metabelian, supersoluble, monomial
Aliases: C9⋊1Dic6, C6.1D18, C18.1D6, Dic9.S3, Dic3.D9, C3⋊1Dic18, C32.2Dic6, (C3×C9)⋊Q8, C6.1S32, C2.4(S3×D9), (C3×C6).22D6, C9⋊Dic3.1C2, (C3×C18).1C22, (C3×Dic9).1C2, (C3×Dic3).1S3, (C9×Dic3).1C2, C3.1(C32⋊2Q8), SmallGroup(216,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9⋊Dic6
G = < a,b,c | a9=b12=1, c2=b6, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C9⋊Dic6
►On 72 points
Generators in S72
(1 20 28 5 24 32 9 16 36)(2 25 17 10 33 13 6 29 21)(3 22 30 7 14 34 11 18 26)(4 27 19 12 35 15 8 31 23)(37 59 67 41 51 71 45 55 63)(38 64 56 46 72 52 42 68 60)(39 49 69 43 53 61 47 57 65)(40 66 58 48 62 54 44 70 50)
(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)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 68 7 62)(2 67 8 61)(3 66 9 72)(4 65 10 71)(5 64 11 70)(6 63 12 69)(13 55 19 49)(14 54 20 60)(15 53 21 59)(16 52 22 58)(17 51 23 57)(18 50 24 56)(25 41 31 47)(26 40 32 46)(27 39 33 45)(28 38 34 44)(29 37 35 43)(30 48 36 42)
G:=sub<Sym(72)| (1,20,28,5,24,32,9,16,36)(2,25,17,10,33,13,6,29,21)(3,22,30,7,14,34,11,18,26)(4,27,19,12,35,15,8,31,23)(37,59,67,41,51,71,45,55,63)(38,64,56,46,72,52,42,68,60)(39,49,69,43,53,61,47,57,65)(40,66,58,48,62,54,44,70,50), (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)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,68,7,62)(2,67,8,61)(3,66,9,72)(4,65,10,71)(5,64,11,70)(6,63,12,69)(13,55,19,49)(14,54,20,60)(15,53,21,59)(16,52,22,58)(17,51,23,57)(18,50,24,56)(25,41,31,47)(26,40,32,46)(27,39,33,45)(28,38,34,44)(29,37,35,43)(30,48,36,42)>;
G:=Group( (1,20,28,5,24,32,9,16,36)(2,25,17,10,33,13,6,29,21)(3,22,30,7,14,34,11,18,26)(4,27,19,12,35,15,8,31,23)(37,59,67,41,51,71,45,55,63)(38,64,56,46,72,52,42,68,60)(39,49,69,43,53,61,47,57,65)(40,66,58,48,62,54,44,70,50), (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)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,68,7,62)(2,67,8,61)(3,66,9,72)(4,65,10,71)(5,64,11,70)(6,63,12,69)(13,55,19,49)(14,54,20,60)(15,53,21,59)(16,52,22,58)(17,51,23,57)(18,50,24,56)(25,41,31,47)(26,40,32,46)(27,39,33,45)(28,38,34,44)(29,37,35,43)(30,48,36,42) );
G=PermutationGroup([[(1,20,28,5,24,32,9,16,36),(2,25,17,10,33,13,6,29,21),(3,22,30,7,14,34,11,18,26),(4,27,19,12,35,15,8,31,23),(37,59,67,41,51,71,45,55,63),(38,64,56,46,72,52,42,68,60),(39,49,69,43,53,61,47,57,65),(40,66,58,48,62,54,44,70,50)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,68,7,62),(2,67,8,61),(3,66,9,72),(4,65,10,71),(5,64,11,70),(6,63,12,69),(13,55,19,49),(14,54,20,60),(15,53,21,59),(16,52,22,58),(17,51,23,57),(18,50,24,56),(25,41,31,47),(26,40,32,46),(27,39,33,45),(28,38,34,44),(29,37,35,43),(30,48,36,42)]])
C9⋊Dic6 is a maximal subgroup of
D9×Dic6 Dic18⋊S3 S3×Dic18 D6.D18 D18.3D6 Dic3.D18 D18.4D6
C9⋊Dic6 is a maximal quotient of Dic9⋊Dic3 C18.Dic6 Dic3⋊Dic9
33 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 18A | 18B | 18C | 18D | 18E | 18F | 36A | ··· | 36F |
| order | 1 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 4 | 6 | 18 | 54 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | + | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | S3 | S3 | Q8 | D6 | D6 | D9 | Dic6 | Dic6 | D18 | Dic18 | S32 | C32⋊2Q8 | S3×D9 | C9⋊Dic6 |
| kernel | C9⋊Dic6 | C3×Dic9 | C9×Dic3 | C9⋊Dic3 | Dic9 | C3×Dic3 | C3×C9 | C18 | C3×C6 | Dic3 | C9 | C32 | C6 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 1 | 1 | 3 | 3 |
Matrix representation of C9⋊Dic6 ►in GL6(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 26 | 0 | 0 |
| 0 | 0 | 11 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 0 | 0 | 0 | 0 | 1 | 36 |
| 15 | 12 | 0 | 0 | 0 | 0 |
| 12 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,11,0,0,0,0,26,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,36,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,36,36],[15,12,0,0,0,0,12,22,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C9⋊Dic6 in GAP, Magma, Sage, TeX
C_9\rtimes {\rm Dic}_6 % in TeX
G:=Group("C9:Dic6"); // GroupNames label
G:=SmallGroup(216,26);
// by ID
G=gap.SmallGroup(216,26);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,31,1065,453,1444,2603]);
// Polycyclic
G:=Group<a,b,c|a^9=b^12=1,c^2=b^6,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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