direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C32⋊C4, C33⋊1C4, C32⋊2C12, C3⋊S3.C6, (C3×C3⋊S3).1C2, SmallGroup(108,36)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3×C32⋊C4 |
Generators and relations for C3×C32⋊C4
G = < a,b,c,d | a3=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Character table of C3×C32⋊C4
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | |
| size | 1 | 9 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ3 | -1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ32 | -1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ32 | -i | i | ζ65 | ζ6 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | linear of order 12 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ3 | -i | i | ζ6 | ζ65 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | linear of order 12 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ32 | i | -i | ζ65 | ζ6 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | linear of order 12 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ3 | i | -i | ζ6 | ζ65 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | linear of order 12 |
| ρ13 | 4 | 0 | 4 | 4 | -2 | -2 | 1 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ14 | 4 | 0 | 4 | 4 | 1 | 1 | -2 | 1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ15 | 4 | 0 | -2-2√-3 | -2+2√-3 | 1-√-3 | 1+√-3 | ζ32 | -2 | 1 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 4 | 0 | -2-2√-3 | -2+2√-3 | ζ3 | ζ32 | 1+√-3 | 1 | -2 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ17 | 4 | 0 | -2+2√-3 | -2-2√-3 | ζ32 | ζ3 | 1-√-3 | 1 | -2 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ18 | 4 | 0 | -2+2√-3 | -2-2√-3 | 1+√-3 | 1-√-3 | ζ3 | -2 | 1 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 12 points - transitive group 12T73
Generators in S12
(1 8 9)(2 5 10)(3 6 11)(4 7 12)
(2 5 10)(4 12 7)
(1 8 9)(2 5 10)(3 11 6)(4 12 7)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
G:=sub<Sym(12)| (1,8,9)(2,5,10)(3,6,11)(4,7,12), (2,5,10)(4,12,7), (1,8,9)(2,5,10)(3,11,6)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12)>;
G:=Group( (1,8,9)(2,5,10)(3,6,11)(4,7,12), (2,5,10)(4,12,7), (1,8,9)(2,5,10)(3,11,6)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12) );
G=PermutationGroup([[(1,8,9),(2,5,10),(3,6,11),(4,7,12)], [(2,5,10),(4,12,7)], [(1,8,9),(2,5,10),(3,11,6),(4,12,7)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)]])
G:=TransitiveGroup(12,73);
►On 18 points - transitive group 18T44
Generators in S18
(1 5 4)(2 6 3)(7 17 12)(8 18 13)(9 15 14)(10 16 11)
(1 17 15)(2 16 18)(3 10 8)(4 7 9)(5 12 14)(6 11 13)
(2 18 16)(3 8 10)(6 13 11)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
G:=sub<Sym(18)| (1,5,4)(2,6,3)(7,17,12)(8,18,13)(9,15,14)(10,16,11), (1,17,15)(2,16,18)(3,10,8)(4,7,9)(5,12,14)(6,11,13), (2,18,16)(3,8,10)(6,13,11), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18)>;
G:=Group( (1,5,4)(2,6,3)(7,17,12)(8,18,13)(9,15,14)(10,16,11), (1,17,15)(2,16,18)(3,10,8)(4,7,9)(5,12,14)(6,11,13), (2,18,16)(3,8,10)(6,13,11), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18) );
G=PermutationGroup([[(1,5,4),(2,6,3),(7,17,12),(8,18,13),(9,15,14),(10,16,11)], [(1,17,15),(2,16,18),(3,10,8),(4,7,9),(5,12,14),(6,11,13)], [(2,18,16),(3,8,10),(6,13,11)], [(1,2),(3,4),(5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)]])
G:=TransitiveGroup(18,44);
►On 27 points - transitive group 27T33
Generators in S27
(1 2 3)(4 25 14)(5 26 15)(6 27 12)(7 24 13)(8 21 18)(9 22 19)(10 23 16)(11 20 17)
(1 4 6)(2 25 27)(3 14 12)(5 11 8)(7 10 9)(13 16 19)(15 17 18)(20 21 26)(22 24 23)
(1 10 8)(2 23 21)(3 16 18)(4 9 5)(6 7 11)(12 13 17)(14 19 15)(20 27 24)(22 26 25)
(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)
G:=sub<Sym(27)| (1,2,3)(4,25,14)(5,26,15)(6,27,12)(7,24,13)(8,21,18)(9,22,19)(10,23,16)(11,20,17), (1,4,6)(2,25,27)(3,14,12)(5,11,8)(7,10,9)(13,16,19)(15,17,18)(20,21,26)(22,24,23), (1,10,8)(2,23,21)(3,16,18)(4,9,5)(6,7,11)(12,13,17)(14,19,15)(20,27,24)(22,26,25), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27)>;
G:=Group( (1,2,3)(4,25,14)(5,26,15)(6,27,12)(7,24,13)(8,21,18)(9,22,19)(10,23,16)(11,20,17), (1,4,6)(2,25,27)(3,14,12)(5,11,8)(7,10,9)(13,16,19)(15,17,18)(20,21,26)(22,24,23), (1,10,8)(2,23,21)(3,16,18)(4,9,5)(6,7,11)(12,13,17)(14,19,15)(20,27,24)(22,26,25), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27) );
G=PermutationGroup([[(1,2,3),(4,25,14),(5,26,15),(6,27,12),(7,24,13),(8,21,18),(9,22,19),(10,23,16),(11,20,17)], [(1,4,6),(2,25,27),(3,14,12),(5,11,8),(7,10,9),(13,16,19),(15,17,18),(20,21,26),(22,24,23)], [(1,10,8),(2,23,21),(3,16,18),(4,9,5),(6,7,11),(12,13,17),(14,19,15),(20,27,24),(22,26,25)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)]])
G:=TransitiveGroup(27,33);
C3×C32⋊C4 is a maximal subgroup of
C3⋊F9 C32⋊2D12 C33⋊Q8
C3×C32⋊C4 is a maximal quotient of He3.3C12
| action | f(x) | Disc(f) |
|---|---|---|
| 12T73 | x12-4x11+38x10-163x9+1073x8-3580x7+16135x6-41958x5+122109x4-219057x3+358551x2-335799x+164511 | 232·332·139·532·614·5032·329392·334976032 |
Matrix representation of C3×C32⋊C4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 2 | 5 | 0 |
| 2 | 3 | 5 | 2 |
| 2 | 2 | 6 | 2 |
| 2 | 4 | 3 | 2 |
| 6 | 0 | 3 | 1 |
| 5 | 3 | 6 | 0 |
| 1 | 6 | 4 | 5 |
| 3 | 3 | 2 | 6 |
| 4 | 5 | 1 | 4 |
| 3 | 1 | 2 | 1 |
| 3 | 2 | 1 | 4 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[4,2,2,2,2,3,2,4,5,5,6,3,0,2,2,2],[6,5,1,3,0,3,6,3,3,6,4,2,1,0,5,6],[4,3,3,0,5,1,2,0,1,2,1,0,4,1,4,1] >;
C3×C32⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes C_4
% in TeX
G:=Group("C3xC3^2:C4"); // GroupNames label
G:=SmallGroup(108,36);
// by ID
G=gap.SmallGroup(108,36);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,3,30,1683,93,2404,314]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations
Export
Subgroup lattice of C3×C32⋊C4 in TeX
Character table of C3×C32⋊C4 in TeX