direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×Dic9, C9⋊3C12, C6.4D9, C18.3C6, C32.2Dic3, (C3×C9)⋊2C4, C2.(C3×D9), C6.1(C3×S3), (C3×C6).5S3, (C3×C18).2C2, C3.1(C3×Dic3), SmallGroup(108,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C3×Dic9 |
Generators and relations for C3×Dic9
G = < a,b,c | a3=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic9
►On 36 points
Generators in S36
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 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 32 10 23)(2 31 11 22)(3 30 12 21)(4 29 13 20)(5 28 14 19)(6 27 15 36)(7 26 16 35)(8 25 17 34)(9 24 18 33)
G:=sub<Sym(36)| (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36), (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), (1,32,10,23)(2,31,11,22)(3,30,12,21)(4,29,13,20)(5,28,14,19)(6,27,15,36)(7,26,16,35)(8,25,17,34)(9,24,18,33)>;
G:=Group( (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36), (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), (1,32,10,23)(2,31,11,22)(3,30,12,21)(4,29,13,20)(5,28,14,19)(6,27,15,36)(7,26,16,35)(8,25,17,34)(9,24,18,33) );
G=PermutationGroup([[(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36)], [(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)], [(1,32,10,23),(2,31,11,22),(3,30,12,21),(4,29,13,20),(5,28,14,19),(6,27,15,36),(7,26,16,35),(8,25,17,34),(9,24,18,33)]])
C3×Dic9 is a maximal subgroup of
C9⋊Dic6 C18.D6 C9⋊D12 C12×D9 C9⋊C36 C27⋊C12 C32⋊2Dic9 He3.3Dic3 He3⋊Dic3 3- 1+2.Dic3 He3.4Dic3 Dic9.2A4
C3×Dic9 is a maximal quotient of
C32⋊Dic9 C27⋊C12
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 18A | ··· | 18I |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | D9 | C3×S3 | Dic9 | C3×Dic3 | C3×D9 | C3×Dic9 |
| kernel | C3×Dic9 | C3×C18 | Dic9 | C3×C9 | C18 | C9 | C3×C6 | C32 | C6 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 3 | 2 | 3 | 2 | 6 | 6 |
Matrix representation of C3×Dic9 ►in GL2(𝔽19) generated by
| 7 | 0 |
| 0 | 7 |
| 15 | 0 |
| 0 | 14 |
| 0 | 18 |
| 1 | 0 |
G:=sub<GL(2,GF(19))| [7,0,0,7],[15,0,0,14],[0,1,18,0] >;
C3×Dic9 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_9 % in TeX
G:=Group("C3xDic9"); // GroupNames label
G:=SmallGroup(108,6);
// by ID
G=gap.SmallGroup(108,6);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-3,30,1203,138,1804]);
// Polycyclic
G:=Group<a,b,c|a^3=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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