9,28,18,19) );
G=PermutationGroup([[(1,3,5,7,9,11,13,15,17),(2,4,6,8,10,12,14,16,18),(19,35,33,31,29,27,25,23,21),(20,36,34,32,30,28,26,24,22)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)], [(1,36,10,27),(2,35,11,26),(3,34,12,25),(4,33,13,24),(5,32,14,23),(6,31,15,22),(7,30,16,21),(8,29,17,20),(9,28,18,19)]])
108 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 9A | ··· | 9F | 9G | ··· | 9AM | 12A | 12B | 12C | 12D | 18A | ··· | 18F | 18G | ··· | 18AM | 36A | ··· | 36L |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | |||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | S3 | Dic3 | D9 | C3×S3 | Dic9 | C3×Dic3 | C3×D9 | S3×C9 | C3×Dic9 | C9×Dic3 | C9×D9 | C9×Dic9 |
| kernel | C9×Dic9 | C9×C18 | C3×Dic9 | C92 | C3×C18 | Dic9 | C3×C9 | C18 | C9 | C3×C18 | C3×C9 | C18 | C3×C6 | C9 | C32 | C6 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 1 | 1 | 3 | 2 | 3 | 2 | 6 | 6 | 6 | 6 | 18 | 18 |
Matrix representation of C9×Dic9 ►in GL2(𝔽19) generated by
| 4 | 0 |
| 0 | 4 |
| 14 | 0 |
| 0 | 15 |
| 0 | 18 |
| 1 | 0 |
G:=sub<GL(2,GF(19))| [4,0,0,4],[14,0,0,15],[0,1,18,0] >;
C9×Dic9 in GAP, Magma, Sage, TeX
C_9\times {\rm Dic}_9 % in TeX
G:=Group("C9xDic9"); // GroupNames label
G:=SmallGroup(324,6);
// by ID
G=gap.SmallGroup(324,6);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,79,5404,208,7781]);
// Polycyclic
G:=Group<a,b,c|a^9=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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