metabelian, supersoluble, monomial
Aliases: C32⋊C12, He3⋊2C4, C32⋊1Dic3, (C3×C6).C6, C3⋊Dic3⋊C3, C6.2(C3×S3), (C3×C6).1S3, C2.(C32⋊C6), (C2×He3).1C2, C3.2(C3×Dic3), SmallGroup(108,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C32⋊C12 |
Generators and relations for C32⋊C12
G = < a,b,c | a3=b3=c12=1, ab=ba, cac-1=a-1b, cbc-1=b-1 >
Character table of C32⋊C12
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 6 | 9 | 9 | 2 | 3 | 3 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | |
| ρ1 | 1 | 1 | 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 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ9 | 1 | -1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | i | -i | -1 | ζ6 | ζ65 | -1 | ζ65 | ζ6 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | linear of order 12 |
| ρ10 | 1 | -1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | i | -i | -1 | ζ65 | ζ6 | -1 | ζ6 | ζ65 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | linear of order 12 |
| ρ11 | 1 | -1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | -i | i | -1 | ζ65 | ζ6 | -1 | ζ6 | ζ65 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | linear of order 12 |
| ρ12 | 1 | -1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | -i | i | -1 | ζ6 | ζ65 | -1 | ζ65 | ζ6 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | linear of order 12 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | 2 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 0 | 0 | -2 | 1-√-3 | 1+√-3 | 1 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | complex lifted from C3×Dic3 |
| ρ16 | 2 | -2 | 2 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 0 | 0 | -2 | 1+√-3 | 1-√-3 | 1 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | complex lifted from C3×Dic3 |
| ρ17 | 2 | 2 | 2 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 2 | -1+√-3 | -1-√-3 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ18 | 2 | 2 | 2 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 2 | -1-√-3 | -1+√-3 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ19 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C6 |
| ρ20 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 36 points
(1 22 5)(2 19 10)(3 32 28)(4 8 13)(6 31 35)(7 16 11)(9 26 34)(12 25 29)(14 30 18)(15 27 23)(17 21 33)(20 36 24)
(1 14 34)(2 35 15)(3 16 36)(4 25 17)(5 18 26)(6 27 19)(7 20 28)(8 29 21)(9 22 30)(10 31 23)(11 24 32)(12 33 13)
(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)
G:=sub<Sym(36)| (1,22,5)(2,19,10)(3,32,28)(4,8,13)(6,31,35)(7,16,11)(9,26,34)(12,25,29)(14,30,18)(15,27,23)(17,21,33)(20,36,24), (1,14,34)(2,35,15)(3,16,36)(4,25,17)(5,18,26)(6,27,19)(7,20,28)(8,29,21)(9,22,30)(10,31,23)(11,24,32)(12,33,13), (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)>;
G:=Group( (1,22,5)(2,19,10)(3,32,28)(4,8,13)(6,31,35)(7,16,11)(9,26,34)(12,25,29)(14,30,18)(15,27,23)(17,21,33)(20,36,24), (1,14,34)(2,35,15)(3,16,36)(4,25,17)(5,18,26)(6,27,19)(7,20,28)(8,29,21)(9,22,30)(10,31,23)(11,24,32)(12,33,13), (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) );
G=PermutationGroup([[(1,22,5),(2,19,10),(3,32,28),(4,8,13),(6,31,35),(7,16,11),(9,26,34),(12,25,29),(14,30,18),(15,27,23),(17,21,33),(20,36,24)], [(1,14,34),(2,35,15),(3,16,36),(4,25,17),(5,18,26),(6,27,19),(7,20,28),(8,29,21),(9,22,30),(10,31,23),(11,24,32),(12,33,13)], [(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)]])
C32⋊C12 is a maximal subgroup of
He3⋊2Q8 C6.S32 He3⋊(C2×C4) He3⋊3D4 He3⋊3Q8 C4×C32⋊C6 He3⋊6D4 C33⋊C12 He3.Dic3 He3.2Dic3 C33⋊Dic3 He3.3Dic3 He3⋊Dic3 C33⋊4C12 He3.4Dic3 C32⋊CSU2(𝔽3) C62⋊5Dic3 C6.(S3×A4) C62⋊4C12
C32⋊C12 is a maximal quotient of
He3⋊3C8 C32⋊C36 C32⋊Dic9 He3⋊C12 C33⋊C12 He3.C12 He3.Dic3 He3.2C12 He3.2Dic3 C33⋊4C12 C62⋊5Dic3 C62⋊4C12
Matrix representation of C32⋊C12 ►in GL6(𝔽13)
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 7 | 12 | 7 | 7 | 7 |
| 6 | 1 | 6 | 1 | 1 | 6 |
| 7 | 12 | 12 | 7 | 7 | 12 |
| 6 | 6 | 6 | 1 | 6 | 6 |
| 7 | 12 | 7 | 7 | 7 | 7 |
| 6 | 6 | 1 | 6 | 1 | 6 |
G:=sub<GL(6,GF(13))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,6,7,6,7,6,7,1,12,6,12,6,12,6,12,6,7,1,7,1,7,1,7,6,7,1,7,6,7,1,7,6,12,6,7,6] >;
C32⋊C12 in GAP, Magma, Sage, TeX
C_3^2\rtimes C_{12} % in TeX
G:=Group("C3^2:C12"); // GroupNames label
G:=SmallGroup(108,8);
// by ID
G=gap.SmallGroup(108,8);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-3,30,483,488,1804]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C32⋊C12 in TeX
Character table of C32⋊C12 in TeX