Aliases: C62.4C32, 3- 1+2.A4, (C2×C6).8He3, C32.4(C3×A4), C3.9(C32⋊A4), C32.A4.2C3, C22⋊2(C3.He3), (C22×3- 1+2).1C3, (C3×C3.A4).2C3, SmallGroup(324,56)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.C32
G = < a,b,c,d | a6=b6=1, c3=d3=b2, ab=ba, cac-1=ab3, dad-1=ab2, cbc-1=a3b4, bd=db, dcd-1=a2b4c >
Character table of C62.C32
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 9J | 9K | 9L | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 36 | 36 | 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 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ10 | 3 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ11 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | 2 | 2 | -1+√-3 | -1-√-3 | complex lifted from C32⋊A4 |
| ρ12 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | complex lifted from C32⋊A4 |
| ρ13 | 3 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | complex lifted from C3×A4 |
| ρ14 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | 2 | 2 | complex lifted from C32⋊A4 |
| ρ15 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | complex lifted from C32⋊A4 |
| ρ16 | 3 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ17 | 3 | -1 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | 2 | 2 | complex lifted from C32⋊A4 |
| ρ18 | 3 | -1 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | 2 | 2 | -1-√-3 | -1+√-3 | complex lifted from C32⋊A4 |
| ρ19 | 3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ20 | 3 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | complex lifted from C3×A4 |
| ρ21 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ95+2ζ92 | 2ζ97+ζ94 | ζ97+2ζ9 | 2ζ94+ζ9 | ζ98+2ζ95 | 2ζ98+ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3.He3 |
| ρ22 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ97+2ζ9 | 2ζ98+ζ92 | ζ98+2ζ95 | ζ95+2ζ92 | 2ζ97+ζ94 | 2ζ94+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3.He3 |
| ρ23 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ98+2ζ95 | 2ζ94+ζ9 | 2ζ97+ζ94 | ζ97+2ζ9 | 2ζ98+ζ92 | ζ95+2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3.He3 |
| ρ24 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 2ζ98+ζ92 | ζ97+2ζ9 | 2ζ94+ζ9 | 2ζ97+ζ94 | ζ95+2ζ92 | ζ98+2ζ95 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3.He3 |
| ρ25 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 2ζ94+ζ9 | ζ98+2ζ95 | ζ95+2ζ92 | 2ζ98+ζ92 | ζ97+2ζ9 | 2ζ97+ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3.He3 |
| ρ26 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 2ζ97+ζ94 | ζ95+2ζ92 | 2ζ98+ζ92 | ζ98+2ζ95 | 2ζ94+ζ9 | ζ97+2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3.He3 |
| ρ27 | 9 | -3 | -9-9√-3/2 | -9+9√-3/2 | 0 | 0 | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 54 points
Generators in S54
(2 14)(3 15)(5 17)(6 18)(8 11)(9 12)(19 48 22 51 25 54)(20 49 23 52 26 46)(21 27 24)(28 31 34)(29 38 35 44 32 41)(30 39 36 45 33 42)(37 40 43)(47 53 50)
(1 10 4 13 7 16)(2 8 5)(3 12 6 15 9 18)(11 17 14)(19 25 22)(20 49 23 52 26 46)(21 50 24 53 27 47)(28 40 31 43 34 37)(29 35 32)(30 42 33 45 36 39)(38 44 41)(48 54 51)
(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)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)
(1 28 27 4 31 21 7 34 24)(2 32 19 5 35 22 8 29 25)(3 36 20 6 30 23 9 33 26)(10 40 47 13 43 50 16 37 53)(11 44 48 14 38 51 17 41 54)(12 39 49 15 42 52 18 45 46)
G:=sub<Sym(54)| (2,14)(3,15)(5,17)(6,18)(8,11)(9,12)(19,48,22,51,25,54)(20,49,23,52,26,46)(21,27,24)(28,31,34)(29,38,35,44,32,41)(30,39,36,45,33,42)(37,40,43)(47,53,50), (1,10,4,13,7,16)(2,8,5)(3,12,6,15,9,18)(11,17,14)(19,25,22)(20,49,23,52,26,46)(21,50,24,53,27,47)(28,40,31,43,34,37)(29,35,32)(30,42,33,45,36,39)(38,44,41)(48,54,51), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,28,27,4,31,21,7,34,24)(2,32,19,5,35,22,8,29,25)(3,36,20,6,30,23,9,33,26)(10,40,47,13,43,50,16,37,53)(11,44,48,14,38,51,17,41,54)(12,39,49,15,42,52,18,45,46)>;
G:=Group( (2,14)(3,15)(5,17)(6,18)(8,11)(9,12)(19,48,22,51,25,54)(20,49,23,52,26,46)(21,27,24)(28,31,34)(29,38,35,44,32,41)(30,39,36,45,33,42)(37,40,43)(47,53,50), (1,10,4,13,7,16)(2,8,5)(3,12,6,15,9,18)(11,17,14)(19,25,22)(20,49,23,52,26,46)(21,50,24,53,27,47)(28,40,31,43,34,37)(29,35,32)(30,42,33,45,36,39)(38,44,41)(48,54,51), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,28,27,4,31,21,7,34,24)(2,32,19,5,35,22,8,29,25)(3,36,20,6,30,23,9,33,26)(10,40,47,13,43,50,16,37,53)(11,44,48,14,38,51,17,41,54)(12,39,49,15,42,52,18,45,46) );
G=PermutationGroup([[(2,14),(3,15),(5,17),(6,18),(8,11),(9,12),(19,48,22,51,25,54),(20,49,23,52,26,46),(21,27,24),(28,31,34),(29,38,35,44,32,41),(30,39,36,45,33,42),(37,40,43),(47,53,50)], [(1,10,4,13,7,16),(2,8,5),(3,12,6,15,9,18),(11,17,14),(19,25,22),(20,49,23,52,26,46),(21,50,24,53,27,47),(28,40,31,43,34,37),(29,35,32),(30,42,33,45,36,39),(38,44,41),(48,54,51)], [(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),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(1,28,27,4,31,21,7,34,24),(2,32,19,5,35,22,8,29,25),(3,36,20,6,30,23,9,33,26),(10,40,47,13,43,50,16,37,53),(11,44,48,14,38,51,17,41,54),(12,39,49,15,42,52,18,45,46)]])
Matrix representation of C62.C32 ►in GL6(𝔽19)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 7 | 0 |
| 0 | 0 | 0 | 10 | 3 | 11 |
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 13 | 13 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 | 0 |
| 0 | 0 | 0 | 8 | 17 | 5 |
| 0 | 0 | 0 | 1 | 0 | 1 |
G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,3,10,0,0,0,0,7,3,0,0,0,0,0,11],[18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,13,0,0,0,0,5,13,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,1,0,0,0,2,17,0,0,0,0,0,5,1] >;
C62.C32 in GAP, Magma, Sage, TeX
C_6^2.C_3^2
% in TeX
G:=Group("C6^2.C3^2"); // GroupNames label
G:=SmallGroup(324,56);
// by ID
G=gap.SmallGroup(324,56);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,162,145,115,224,4864,8753]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^3=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.C32 in TeX
Character table of C62.C32 in TeX