direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C6×D9, C18⋊3C6, C32.3D6, C9⋊3(C2×C6), (C3×C18)⋊2C2, C3.1(S3×C6), C6.4(C3×S3), (C3×C6).7S3, (C3×C9)⋊3C22, SmallGroup(108,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C6×D9 |
Generators and relations for C6×D9
G = < a,b,c | a6=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D9
►On 36 points
Generators in S36
(1 11 4 14 7 17)(2 12 5 15 8 18)(3 13 6 16 9 10)(19 31 25 28 22 34)(20 32 26 29 23 35)(21 33 27 30 24 36)
(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 21)(2 20)(3 19)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 36)(18 35)
G:=sub<Sym(36)| (1,11,4,14,7,17)(2,12,5,15,8,18)(3,13,6,16,9,10)(19,31,25,28,22,34)(20,32,26,29,23,35)(21,33,27,30,24,36), (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), (1,21)(2,20)(3,19)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,36)(18,35)>;
G:=Group( (1,11,4,14,7,17)(2,12,5,15,8,18)(3,13,6,16,9,10)(19,31,25,28,22,34)(20,32,26,29,23,35)(21,33,27,30,24,36), (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), (1,21)(2,20)(3,19)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,36)(18,35) );
G=PermutationGroup([[(1,11,4,14,7,17),(2,12,5,15,8,18),(3,13,6,16,9,10),(19,31,25,28,22,34),(20,32,26,29,23,35),(21,33,27,30,24,36)], [(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)], [(1,21),(2,20),(3,19),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,36),(18,35)]])
C6×D9 is a maximal subgroup of
C3⋊D36 D6⋊D9
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 9A | ··· | 9I | 18A | ··· | 18I |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | D9 | C3×S3 | D18 | S3×C6 | C3×D9 | C6×D9 |
| kernel | C6×D9 | C3×D9 | C3×C18 | D18 | D9 | C18 | C3×C6 | C32 | C6 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 3 | 2 | 3 | 2 | 6 | 6 |
Matrix representation of C6×D9 ►in GL2(𝔽19) generated by
| 8 | 0 |
| 0 | 8 |
| 9 | 0 |
| 4 | 17 |
| 9 | 18 |
| 4 | 10 |
G:=sub<GL(2,GF(19))| [8,0,0,8],[9,4,0,17],[9,4,18,10] >;
C6×D9 in GAP, Magma, Sage, TeX
C_6\times D_9
% in TeX
G:=Group("C6xD9"); // GroupNames label
G:=SmallGroup(108,23);
// by ID
G=gap.SmallGroup(108,23);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-3,1203,138,1804]);
// Polycyclic
G:=Group<a,b,c|a^6=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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