metabelian, supersoluble, monomial
Aliases: C9⋊S3⋊2C9, (C3×C9)⋊2C18, C32⋊C9.5S3, (C32×C9).4C6, C32.7(S3×C9), C33.50(C3×S3), C3.3(C32⋊C18), C32.19He3⋊1C2, C32.36(C32⋊C6), C3.2(He3.S3), (C3×C9⋊S3).2C3, SmallGroup(486,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — (C3×C9)⋊C18 |
Generators and relations for (C3×C9)⋊C18
G = < a,b,c | a3=b9=c18=1, ab=ba, cac-1=a-1b3, cbc-1=ab2 >
Smallest permutation representation of (C3×C9)⋊C18
►On 54 points
Generators in S54
(2 14 8)(3 15 9)(5 11 17)(6 12 18)(19 25 31)(20 26 32)(22 34 28)(23 35 29)(37 43 49)(39 51 45)(40 52 46)(42 48 54)
(1 27 44 13 21 38 7 33 50)(2 45 22 8 51 28 14 39 34)(3 35 40 15 29 52 9 23 46)(4 53 36 10 41 24 16 47 30)(5 19 42 17 31 54 11 25 48)(6 49 26 12 37 32 18 43 20)
(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)
G:=sub<Sym(54)| (2,14,8)(3,15,9)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(22,34,28)(23,35,29)(37,43,49)(39,51,45)(40,52,46)(42,48,54), (1,27,44,13,21,38,7,33,50)(2,45,22,8,51,28,14,39,34)(3,35,40,15,29,52,9,23,46)(4,53,36,10,41,24,16,47,30)(5,19,42,17,31,54,11,25,48)(6,49,26,12,37,32,18,43,20), (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)>;
G:=Group( (2,14,8)(3,15,9)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(22,34,28)(23,35,29)(37,43,49)(39,51,45)(40,52,46)(42,48,54), (1,27,44,13,21,38,7,33,50)(2,45,22,8,51,28,14,39,34)(3,35,40,15,29,52,9,23,46)(4,53,36,10,41,24,16,47,30)(5,19,42,17,31,54,11,25,48)(6,49,26,12,37,32,18,43,20), (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) );
G=PermutationGroup([[(2,14,8),(3,15,9),(5,11,17),(6,12,18),(19,25,31),(20,26,32),(22,34,28),(23,35,29),(37,43,49),(39,51,45),(40,52,46),(42,48,54)], [(1,27,44,13,21,38,7,33,50),(2,45,22,8,51,28,14,39,34),(3,35,40,15,29,52,9,23,46),(4,53,36,10,41,24,16,47,30),(5,19,42,17,31,54,11,25,48),(6,49,26,12,37,32,18,43,20)], [(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)]])
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 9A | ··· | 9I | 9J | ··· | 9O | 9P | ··· | 9U | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 27 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 27 | 27 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | S3 | C3×S3 | S3×C9 | C32⋊C6 | C32⋊C18 | He3.S3 | (C3×C9)⋊C18 |
| kernel | (C3×C9)⋊C18 | C32.19He3 | C3×C9⋊S3 | C32×C9 | C9⋊S3 | C3×C9 | C32⋊C9 | C33 | C32 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 6 | 1 | 2 | 3 | 6 |
Matrix representation of (C3×C9)⋊C18 ►in GL6(𝔽19)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 18 | 11 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 12 | 8 | 0 | 7 | 11 | 0 |
| 8 | 0 | 0 | 18 | 0 | 7 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 17 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 3 | 3 | 0 | 4 | 9 | 0 |
| 16 | 16 | 0 | 9 | 0 | 9 |
| 11 | 12 | 0 | 1 | 0 | 13 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 1 | 8 | 0 | 7 | 11 | 7 |
| 12 | 12 | 13 | 0 | 0 | 0 |
| 8 | 0 | 8 | 11 | 0 | 11 |
| 0 | 8 | 18 | 8 | 0 | 8 |
G:=sub<GL(6,GF(19))| [1,0,18,0,12,8,0,7,11,0,8,0,0,0,11,0,0,0,0,0,0,1,7,18,0,0,0,0,11,0,0,0,0,0,0,7],[16,0,17,0,3,16,0,17,0,0,3,16,0,0,17,0,0,0,0,0,0,6,4,9,0,0,0,0,9,0,0,0,0,0,0,9],[11,0,1,12,8,0,12,0,8,12,0,8,0,0,0,13,8,18,1,11,7,0,11,8,0,0,11,0,0,0,13,0,7,0,11,8] >;
(C3×C9)⋊C18 in GAP, Magma, Sage, TeX
(C_3\times C_9)\rtimes C_{18} % in TeX
G:=Group("(C3xC9):C18"); // GroupNames label
G:=SmallGroup(486,20);
// by ID
G=gap.SmallGroup(486,20);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,8643,873,237,3244,3250,11669]);
// Polycyclic
G:=Group<a,b,c|a^3=b^9=c^18=1,a*b=b*a,c*a*c^-1=a^-1*b^3,c*b*c^-1=a*b^2>;
// generators/relations
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