metabelian, supersoluble, monomial, A-group
Aliases: C52⋊5C8, C10.4F5, C5⋊2(C5⋊C8), (C5×C10).4C4, C2.(C52⋊C4), C52⋊6C4.4C2, SmallGroup(200,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊5C8 |
Generators and relations for C52⋊5C8
G = < a,b,c | a5=b5=c8=1, ab=ba, cac-1=a2, cbc-1=b3 >
Character table of C52⋊5C8
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | |
| size | 1 | 1 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | 25 | 25 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | ζ87 | ζ85 | ζ83 | ζ8 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ6 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | ζ83 | ζ8 | ζ87 | ζ85 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ7 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | ζ85 | ζ87 | ζ8 | ζ83 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ8 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | ζ8 | ζ83 | ζ85 | ζ87 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ9 | 4 | 4 | 0 | 0 | -1+√5 | -1-√5 | 3-√5/2 | -1 | -1 | 3+√5/2 | 0 | 0 | 0 | 0 | -1 | -1 | 3+√5/2 | -1+√5 | -1-√5 | 3-√5/2 | orthogonal lifted from C52⋊C4 |
| ρ10 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | 0 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ11 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ12 | 4 | 4 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1-√5 | -1 | -1 | -1+√5 | 0 | 0 | 0 | 0 | -1 | -1 | -1+√5 | 3-√5/2 | 3+√5/2 | -1-√5 | orthogonal lifted from C52⋊C4 |
| ρ13 | 4 | 4 | 0 | 0 | -1-√5 | -1+√5 | 3+√5/2 | -1 | -1 | 3-√5/2 | 0 | 0 | 0 | 0 | -1 | -1 | 3-√5/2 | -1-√5 | -1+√5 | 3+√5/2 | orthogonal lifted from C52⋊C4 |
| ρ14 | 4 | 4 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1+√5 | -1 | -1 | -1-√5 | 0 | 0 | 0 | 0 | -1 | -1 | -1-√5 | 3+√5/2 | 3-√5/2 | -1+√5 | orthogonal lifted from C52⋊C4 |
| ρ15 | 4 | -4 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | 0 | 0 | 0 | 0 | 1 | -4 | 1 | 1 | 1 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ16 | 4 | -4 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | -4 | 1 | 1 | 1 | 1 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ17 | 4 | -4 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1+√5 | -1 | -1 | -1-√5 | 0 | 0 | 0 | 0 | 1 | 1 | 1+√5 | -3-√5/2 | -3+√5/2 | 1-√5 | symplectic faithful, Schur index 2 |
| ρ18 | 4 | -4 | 0 | 0 | -1+√5 | -1-√5 | 3-√5/2 | -1 | -1 | 3+√5/2 | 0 | 0 | 0 | 0 | 1 | 1 | -3-√5/2 | 1-√5 | 1+√5 | -3+√5/2 | symplectic faithful, Schur index 2 |
| ρ19 | 4 | -4 | 0 | 0 | -1-√5 | -1+√5 | 3+√5/2 | -1 | -1 | 3-√5/2 | 0 | 0 | 0 | 0 | 1 | 1 | -3+√5/2 | 1+√5 | 1-√5 | -3-√5/2 | symplectic faithful, Schur index 2 |
| ρ20 | 4 | -4 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1-√5 | -1 | -1 | -1+√5 | 0 | 0 | 0 | 0 | 1 | 1 | 1-√5 | -3+√5/2 | -3-√5/2 | 1+√5 | symplectic faithful, Schur index 2 |
►On 40 points
Generators in S40
(1 20 35 11 27)(2 36 28 21 12)(3 29 13 37 22)(4 14 23 30 38)(5 24 39 15 31)(6 40 32 17 16)(7 25 9 33 18)(8 10 19 26 34)
(1 35 27 20 11)(2 21 36 12 28)(3 13 22 29 37)(4 30 14 38 23)(5 39 31 24 15)(6 17 40 16 32)(7 9 18 25 33)(8 26 10 34 19)
(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 37 38 39 40)
G:=sub<Sym(40)| (1,20,35,11,27)(2,36,28,21,12)(3,29,13,37,22)(4,14,23,30,38)(5,24,39,15,31)(6,40,32,17,16)(7,25,9,33,18)(8,10,19,26,34), (1,35,27,20,11)(2,21,36,12,28)(3,13,22,29,37)(4,30,14,38,23)(5,39,31,24,15)(6,17,40,16,32)(7,9,18,25,33)(8,26,10,34,19), (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)(33,34,35,36,37,38,39,40)>;
G:=Group( (1,20,35,11,27)(2,36,28,21,12)(3,29,13,37,22)(4,14,23,30,38)(5,24,39,15,31)(6,40,32,17,16)(7,25,9,33,18)(8,10,19,26,34), (1,35,27,20,11)(2,21,36,12,28)(3,13,22,29,37)(4,30,14,38,23)(5,39,31,24,15)(6,17,40,16,32)(7,9,18,25,33)(8,26,10,34,19), (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)(33,34,35,36,37,38,39,40) );
G=PermutationGroup([[(1,20,35,11,27),(2,36,28,21,12),(3,29,13,37,22),(4,14,23,30,38),(5,24,39,15,31),(6,40,32,17,16),(7,25,9,33,18),(8,10,19,26,34)], [(1,35,27,20,11),(2,21,36,12,28),(3,13,22,29,37),(4,30,14,38,23),(5,39,31,24,15),(6,17,40,16,32),(7,9,18,25,33),(8,26,10,34,19)], [(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),(33,34,35,36,37,38,39,40)]])
C52⋊5C8 is a maximal subgroup of
D10.2F5 C52⋊4M4(2) C52⋊D8 C52⋊SD16 C52⋊Q16 C20.11F5 C52⋊8M4(2) C52⋊14M4(2)
C52⋊5C8 is a maximal quotient of
C52⋊5C16
Matrix representation of C52⋊5C8 ►in GL4(𝔽41) generated by
| 0 | 7 | 0 | 0 |
| 35 | 6 | 0 | 0 |
| 13 | 36 | 34 | 40 |
| 35 | 37 | 1 | 0 |
| 40 | 1 | 0 | 0 |
| 5 | 35 | 0 | 0 |
| 37 | 11 | 40 | 34 |
| 27 | 18 | 7 | 7 |
| 0 | 0 | 40 | 1 |
| 19 | 40 | 39 | 34 |
| 30 | 12 | 30 | 12 |
| 10 | 29 | 30 | 12 |
G:=sub<GL(4,GF(41))| [0,35,13,35,7,6,36,37,0,0,34,1,0,0,40,0],[40,5,37,27,1,35,11,18,0,0,40,7,0,0,34,7],[0,19,30,10,0,40,12,29,40,39,30,30,1,34,12,12] >;
C52⋊5C8 in GAP, Magma, Sage, TeX
C_5^2\rtimes_5C_8
% in TeX
G:=Group("C5^2:5C8"); // GroupNames label
G:=SmallGroup(200,21);
// by ID
G=gap.SmallGroup(200,21);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,10,26,483,328,2004,2009]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^8=1,a*b=b*a,c*a*c^-1=a^2,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C52⋊5C8 in TeX
Character table of C52⋊5C8 in TeX