direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C9×Dic3, C3⋊C36, C6.C18, C18.4S3, C32.2C12, (C3×C9)⋊1C4, C2.(S3×C9), C6.8(C3×S3), (C3×C6).5C6, (C3×C18).1C2, C18○(C3×Dic3), (C3×Dic3).C3, C3.4(C3×Dic3), SmallGroup(108,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C9×Dic3 |
Generators and relations for C9×Dic3
G = < a,b,c | a9=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C9×Dic3
►On 36 points
Generators in S36
(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 26 4 20 7 23)(2 27 5 21 8 24)(3 19 6 22 9 25)(10 34 16 31 13 28)(11 35 17 32 14 29)(12 36 18 33 15 30)
(1 10 20 31)(2 11 21 32)(3 12 22 33)(4 13 23 34)(5 14 24 35)(6 15 25 36)(7 16 26 28)(8 17 27 29)(9 18 19 30)
G:=sub<Sym(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), (1,26,4,20,7,23)(2,27,5,21,8,24)(3,19,6,22,9,25)(10,34,16,31,13,28)(11,35,17,32,14,29)(12,36,18,33,15,30), (1,10,20,31)(2,11,21,32)(3,12,22,33)(4,13,23,34)(5,14,24,35)(6,15,25,36)(7,16,26,28)(8,17,27,29)(9,18,19,30)>;
G:=Group( (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), (1,26,4,20,7,23)(2,27,5,21,8,24)(3,19,6,22,9,25)(10,34,16,31,13,28)(11,35,17,32,14,29)(12,36,18,33,15,30), (1,10,20,31)(2,11,21,32)(3,12,22,33)(4,13,23,34)(5,14,24,35)(6,15,25,36)(7,16,26,28)(8,17,27,29)(9,18,19,30) );
G=PermutationGroup([[(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)], [(1,26,4,20,7,23),(2,27,5,21,8,24),(3,19,6,22,9,25),(10,34,16,31,13,28),(11,35,17,32,14,29),(12,36,18,33,15,30)], [(1,10,20,31),(2,11,21,32),(3,12,22,33),(4,13,23,34),(5,14,24,35),(6,15,25,36),(7,16,26,28),(8,17,27,29),(9,18,19,30)]])
C9×Dic3 is a maximal subgroup of
C9⋊Dic6 C18.D6 C3⋊D36 S3×C36 C32⋊C36 C9⋊C36 He3.C12 He3.2C12 He3.5C12 Q8⋊C9⋊3S3
C9×Dic3 is a maximal quotient of
C32⋊C36 C9⋊C36
54 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 18A | ··· | 18F | 18G | ··· | 18L | 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 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | S3 | Dic3 | C3×S3 | C3×Dic3 | S3×C9 | C9×Dic3 |
| kernel | C9×Dic3 | C3×C18 | C3×Dic3 | C3×C9 | C3×C6 | Dic3 | C32 | C6 | C3 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 1 | 1 | 2 | 2 | 6 | 6 |
Matrix representation of C9×Dic3 ►in GL2(𝔽19) generated by
| 9 | 0 |
| 0 | 9 |
| 12 | 0 |
| 0 | 8 |
| 0 | 18 |
| 1 | 0 |
G:=sub<GL(2,GF(19))| [9,0,0,9],[12,0,0,8],[0,1,18,0] >;
C9×Dic3 in GAP, Magma, Sage, TeX
C_9\times {\rm Dic}_3 % in TeX
G:=Group("C9xDic3"); // GroupNames label
G:=SmallGroup(108,7);
// by ID
G=gap.SmallGroup(108,7);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-3,30,66,1804]);
// Polycyclic
G:=Group<a,b,c|a^9=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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