Aliases: C32.A4, C62.2C3, C22⋊23- 1+2, C3.A4⋊2C3, C3.4(C3×A4), (C2×C6).4C32, SmallGroup(108,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32.A4
G = < a,b,c,d,e | a3=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, eae-1=ab-1, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
Character table of C32.A4
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 9A | 9B | 9C | 9D | 9E | 9F | |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 12 | 12 | 12 | 12 | 12 | 12 | |
| 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 | ζ32 | ζ3 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | linear of order 3 |
| ρ10 | 3 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ11 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -3-3√-3/2 | 0 | 0 | 0 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from 3- 1+2 |
| ρ12 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ6 | 2 | -1+√-3 | -1-√-3 | ζ65 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ13 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ6 | -1+√-3 | -1-√-3 | 2 | ζ65 | -1-√-3 | -1+√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ14 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | ζ65 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×A4 |
| ρ15 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ6 | -1-√-3 | 2 | -1+√-3 | ζ65 | -1+√-3 | 2 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ65 | 2 | -1-√-3 | -1+√-3 | ζ6 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ17 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -3+3√-3/2 | 0 | 0 | 0 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from 3- 1+2 |
| ρ18 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ65 | -1+√-3 | 2 | -1-√-3 | ζ6 | -1-√-3 | 2 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ19 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ65 | -1-√-3 | -1+√-3 | 2 | ζ6 | -1+√-3 | -1-√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | ζ6 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×A4 |
►On 18 points - transitive group 18T47
(2 8 5)(3 6 9)(11 17 14)(12 15 18)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 16)(3 18)(4 10)(6 12)(7 13)(9 15)
(1 16)(2 17)(4 10)(5 11)(7 13)(8 14)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
G:=sub<Sym(18)| (2,8,5)(3,6,9)(11,17,14)(12,15,18), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,16)(3,18)(4,10)(6,12)(7,13)(9,15), (1,16)(2,17)(4,10)(5,11)(7,13)(8,14), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)>;
G:=Group( (2,8,5)(3,6,9)(11,17,14)(12,15,18), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,16)(3,18)(4,10)(6,12)(7,13)(9,15), (1,16)(2,17)(4,10)(5,11)(7,13)(8,14), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18) );
G=PermutationGroup([[(2,8,5),(3,6,9),(11,17,14),(12,15,18)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18)], [(1,16),(3,18),(4,10),(6,12),(7,13),(9,15)], [(1,16),(2,17),(4,10),(5,11),(7,13),(8,14)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)]])
G:=TransitiveGroup(18,47);
C32.A4 is a maximal subgroup of
C32.S4 C62.13C32 C62.15C32 He3.A4 He3⋊2A4 C62.C32 3- 1+2⋊A4 C62.6C32 C33⋊2A4 C62.25C32 He3.2A4 A4×3- 1+2 C62.9C32 C122.C3 C24⋊3- 1+2 C62.A4
C32.A4 is a maximal quotient of
Q8⋊3- 1+2 C62.11C32 C62⋊C9 C122.C3 C24⋊3- 1+2 C62.A4
Matrix representation of C32.A4 ►in GL3(𝔽7) generated by
| 4 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 1 |
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 6 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 6 |
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 1 |
| 0 | 0 | 4 |
| 4 | 0 | 0 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(7))| [4,0,0,0,2,0,0,0,1],[2,0,0,0,2,0,0,0,2],[6,0,0,0,1,0,0,0,6],[6,0,0,0,6,0,0,0,1],[0,4,0,0,0,1,4,0,0] >;
C32.A4 in GAP, Magma, Sage, TeX
C_3^2.A_4
% in TeX
G:=Group("C3^2.A4"); // GroupNames label
G:=SmallGroup(108,21);
// by ID
G=gap.SmallGroup(108,21);
# by ID
G:=PCGroup([5,-3,-3,-3,-2,2,45,121,1083,2029]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^-1,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
Export
Subgroup lattice of C32.A4 in TeX
Character table of C32.A4 in TeX