metabelian, soluble, monomial, A-group
Aliases: C52⋊C8, C5⋊D5.C4, C5⋊F5.C2, SmallGroup(200,40)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊C8 |
Generators and relations for C52⋊C8
G = < a,b,c | a5=b5=c8=1, ab=ba, cac-1=ab2, cbc-1=ab-1 >
Character table of C52⋊C8
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 8A | 8B | 8C | 8D | |
| size | 1 | 25 | 25 | 25 | 8 | 8 | 8 | 25 | 25 | 25 | 25 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | linear of order 4 |
| ρ5 | 1 | -1 | i | -i | 1 | 1 | 1 | ζ83 | ζ8 | ζ87 | ζ85 | linear of order 8 |
| ρ6 | 1 | -1 | -i | i | 1 | 1 | 1 | ζ8 | ζ83 | ζ85 | ζ87 | linear of order 8 |
| ρ7 | 1 | -1 | -i | i | 1 | 1 | 1 | ζ85 | ζ87 | ζ8 | ζ83 | linear of order 8 |
| ρ8 | 1 | -1 | i | -i | 1 | 1 | 1 | ζ87 | ζ85 | ζ83 | ζ8 | linear of order 8 |
| ρ9 | 8 | 0 | 0 | 0 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ10 | 8 | 0 | 0 | 0 | -2 | 3 | -2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ11 | 8 | 0 | 0 | 0 | 3 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 10 points - transitive group 10T18
Generators in S10
(1 6 8 4 10)(2 5 7 3 9)
(1 6 8 4 10)
(1 2)(3 4 5 6 7 8 9 10)
G:=sub<Sym(10)| (1,6,8,4,10)(2,5,7,3,9), (1,6,8,4,10), (1,2)(3,4,5,6,7,8,9,10)>;
G:=Group( (1,6,8,4,10)(2,5,7,3,9), (1,6,8,4,10), (1,2)(3,4,5,6,7,8,9,10) );
G=PermutationGroup([[(1,6,8,4,10),(2,5,7,3,9)], [(1,6,8,4,10)], [(1,2),(3,4,5,6,7,8,9,10)]])
G:=TransitiveGroup(10,18);
►On 20 points - transitive group 20T56
Generators in S20
(1 17 6 10 13)(2 11 18 14 7)(3 12 19 15 8)(4 16 5 9 20)
(1 17 6 10 13)(3 12 19 15 8)
(1 2 3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)
G:=sub<Sym(20)| (1,17,6,10,13)(2,11,18,14,7)(3,12,19,15,8)(4,16,5,9,20), (1,17,6,10,13)(3,12,19,15,8), (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20)>;
G:=Group( (1,17,6,10,13)(2,11,18,14,7)(3,12,19,15,8)(4,16,5,9,20), (1,17,6,10,13)(3,12,19,15,8), (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20) );
G=PermutationGroup([[(1,17,6,10,13),(2,11,18,14,7),(3,12,19,15,8),(4,16,5,9,20)], [(1,17,6,10,13),(3,12,19,15,8)], [(1,2,3,4),(5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20)]])
G:=TransitiveGroup(20,56);
►On 25 points: primitive - transitive group 25T20
Generators in S25
(1 13 15 11 17)(2 22 16 7 21)(3 12 18 6 25)(4 9 24 23 10)(5 8 14 19 20)
(1 9 3 7 5)(2 14 15 23 18)(4 25 16 20 17)(6 22 19 11 10)(8 13 24 12 21)
(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)| (1,13,15,11,17)(2,22,16,7,21)(3,12,18,6,25)(4,9,24,23,10)(5,8,14,19,20), (1,9,3,7,5)(2,14,15,23,18)(4,25,16,20,17)(6,22,19,11,10)(8,13,24,12,21), (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( (1,13,15,11,17)(2,22,16,7,21)(3,12,18,6,25)(4,9,24,23,10)(5,8,14,19,20), (1,9,3,7,5)(2,14,15,23,18)(4,25,16,20,17)(6,22,19,11,10)(8,13,24,12,21), (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([[(1,13,15,11,17),(2,22,16,7,21),(3,12,18,6,25),(4,9,24,23,10),(5,8,14,19,20)], [(1,9,3,7,5),(2,14,15,23,18),(4,25,16,20,17),(6,22,19,11,10),(8,13,24,12,21)], [(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(25,20);
C52⋊C8 is a maximal subgroup of
C52⋊M4(2)
C52⋊C8 is a maximal quotient of C52⋊C16
| action | f(x) | Disc(f) |
|---|---|---|
| 10T18 | x10+2x9-32x8-56x7+288x6+476x5-800x4-1152x3+816x2+704x-416 | 240·36·132·178·434 |
Matrix representation of C52⋊C8 ►in GL8(ℤ)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
G:=sub<GL(8,Integers())| [0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0],[0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0],[0,0,0,0,1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
C52⋊C8 in GAP, Magma, Sage, TeX
C_5^2\rtimes C_8
% in TeX
G:=Group("C5^2:C8"); // GroupNames label
G:=SmallGroup(200,40);
// by ID
G=gap.SmallGroup(200,40);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,5,10,26,3523,168,173,3404,1009,1014]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^8=1,a*b=b*a,c*a*c^-1=a*b^2,c*b*c^-1=a*b^-1>;
// generators/relations
Export
Subgroup lattice of C52⋊C8 in TeX
Character table of C52⋊C8 in TeX