direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C32⋊4D6, C3≀C22, C33⋊14D6, C34⋊3C22, C32⋊9S32, C33⋊8(C2×C6), C32⋊7(S3×C6), C3⋊2(C3×S32), (C3×C3⋊S3)⋊3S3, (C3×C3⋊S3)⋊5C6, C3⋊S3⋊3(C3×S3), (C32×C3⋊S3)⋊3C2, SmallGroup(324,167)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C3×C32⋊4D6 |
Generators and relations for C3×C32⋊4D6
G = < a,b,c,d,e | a3=b3=c3=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 736 in 174 conjugacy classes, 30 normal (6 characteristic)
C1, C2, C3, C3, C3, C22, S3, C6, C32, C32, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C33, C33, C33, S32, S3×C6, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C34, C3×S32, C32⋊4D6, C32×C3⋊S3, C3×C32⋊4D6
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S32, S3×C6, C3×S32, C32⋊4D6, C3×C32⋊4D6
►On 12 points - transitive group 12T130
(1 5 3)(2 6 4)(7 9 11)(8 10 12)
(1 5 3)(2 4 6)(7 11 9)(8 10 12)
(1 3 5)(2 6 4)(7 11 9)(8 10 12)
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 10)(2 9)(3 8)(4 7)(5 12)(6 11)
G:=sub<Sym(12)| (1,5,3)(2,6,4)(7,9,11)(8,10,12), (1,5,3)(2,4,6)(7,11,9)(8,10,12), (1,3,5)(2,6,4)(7,11,9)(8,10,12), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10)(2,9)(3,8)(4,7)(5,12)(6,11)>;
G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12), (1,5,3)(2,4,6)(7,11,9)(8,10,12), (1,3,5)(2,6,4)(7,11,9)(8,10,12), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10)(2,9)(3,8)(4,7)(5,12)(6,11) );
G=PermutationGroup([[(1,5,3),(2,6,4),(7,9,11),(8,10,12)], [(1,5,3),(2,4,6),(7,11,9),(8,10,12)], [(1,3,5),(2,6,4),(7,11,9),(8,10,12)], [(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,10),(2,9),(3,8),(4,7),(5,12),(6,11)]])
G:=TransitiveGroup(12,130);
►On 18 points - transitive group 18T120
(1 15 12)(2 16 7)(3 17 8)(4 18 9)(5 13 10)(6 14 11)
(1 3 5)(2 6 4)(7 11 9)(8 10 12)(13 15 17)(14 18 16)
(1 15 12)(2 7 16)(3 17 8)(4 9 18)(5 13 10)(6 11 14)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 6)(2 5)(3 4)(7 10)(8 9)(11 12)(13 16)(14 15)(17 18)
G:=sub<Sym(18)| (1,15,12)(2,16,7)(3,17,8)(4,18,9)(5,13,10)(6,14,11), (1,3,5)(2,6,4)(7,11,9)(8,10,12)(13,15,17)(14,18,16), (1,15,12)(2,7,16)(3,17,8)(4,9,18)(5,13,10)(6,11,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,10)(8,9)(11,12)(13,16)(14,15)(17,18)>;
G:=Group( (1,15,12)(2,16,7)(3,17,8)(4,18,9)(5,13,10)(6,14,11), (1,3,5)(2,6,4)(7,11,9)(8,10,12)(13,15,17)(14,18,16), (1,15,12)(2,7,16)(3,17,8)(4,9,18)(5,13,10)(6,11,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,10)(8,9)(11,12)(13,16)(14,15)(17,18) );
G=PermutationGroup([[(1,15,12),(2,16,7),(3,17,8),(4,18,9),(5,13,10),(6,14,11)], [(1,3,5),(2,6,4),(7,11,9),(8,10,12),(13,15,17),(14,18,16)], [(1,15,12),(2,7,16),(3,17,8),(4,9,18),(5,13,10),(6,11,14)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12),(13,16),(14,15),(17,18)]])
G:=TransitiveGroup(18,120);
Polynomial with Galois group C3×C32⋊4D6 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 12T130 | x12-x9+5x6-8x3+4 | 222·318·56 |
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3K | 3L | ··· | 3Z | 6A | ··· | 6F | 6G | ··· | 6O |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 |
| size | 1 | 9 | 9 | 9 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 9 | ··· | 9 | 18 | ··· | 18 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C3 | C6 | S3 | D6 | C3×S3 | S3×C6 | S32 | C3×S32 | C32⋊4D6 | C3×C32⋊4D6 |
| kernel | C3×C32⋊4D6 | C32×C3⋊S3 | C32⋊4D6 | C3×C3⋊S3 | C3×C3⋊S3 | C33 | C3⋊S3 | C32 | C32 | C3 | C3 | C1 |
| # reps | 1 | 3 | 2 | 6 | 3 | 3 | 6 | 6 | 3 | 6 | 2 | 4 |
Matrix representation of C3×C32⋊4D6 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 5 | 3 | 2 | 3 |
| 1 | 3 | 3 | 0 |
| 4 | 4 | 0 | 6 |
| 0 | 0 | 0 | 4 |
| 3 | 6 | 3 | 2 |
| 6 | 3 | 4 | 2 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 5 | 6 | 6 |
| 3 | 2 | 5 | 2 |
| 2 | 5 | 6 | 3 |
| 1 | 1 | 6 | 5 |
| 6 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 6 | 1 | 4 | 1 |
| 2 | 2 | 6 | 3 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[5,1,4,0,3,3,4,0,2,3,0,0,3,0,6,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[1,3,2,1,5,2,5,1,6,5,6,6,6,2,3,5],[6,1,6,2,0,1,1,2,0,0,4,6,0,0,1,3] >;
C3×C32⋊4D6 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes_4D_6
% in TeX
G:=Group("C3xC3^2:4D6"); // GroupNames label
G:=SmallGroup(324,167);
// by ID
G=gap.SmallGroup(324,167);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,579,297,1090,7781]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations