Aliases: C32⋊A4, C22⋊He3, C62⋊1C3, (C3×A4)⋊C3, C3.5(C3×A4), (C2×C6).5C32, SmallGroup(108,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊A4
G = < a,b,c,d,e | a3=b3=c2=d2=e3=1, 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 | 3E | 3F | 3G | 3H | 3I | 3J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 12 | 12 | 12 | 12 | 12 | 12 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| 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 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | linear of order 3 |
| ρ10 | 3 | -1 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ11 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-3 | ζ6 | -1-√-3 | 2 | -1+√-3 | ζ65 | -1+√-3 | 2 | complex faithful |
| ρ12 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | ζ6 | -1+√-3 | -1-√-3 | 2 | ζ65 | -1-√-3 | -1+√-3 | complex faithful |
| ρ13 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3+3√-3/2 | 0 | 0 | 0 | -3-3√-3/2 | 0 | 0 | complex lifted from He3 |
| ρ14 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | ζ65 | -1-√-3 | -1+√-3 | 2 | ζ6 | -1+√-3 | -1-√-3 | complex faithful |
| ρ15 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-3 | ζ65 | -1+√-3 | 2 | -1-√-3 | ζ6 | -1-√-3 | 2 | complex faithful |
| ρ16 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | -1 | ζ65 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | complex lifted from C3×A4 |
| ρ17 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3-3√-3/2 | 0 | 0 | 0 | -3+3√-3/2 | 0 | 0 | complex lifted from He3 |
| ρ18 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-3 | ζ6 | 2 | -1+√-3 | -1-√-3 | ζ65 | 2 | -1-√-3 | complex faithful |
| ρ19 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-3 | ζ65 | 2 | -1-√-3 | -1+√-3 | ζ6 | 2 | -1+√-3 | complex faithful |
| ρ20 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | -1 | ζ6 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | complex lifted from C3×A4 |
►On 18 points - transitive group 18T48
(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 3)(2 6 4)(7 9 8)(10 11 12)(13 14 15)(16 18 17)
(1 2)(3 4)(5 6)(10 14)(11 15)(12 13)
(1 2)(3 4)(5 6)(7 17)(8 18)(9 16)
(1 9 14)(2 16 10)(3 7 13)(4 17 12)(5 8 15)(6 18 11)
G:=sub<Sym(18)| (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,3)(2,6,4)(7,9,8)(10,11,12)(13,14,15)(16,18,17), (1,2)(3,4)(5,6)(10,14)(11,15)(12,13), (1,2)(3,4)(5,6)(7,17)(8,18)(9,16), (1,9,14)(2,16,10)(3,7,13)(4,17,12)(5,8,15)(6,18,11)>;
G:=Group( (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,3)(2,6,4)(7,9,8)(10,11,12)(13,14,15)(16,18,17), (1,2)(3,4)(5,6)(10,14)(11,15)(12,13), (1,2)(3,4)(5,6)(7,17)(8,18)(9,16), (1,9,14)(2,16,10)(3,7,13)(4,17,12)(5,8,15)(6,18,11) );
G=PermutationGroup([[(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,5,3),(2,6,4),(7,9,8),(10,11,12),(13,14,15),(16,18,17)], [(1,2),(3,4),(5,6),(10,14),(11,15),(12,13)], [(1,2),(3,4),(5,6),(7,17),(8,18),(9,16)], [(1,9,14),(2,16,10),(3,7,13),(4,17,12),(5,8,15),(6,18,11)]])
G:=TransitiveGroup(18,48);
C32⋊A4 is a maximal subgroup of
C62⋊S3 C32⋊S4 C62⋊C6 C62.13C32 C62.14C32 He3⋊A4 3- 1+2⋊A4 C62.6C32 C33⋊2A4 C62.25C32 A4×He3 C62.9C32 C42⋊He3 C24⋊He3 C62⋊A4
C32⋊A4 is a maximal quotient of
Q8⋊He3 C62.13C32 C62.14C32 C62.15C32 C62.16C32 He3.A4 He3⋊A4 He3⋊2A4 C62.C32 3- 1+2⋊A4 C62.6C32 C62⋊C9 C33⋊2A4 C42⋊He3 C24⋊He3 C62⋊A4
Matrix representation of C32⋊A4 ►in GL3(𝔽7) generated by
| 1 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 4 |
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 6 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 6 |
| 1 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(7))| [1,0,0,0,2,0,0,0,4],[2,0,0,0,2,0,0,0,2],[6,0,0,0,1,0,0,0,6],[1,0,0,0,6,0,0,0,6],[0,0,1,1,0,0,0,1,0] >;
C32⋊A4 in GAP, Magma, Sage, TeX
C_3^2\rtimes A_4
% in TeX
G:=Group("C3^2:A4"); // GroupNames label
G:=SmallGroup(108,22);
// by ID
G=gap.SmallGroup(108,22);
# by ID
G:=PCGroup([5,-3,-3,-3,-2,2,121,1083,2029]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^2=e^3=1,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