direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C33⋊C4, C3≀C4, AΣL1(𝔽81), C34⋊2C4, C33⋊6C12, C33⋊7Dic3, C32⋊2(C32⋊C4), C32⋊4(C3×Dic3), C3⋊(C3×C32⋊C4), C3⋊S3.(C3×S3), (C3×C3⋊S3).2S3, (C3×C3⋊S3).5C6, (C32×C3⋊S3).2C2, SmallGroup(324,162)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C3×C33⋊C4 |
Generators and relations for C3×C33⋊C4
G = < a,b,c,d,e | a3=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc-1, cd=dc, ece-1=b-1c-1, ede-1=d-1 >
Subgroups: 424 in 96 conjugacy classes, 14 normal (all characteristic)
C1, C2, C3, C3, C4, S3, C6, C32, C32, Dic3, C12, C3×S3, C3⋊S3, C3×C6, C33, C33, C3×Dic3, C32⋊C4, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C34, C3×C32⋊C4, C33⋊C4, C32×C3⋊S3, C3×C33⋊C4
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, C3×Dic3, C32⋊C4, C3×C32⋊C4, C33⋊C4, C3×C33⋊C4
►On 12 points - transitive group 12T131
(1 9 8)(2 10 5)(3 11 6)(4 12 7)
(1 9 8)(2 5 10)(3 6 11)(4 12 7)
(1 8 9)(3 11 6)
(1 9 8)(2 5 10)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
G:=sub<Sym(12)| (1,9,8)(2,10,5)(3,11,6)(4,12,7), (1,9,8)(2,5,10)(3,6,11)(4,12,7), (1,8,9)(3,11,6), (1,9,8)(2,5,10)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12)>;
G:=Group( (1,9,8)(2,10,5)(3,11,6)(4,12,7), (1,9,8)(2,5,10)(3,6,11)(4,12,7), (1,8,9)(3,11,6), (1,9,8)(2,5,10)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12) );
G=PermutationGroup([[(1,9,8),(2,10,5),(3,11,6),(4,12,7)], [(1,9,8),(2,5,10),(3,6,11),(4,12,7)], [(1,8,9),(3,11,6)], [(1,9,8),(2,5,10),(3,11,6),(4,7,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)]])
G:=TransitiveGroup(12,131);
►On 18 points - transitive group 18T123
(1 3 6)(2 4 5)(7 13 18)(8 14 15)(9 11 16)(10 12 17)
(1 7 9)(3 13 11)(6 18 16)
(1 7 9)(2 8 10)(3 13 11)(4 14 12)(5 15 17)(6 18 16)
(1 6 3)(2 4 5)(7 18 13)(8 14 15)(9 16 11)(10 12 17)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
G:=sub<Sym(18)| (1,3,6)(2,4,5)(7,13,18)(8,14,15)(9,11,16)(10,12,17), (1,7,9)(3,13,11)(6,18,16), (1,7,9)(2,8,10)(3,13,11)(4,14,12)(5,15,17)(6,18,16), (1,6,3)(2,4,5)(7,18,13)(8,14,15)(9,16,11)(10,12,17), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18)>;
G:=Group( (1,3,6)(2,4,5)(7,13,18)(8,14,15)(9,11,16)(10,12,17), (1,7,9)(3,13,11)(6,18,16), (1,7,9)(2,8,10)(3,13,11)(4,14,12)(5,15,17)(6,18,16), (1,6,3)(2,4,5)(7,18,13)(8,14,15)(9,16,11)(10,12,17), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18) );
G=PermutationGroup([[(1,3,6),(2,4,5),(7,13,18),(8,14,15),(9,11,16),(10,12,17)], [(1,7,9),(3,13,11),(6,18,16)], [(1,7,9),(2,8,10),(3,13,11),(4,14,12),(5,15,17),(6,18,16)], [(1,6,3),(2,4,5),(7,18,13),(8,14,15),(9,16,11),(10,12,17)], [(1,2),(3,4),(5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)]])
G:=TransitiveGroup(18,123);
Polynomial with Galois group C3×C33⋊C4 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 12T131 | x12-69x10-72x9+1566x8+2466x7-14530x6-27216x5+49953x4+102474x3-21636x2-43272x+9616 | 220·334·59·292·592·4992·6012·32394292 |
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | ··· | 3W | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 9 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 27 | 27 | 9 | 9 | 18 | 18 | 18 | 27 | 27 | 27 | 27 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | C3×S3 | C3×Dic3 | C32⋊C4 | C3×C32⋊C4 | C33⋊C4 | C3×C33⋊C4 |
| kernel | C3×C33⋊C4 | C32×C3⋊S3 | C33⋊C4 | C34 | C3×C3⋊S3 | C33 | C3×C3⋊S3 | C33 | C3⋊S3 | C32 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3×C33⋊C4 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 3 | 2 | 4 | 3 |
| 4 | 5 | 5 | 6 |
| 3 | 3 | 6 | 1 |
| 0 | 0 | 0 | 1 |
| 0 | 5 | 2 | 6 |
| 0 | 2 | 0 | 2 |
| 3 | 3 | 6 | 1 |
| 0 | 0 | 0 | 4 |
| 3 | 6 | 3 | 2 |
| 6 | 3 | 4 | 2 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 |
| 5 | 3 | 5 | 3 |
| 1 | 1 | 0 | 4 |
| 2 | 5 | 6 | 3 |
| 6 | 6 | 1 | 2 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[0,0,3,0,5,2,3,0,2,0,6,0,6,2,1,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[5,1,2,6,3,1,5,6,5,0,6,1,3,4,3,2] >;
C3×C33⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_3^3\rtimes C_4
% in TeX
G:=Group("C3xC3^3:C4"); // GroupNames label
G:=SmallGroup(324,162);
// by ID
G=gap.SmallGroup(324,162);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,3,-3,36,1443,111,1444,376,7781]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c^-1,c*d=d*c,e*c*e^-1=b^-1*c^-1,e*d*e^-1=d^-1>;
// generators/relations