metabelian, soluble, monomial, A-group
Aliases: C52⋊C12, C5⋊D5.C6, C5⋊F5⋊C3, C52⋊C3⋊2C4, C52⋊C6.1C2, SmallGroup(300,24)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊C12 |
Generators and relations for C52⋊C12
G = < a,b,c | a5=b5=c12=1, ab=ba, cac-1=a-1b, cbc-1=a3b3 >
Character table of C52⋊C12
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 12A | 12B | 12C | 12D | |
| size | 1 | 25 | 25 | 25 | 25 | 25 | 12 | 12 | 25 | 25 | 25 | 25 | 25 | 25 | |
| ρ1 | 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 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | 1 | -i | i | 1 | 1 | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | i | -i | 1 | 1 | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | -i | i | 1 | 1 | ζ6 | ζ65 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | linear of order 12 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | -i | i | 1 | 1 | ζ65 | ζ6 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | linear of order 12 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | i | -i | 1 | 1 | ζ6 | ζ65 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | linear of order 12 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | i | -i | 1 | 1 | ζ65 | ζ6 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | linear of order 12 |
| ρ13 | 12 | 0 | 0 | 0 | 0 | 0 | 2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ14 | 12 | 0 | 0 | 0 | 0 | 0 | -3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 15 points - transitive group 15T19
Generators in S15
(1 9 6 12 15)(2 10 7 13 4)(3 14 11 5 8)
(1 6 15 9 12)(2 4 13 7 10)
(1 2 3)(4 5 6 7 8 9 10 11 12 13 14 15)
G:=sub<Sym(15)| (1,9,6,12,15)(2,10,7,13,4)(3,14,11,5,8), (1,6,15,9,12)(2,4,13,7,10), (1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15)>;
G:=Group( (1,9,6,12,15)(2,10,7,13,4)(3,14,11,5,8), (1,6,15,9,12)(2,4,13,7,10), (1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15) );
G=PermutationGroup([[(1,9,6,12,15),(2,10,7,13,4),(3,14,11,5,8)], [(1,6,15,9,12),(2,4,13,7,10)], [(1,2,3),(4,5,6,7,8,9,10,11,12,13,14,15)]])
G:=TransitiveGroup(15,19);
►On 25 points: primitive - transitive group 25T26
Generators in S25
(1 11 8 2 5)(3 13 20 25 24)(4 16 6 23 15)(7 9 18 19 14)(10 21 17 12 22)
(1 21 18 24 15)(2 22 7 20 6)(3 4 11 17 19)(5 10 9 25 23)(8 12 14 13 16)
(2 3 4 5 6 7 8 9 10 11 12 13)(14 15 16 17 18 19 20 21 22 23 24 25)
G:=sub<Sym(25)| (1,11,8,2,5)(3,13,20,25,24)(4,16,6,23,15)(7,9,18,19,14)(10,21,17,12,22), (1,21,18,24,15)(2,22,7,20,6)(3,4,11,17,19)(5,10,9,25,23)(8,12,14,13,16), (2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25)>;
G:=Group( (1,11,8,2,5)(3,13,20,25,24)(4,16,6,23,15)(7,9,18,19,14)(10,21,17,12,22), (1,21,18,24,15)(2,22,7,20,6)(3,4,11,17,19)(5,10,9,25,23)(8,12,14,13,16), (2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25) );
G=PermutationGroup([[(1,11,8,2,5),(3,13,20,25,24),(4,16,6,23,15),(7,9,18,19,14),(10,21,17,12,22)], [(1,21,18,24,15),(2,22,7,20,6),(3,4,11,17,19),(5,10,9,25,23),(8,12,14,13,16)], [(2,3,4,5,6,7,8,9,10,11,12,13),(14,15,16,17,18,19,20,21,22,23,24,25)]])
G:=TransitiveGroup(25,26);
►On 30 points - transitive group 30T78
Generators in S30
(2 17 26 20 11)(3 27 12 18 21)(5 29 14 8 23)(6 15 24 30 9)
(1 16 25 19 10)(2 20 17 11 26)(3 27 12 18 21)(4 28 13 7 22)(5 8 29 23 14)(6 15 24 30 9)
(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)
G:=sub<Sym(30)| (2,17,26,20,11)(3,27,12,18,21)(5,29,14,8,23)(6,15,24,30,9), (1,16,25,19,10)(2,20,17,11,26)(3,27,12,18,21)(4,28,13,7,22)(5,8,29,23,14)(6,15,24,30,9), (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)>;
G:=Group( (2,17,26,20,11)(3,27,12,18,21)(5,29,14,8,23)(6,15,24,30,9), (1,16,25,19,10)(2,20,17,11,26)(3,27,12,18,21)(4,28,13,7,22)(5,8,29,23,14)(6,15,24,30,9), (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) );
G=PermutationGroup([[(2,17,26,20,11),(3,27,12,18,21),(5,29,14,8,23),(6,15,24,30,9)], [(1,16,25,19,10),(2,20,17,11,26),(3,27,12,18,21),(4,28,13,7,22),(5,8,29,23,14),(6,15,24,30,9)], [(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)]])
G:=TransitiveGroup(30,78);
Polynomial with Galois group C52⋊C12 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 15T19 | x15+x10-2x5-1 | 515·710 |
Matrix representation of C52⋊C12 ►in GL12(ℤ)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(12,Integers())| [0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0] >;
C52⋊C12 in GAP, Magma, Sage, TeX
C_5^2\rtimes C_{12} % in TeX
G:=Group("C5^2:C12"); // GroupNames label
G:=SmallGroup(300,24);
// by ID
G=gap.SmallGroup(300,24);
# by ID
G:=PCGroup([5,-2,-3,-2,-5,5,30,483,1928,173,3004,2859,1014]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^3*b^3>;
// generators/relations
Export
Subgroup lattice of C52⋊C12 in TeX
Character table of C52⋊C12 in TeX