direct product, abelian, monomial
Aliases: C62, SmallGroup(36,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C62 |
| C1 — C62 |
| C1 — C62 |
Generators and relations for C62
G = < a,b | a6=b6=1, ab=ba >
Smallest permutation representation of C62
►Regular action on 36 points
Generators in S36
(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)
(1 32 30 13 8 23)(2 33 25 14 9 24)(3 34 26 15 10 19)(4 35 27 16 11 20)(5 36 28 17 12 21)(6 31 29 18 7 22)
G:=sub<Sym(36)| (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), (1,32,30,13,8,23)(2,33,25,14,9,24)(3,34,26,15,10,19)(4,35,27,16,11,20)(5,36,28,17,12,21)(6,31,29,18,7,22)>;
G:=Group( (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), (1,32,30,13,8,23)(2,33,25,14,9,24)(3,34,26,15,10,19)(4,35,27,16,11,20)(5,36,28,17,12,21)(6,31,29,18,7,22) );
G=PermutationGroup([(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)], [(1,32,30,13,8,23),(2,33,25,14,9,24),(3,34,26,15,10,19),(4,35,27,16,11,20),(5,36,28,17,12,21),(6,31,29,18,7,22)])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 6A | ··· | 6X |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C3 | C6 |
| kernel | C62 | C3×C6 | C2×C6 | C6 |
| # reps | 1 | 3 | 8 | 24 |
Matrix representation of C62 ►in GL2(𝔽7) generated by
| 5 | 0 |
| 0 | 1 |
| 6 | 0 |
| 0 | 3 |
G:=sub<GL(2,GF(7))| [5,0,0,1],[6,0,0,3] >;
C62 in GAP, Magma, Sage, TeX
C_6^2
% in TeX
G:=Group("C6^2"); // GroupNames label
G:=SmallGroup(36,14);
// by ID
G=gap.SmallGroup(36,14);
# by ID
G:=PCGroup([4,-2,-2,-3,-3]);
// Polycyclic
G:=Group<a,b|a^6=b^6=1,a*b=b*a>;
// generators/relations