Aliases: C62.6C32, 3- 1+2⋊2A4, C22⋊2C3≀C3, C32⋊A4⋊3C3, (C32×A4)⋊2C3, C32.A4⋊5C3, C32.6(C3×A4), (C2×C6).10He3, C3.11(C32⋊A4), (C22×3- 1+2)⋊2C3, SmallGroup(324,58)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.6C32
G = < a,b,c,d | a6=b6=c3=1, d3=b2, ab=ba, cac-1=ab3, dad-1=ab2, cbc-1=a3b4, bd=db, dcd-1=a2b4c >
3C2
3C3
12C3
12C3
12C3
36C3
3C6
9C6
3C9
12C32
12C32
12C32
12C9
12C32
12C32
3A4
3A4
3A4
9A4
3C18
3C18
3C18
4He3
4C33
Character table of C62.6C32
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 3L | 6A | 6B | 6C | 6D | 9A | 9B | 9C | 9D | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 12 | 12 | 12 | 12 | 12 | 12 | 36 | 36 | 3 | 3 | 9 | 9 | 9 | 9 | 36 | 36 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ10 | 3 | -1 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ11 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 3+√-3/2 | 3-√-3/2 | √-3 | -3+√-3/2 | -√-3 | -3-√-3/2 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3≀C3 |
| ρ12 | 3 | -1 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | complex lifted from C3×A4 |
| ρ13 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 2 | -1+√-3 | -1+√-3 | -1-√-3 | -1-√-3 | 2 | complex lifted from C32⋊A4 |
| ρ14 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -3+√-3/2 | -3-√-3/2 | 3-√-3/2 | -√-3 | 3+√-3/2 | √-3 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3≀C3 |
| ρ15 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | -1+√-3 | 2 | -1+√-3 | 2 | -1-√-3 | -1-√-3 | complex lifted from C32⋊A4 |
| ρ16 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | 2 | -1-√-3 | 2 | -1+√-3 | complex lifted from C32⋊A4 |
| ρ17 | 3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ18 | 3 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ19 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | -1-√-3 | 2 | -1-√-3 | 2 | -1+√-3 | -1+√-3 | complex lifted from C32⋊A4 |
| ρ20 | 3 | -1 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | complex lifted from C3×A4 |
| ρ21 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | √-3 | -√-3 | -3+√-3/2 | 3-√-3/2 | -3-√-3/2 | 3+√-3/2 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3≀C3 |
| ρ22 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 3-√-3/2 | 3+√-3/2 | -√-3 | -3-√-3/2 | √-3 | -3+√-3/2 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3≀C3 |
| ρ23 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -3-√-3/2 | -3+√-3/2 | 3+√-3/2 | √-3 | 3-√-3/2 | -√-3 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3≀C3 |
| ρ24 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 2 | -1-√-3 | -1-√-3 | -1+√-3 | -1+√-3 | 2 | complex lifted from C32⋊A4 |
| ρ25 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -√-3 | √-3 | -3-√-3/2 | 3+√-3/2 | -3+√-3/2 | 3-√-3/2 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3≀C3 |
| ρ26 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | 2 | -1+√-3 | 2 | -1-√-3 | complex lifted from C32⋊A4 |
| ρ27 | 9 | -3 | -9+9√-3/2 | -9-9√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ28 | 9 | -3 | -9-9√-3/2 | -9+9√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 36 points
Generators in S36
(1 23)(2 27 8 24 5 21)(3 22 6 25 9 19)(4 26)(7 20)(10 33 16 30 13 36)(11 28 14 31 17 34)(12 32)(15 35)(18 29)
(1 18 4 12 7 15)(2 10 5 13 8 16)(3 11 6 14 9 17)(19 34 22 28 25 31)(20 35 23 29 26 32)(21 36 24 30 27 33)
(2 8 5)(10 36 24)(11 31 22)(12 32 23)(13 30 27)(14 34 25)(15 35 26)(16 33 21)(17 28 19)(18 29 20)
(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)| (1,23)(2,27,8,24,5,21)(3,22,6,25,9,19)(4,26)(7,20)(10,33,16,30,13,36)(11,28,14,31,17,34)(12,32)(15,35)(18,29), (1,18,4,12,7,15)(2,10,5,13,8,16)(3,11,6,14,9,17)(19,34,22,28,25,31)(20,35,23,29,26,32)(21,36,24,30,27,33), (2,8,5)(10,36,24)(11,31,22)(12,32,23)(13,30,27)(14,34,25)(15,35,26)(16,33,21)(17,28,19)(18,29,20), (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( (1,23)(2,27,8,24,5,21)(3,22,6,25,9,19)(4,26)(7,20)(10,33,16,30,13,36)(11,28,14,31,17,34)(12,32)(15,35)(18,29), (1,18,4,12,7,15)(2,10,5,13,8,16)(3,11,6,14,9,17)(19,34,22,28,25,31)(20,35,23,29,26,32)(21,36,24,30,27,33), (2,8,5)(10,36,24)(11,31,22)(12,32,23)(13,30,27)(14,34,25)(15,35,26)(16,33,21)(17,28,19)(18,29,20), (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([[(1,23),(2,27,8,24,5,21),(3,22,6,25,9,19),(4,26),(7,20),(10,33,16,30,13,36),(11,28,14,31,17,34),(12,32),(15,35),(18,29)], [(1,18,4,12,7,15),(2,10,5,13,8,16),(3,11,6,14,9,17),(19,34,22,28,25,31),(20,35,23,29,26,32),(21,36,24,30,27,33)], [(2,8,5),(10,36,24),(11,31,22),(12,32,23),(13,30,27),(14,34,25),(15,35,26),(16,33,21),(17,28,19),(18,29,20)], [(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)]])
Matrix representation of C62.6C32 ►in GL6(𝔽19)
| 0 | 18 | 1 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 7 | 0 |
| 0 | 0 | 0 | 1 | 0 | 11 |
| 0 | 1 | 18 | 0 | 0 | 0 |
| 1 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 18 | 1 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 | 0 | 7 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 10 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 |
| 0 | 0 | 0 | 1 | 18 | 0 |
G:=sub<GL(6,GF(19))| [0,0,1,0,0,0,18,18,18,0,0,0,1,0,0,0,0,0,0,0,0,1,12,1,0,0,0,0,7,0,0,0,0,0,0,11],[0,1,0,0,0,0,1,0,0,0,0,0,18,18,18,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,18,18,18,0,0,0,0,1,0,0,0,0,0,0,0,1,0,12,0,0,0,0,1,0,0,0,0,0,0,7],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,1,0,0,0,10,18,18,0,0,0,0,1,0] >;
C62.6C32 in GAP, Magma, Sage, TeX
C_6^2._6C_3^2
% in TeX
G:=Group("C6^2.6C3^2"); // GroupNames label
G:=SmallGroup(324,58);
// by ID
G=gap.SmallGroup(324,58);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,145,115,224,4864,8753]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^3=1,d^3=b^2,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a*b^2,c*b*c^-1=a^3*b^4,b*d=d*b,d*c*d^-1=a^2*b^4*c>;
// generators/relations
Export
Subgroup lattice of C62.6C32 in TeX
Character table of C62.6C32 in TeX