metabelian, soluble, monomial, A-group
Aliases: C52⋊C6, C5⋊D5⋊C3, C52⋊C3⋊2C2, SmallGroup(150,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊C6 |
Generators and relations for C52⋊C6
G = < a,b,c | a5=b5=c6=1, ab=ba, cac-1=a2b3, cbc-1=a-1b-1 >
Character table of C52⋊C6
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 6A | 6B | |
| size | 1 | 25 | 25 | 25 | 6 | 6 | 6 | 6 | 25 | 25 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | linear of order 6 |
| ρ5 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 6 | 0 | 0 | 0 | -3-√5/2 | 1-√5 | 1+√5 | -3+√5/2 | 0 | 0 | orthogonal faithful |
| ρ8 | 6 | 0 | 0 | 0 | 1+√5 | -3-√5/2 | -3+√5/2 | 1-√5 | 0 | 0 | orthogonal faithful |
| ρ9 | 6 | 0 | 0 | 0 | -3+√5/2 | 1+√5 | 1-√5 | -3-√5/2 | 0 | 0 | orthogonal faithful |
| ρ10 | 6 | 0 | 0 | 0 | 1-√5 | -3+√5/2 | -3-√5/2 | 1+√5 | 0 | 0 | orthogonal faithful |
►On 15 points - transitive group 15T12
Generators in S15
(1 4 11 14 7)(2 12 8 5 15)(3 6 13 10 9)
(2 8 15 12 5)(3 9 10 13 6)
(1 2 3)(4 5 6 7 8 9)(10 11 12 13 14 15)
G:=sub<Sym(15)| (1,4,11,14,7)(2,12,8,5,15)(3,6,13,10,9), (2,8,15,12,5)(3,9,10,13,6), (1,2,3)(4,5,6,7,8,9)(10,11,12,13,14,15)>;
G:=Group( (1,4,11,14,7)(2,12,8,5,15)(3,6,13,10,9), (2,8,15,12,5)(3,9,10,13,6), (1,2,3)(4,5,6,7,8,9)(10,11,12,13,14,15) );
G=PermutationGroup([[(1,4,11,14,7),(2,12,8,5,15),(3,6,13,10,9)], [(2,8,15,12,5),(3,9,10,13,6)], [(1,2,3),(4,5,6,7,8,9),(10,11,12,13,14,15)]])
G:=TransitiveGroup(15,12);
►On 25 points: primitive - transitive group 25T15
Generators in S25
(1 15 5 2 18)(3 10 4 23 25)(6 22 20 7 13)(8 17 16 12 24)(9 19 14 11 21)
(1 23 12 9 20)(2 10 17 11 6)(3 8 14 13 5)(4 16 21 22 18)(7 15 25 24 19)
(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)| (1,15,5,2,18)(3,10,4,23,25)(6,22,20,7,13)(8,17,16,12,24)(9,19,14,11,21), (1,23,12,9,20)(2,10,17,11,6)(3,8,14,13,5)(4,16,21,22,18)(7,15,25,24,19), (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( (1,15,5,2,18)(3,10,4,23,25)(6,22,20,7,13)(8,17,16,12,24)(9,19,14,11,21), (1,23,12,9,20)(2,10,17,11,6)(3,8,14,13,5)(4,16,21,22,18)(7,15,25,24,19), (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([[(1,15,5,2,18),(3,10,4,23,25),(6,22,20,7,13),(8,17,16,12,24),(9,19,14,11,21)], [(1,23,12,9,20),(2,10,17,11,6),(3,8,14,13,5),(4,16,21,22,18),(7,15,25,24,19)], [(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(25,15);
►On 30 points - transitive group 30T35
Generators in S30
(1 22 10 13 30)(2 11 25 23 14)(3 24 12 15 26)(4 27 16 7 19)(5 17 20 28 8)(6 29 18 9 21)
(2 25 14 11 23)(3 26 15 12 24)(5 20 8 17 28)(6 21 9 18 29)
(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)| (1,22,10,13,30)(2,11,25,23,14)(3,24,12,15,26)(4,27,16,7,19)(5,17,20,28,8)(6,29,18,9,21), (2,25,14,11,23)(3,26,15,12,24)(5,20,8,17,28)(6,21,9,18,29), (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( (1,22,10,13,30)(2,11,25,23,14)(3,24,12,15,26)(4,27,16,7,19)(5,17,20,28,8)(6,29,18,9,21), (2,25,14,11,23)(3,26,15,12,24)(5,20,8,17,28)(6,21,9,18,29), (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([[(1,22,10,13,30),(2,11,25,23,14),(3,24,12,15,26),(4,27,16,7,19),(5,17,20,28,8),(6,29,18,9,21)], [(2,25,14,11,23),(3,26,15,12,24),(5,20,8,17,28),(6,21,9,18,29)], [(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(30,35);
C52⋊C6 is a maximal subgroup of
C52⋊Dic3 C52⋊C12 C52⋊D6 C5⋊D15⋊C3
C52⋊C6 is a maximal quotient of C52⋊2C12 C52⋊C18 C5⋊D15⋊C3
| action | f(x) | Disc(f) |
|---|---|---|
| 15T12 | x15-235x13+70x12+15930x11-14493x10-325950x9+112750x8+2876560x7+708890x6-11702794x5-8278925x4+19171805x3+20087950x2-6362585x-9179941 | 312·518·722·132·4118·22672·48712·5483031859935751932 |
Matrix representation of C52⋊C6 ►in GL6(𝔽31)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 13 | 1 | 30 | 18 | 0 | 0 |
| 30 | 19 | 13 | 13 | 0 | 0 |
| 30 | 19 | 0 | 0 | 13 | 13 |
| 13 | 1 | 0 | 0 | 18 | 30 |
| 30 | 1 | 0 | 0 | 0 | 0 |
| 11 | 19 | 0 | 0 | 0 | 0 |
| 30 | 0 | 0 | 1 | 0 | 0 |
| 12 | 1 | 30 | 18 | 0 | 0 |
| 29 | 19 | 0 | 0 | 13 | 13 |
| 12 | 1 | 0 | 0 | 18 | 30 |
| 13 | 1 | 0 | 0 | 17 | 30 |
| 30 | 19 | 0 | 0 | 12 | 13 |
| 0 | 0 | 0 | 0 | 30 | 0 |
| 12 | 1 | 0 | 0 | 30 | 0 |
| 0 | 0 | 1 | 0 | 30 | 0 |
| 13 | 1 | 18 | 30 | 30 | 0 |
G:=sub<GL(6,GF(31))| [1,0,13,30,30,13,0,1,1,19,19,1,0,0,30,13,0,0,0,0,18,13,0,0,0,0,0,0,13,18,0,0,0,0,13,30],[30,11,30,12,29,12,1,19,0,1,19,1,0,0,0,30,0,0,0,0,1,18,0,0,0,0,0,0,13,18,0,0,0,0,13,30],[13,30,0,12,0,13,1,19,0,1,0,1,0,0,0,0,1,18,0,0,0,0,0,30,17,12,30,30,30,30,30,13,0,0,0,0] >;
C52⋊C6 in GAP, Magma, Sage, TeX
C_5^2\rtimes C_6
% in TeX
G:=Group("C5^2:C6"); // GroupNames label
G:=SmallGroup(150,6);
// by ID
G=gap.SmallGroup(150,6);
# by ID
G:=PCGroup([4,-2,-3,-5,5,290,474,1923,295]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^6=1,a*b=b*a,c*a*c^-1=a^2*b^3,c*b*c^-1=a^-1*b^-1>;
// generators/relations
Export
Subgroup lattice of C52⋊C6 in TeX
Character table of C52⋊C6 in TeX