Aliases: C62.2A4, C122.2C3, C42⋊23- 1+2, C42⋊C9⋊2C3, C32.(C42⋊C3), (C4×C12).4C32, C22.(C32.A4), (C2×C6).9(C3×A4), C3.4(C3×C42⋊C3), SmallGroup(432,102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C122.C3
G = < a,b,c | a12=b12=1, c3=b4, ab=ba, cac-1=ab-1, cbc-1=a3b10 >
Smallest permutation representation of C122.C3
►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 9 5)(2 10 6)(3 11 7)(4 12 8)(13 24 18 23 17 22 16 21 15 20 14 19)(25 36 35 34 33 32 31 30 29 28 27 26)
(1 26 14 9 34 18 5 30 16)(2 35 21 10 31 19 6 27 23)(3 32 17 11 28 15 7 36 13)(4 29 24 12 25 22 8 33 20)
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), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,24,18,23,17,22,16,21,15,20,14,19)(25,36,35,34,33,32,31,30,29,28,27,26), (1,26,14,9,34,18,5,30,16)(2,35,21,10,31,19,6,27,23)(3,32,17,11,28,15,7,36,13)(4,29,24,12,25,22,8,33,20)>;
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), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,24,18,23,17,22,16,21,15,20,14,19)(25,36,35,34,33,32,31,30,29,28,27,26), (1,26,14,9,34,18,5,30,16)(2,35,21,10,31,19,6,27,23)(3,32,17,11,28,15,7,36,13)(4,29,24,12,25,22,8,33,20) );
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)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,24,18,23,17,22,16,21,15,20,14,19),(25,36,35,34,33,32,31,30,29,28,27,26)], [(1,26,14,9,34,18,5,30,16),(2,35,21,10,31,19,6,27,23),(3,32,17,11,28,15,7,36,13),(4,29,24,12,25,22,8,33,20)]])
56 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6H | 9A | ··· | 9F | 12A | ··· | 12AF |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 48 | ··· | 48 | 3 | ··· | 3 |
56 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C3 | C3 | A4 | 3- 1+2 | C3×A4 | C42⋊C3 | C32.A4 | C3×C42⋊C3 | C122.C3 |
| kernel | C122.C3 | C42⋊C9 | C122 | C62 | C42 | C2×C6 | C32 | C22 | C3 | C1 |
| # reps | 1 | 6 | 2 | 1 | 2 | 2 | 4 | 6 | 8 | 24 |
Matrix representation of C122.C3 ►in GL3(𝔽13) generated by
| 7 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 1 |
| 10 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 0 | 0 | 9 |
| 9 | 0 | 0 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(13))| [7,0,0,0,2,0,0,0,1],[10,0,0,0,2,0,0,0,2],[0,9,0,0,0,1,9,0,0] >;
C122.C3 in GAP, Magma, Sage, TeX
C_{12}^2.C_3 % in TeX
G:=Group("C12^2.C3"); // GroupNames label
G:=SmallGroup(432,102);
// by ID
G=gap.SmallGroup(432,102);
# by ID
G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,63,169,1515,360,10399,102,9077,15882]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^3=b^4,a*b=b*a,c*a*c^-1=a*b^-1,c*b*c^-1=a^3*b^10>;
// generators/relations
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