metabelian, supersoluble, monomial, A-group
Aliases: C52⋊3C8, C10.Dic5, C10.5F5, Dic5.2D5, C5⋊(C5⋊2C8), C5⋊3(C5⋊C8), (C5×C10).2C4, C2.(D5.D5), (C5×Dic5).3C2, SmallGroup(200,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊3C8 |
Generators and relations for C52⋊3C8
G = < a,b,c | a5=b5=c8=1, ab=ba, cac-1=a-1, cbc-1=b2 >
Character table of C52⋊3C8
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 25 | 25 | 25 | 25 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ87 | ζ85 | ζ83 | ζ8 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | i | i | -i | -i | linear of order 8 |
| ρ6 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ83 | ζ8 | ζ87 | ζ85 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | i | i | -i | -i | linear of order 8 |
| ρ7 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ8 | ζ83 | ζ85 | ζ87 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | -i | i | i | linear of order 8 |
| ρ8 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ85 | ζ87 | ζ8 | ζ83 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | -i | i | i | linear of order 8 |
| ρ9 | 2 | 2 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ10 | 2 | 2 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ11 | 2 | 2 | -2 | -2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ12 | 2 | 2 | -2 | -2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ13 | 2 | -2 | -2i | 2i | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | -2 | 1+√5/2 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | complex lifted from C5⋊2C8, Schur index 2 |
| ρ14 | 2 | -2 | 2i | -2i | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | -2 | 1-√5/2 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | complex lifted from C5⋊2C8, Schur index 2 |
| ρ15 | 2 | -2 | 2i | -2i | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | -2 | 1+√5/2 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | complex lifted from C5⋊2C8, Schur index 2 |
| ρ16 | 2 | -2 | -2i | 2i | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | -2 | 1-√5/2 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | complex lifted from C5⋊2C8, Schur index 2 |
| ρ17 | 4 | 4 | 0 | 0 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from F5 |
| ρ18 | 4 | -4 | 0 | 0 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -4 | -4 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ19 | 4 | 4 | 0 | 0 | -1-√5 | -1+√5 | 2ζ54+ζ52+1 | 2ζ52+ζ5+1 | -1 | ζ54+2ζ53+1 | ζ53+2ζ5+1 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2ζ52+ζ5+1 | ζ53+2ζ5+1 | 2ζ54+ζ52+1 | -1 | ζ54+2ζ53+1 | 0 | 0 | 0 | 0 | complex lifted from D5.D5 |
| ρ20 | 4 | -4 | 0 | 0 | -1+√5 | -1-√5 | 2ζ52+ζ5+1 | ζ53+2ζ5+1 | -1 | 2ζ54+ζ52+1 | ζ54+2ζ53+1 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | ζ54+ζ52-ζ5 | -ζ53+ζ52+ζ5 | ζ54+ζ53-ζ52 | 1 | -ζ54+ζ53+ζ5 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | -1-√5 | -1+√5 | ζ53+2ζ5+1 | ζ54+2ζ53+1 | -1 | 2ζ52+ζ5+1 | 2ζ54+ζ52+1 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | -ζ53+ζ52+ζ5 | -ζ54+ζ53+ζ5 | ζ54+ζ52-ζ5 | 1 | ζ54+ζ53-ζ52 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | -1-√5 | -1+√5 | 2ζ54+ζ52+1 | 2ζ52+ζ5+1 | -1 | ζ54+2ζ53+1 | ζ53+2ζ5+1 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | ζ54+ζ53-ζ52 | ζ54+ζ52-ζ5 | -ζ54+ζ53+ζ5 | 1 | -ζ53+ζ52+ζ5 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | -1+√5 | -1-√5 | ζ54+2ζ53+1 | 2ζ54+ζ52+1 | -1 | ζ53+2ζ5+1 | 2ζ52+ζ5+1 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | -ζ54+ζ53+ζ5 | ζ54+ζ53-ζ52 | -ζ53+ζ52+ζ5 | 1 | ζ54+ζ52-ζ5 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | 4 | 0 | 0 | -1+√5 | -1-√5 | 2ζ52+ζ5+1 | ζ53+2ζ5+1 | -1 | 2ζ54+ζ52+1 | ζ54+2ζ53+1 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | ζ53+2ζ5+1 | ζ54+2ζ53+1 | 2ζ52+ζ5+1 | -1 | 2ζ54+ζ52+1 | 0 | 0 | 0 | 0 | complex lifted from D5.D5 |
| ρ25 | 4 | 4 | 0 | 0 | -1-√5 | -1+√5 | ζ53+2ζ5+1 | ζ54+2ζ53+1 | -1 | 2ζ52+ζ5+1 | 2ζ54+ζ52+1 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | ζ54+2ζ53+1 | 2ζ54+ζ52+1 | ζ53+2ζ5+1 | -1 | 2ζ52+ζ5+1 | 0 | 0 | 0 | 0 | complex lifted from D5.D5 |
| ρ26 | 4 | 4 | 0 | 0 | -1+√5 | -1-√5 | ζ54+2ζ53+1 | 2ζ54+ζ52+1 | -1 | ζ53+2ζ5+1 | 2ζ52+ζ5+1 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2ζ54+ζ52+1 | 2ζ52+ζ5+1 | ζ54+2ζ53+1 | -1 | ζ53+2ζ5+1 | 0 | 0 | 0 | 0 | complex lifted from D5.D5 |
►On 40 points
Generators in S40
(1 11 20 27 35)(2 36 28 21 12)(3 13 22 29 37)(4 38 30 23 14)(5 15 24 31 39)(6 40 32 17 16)(7 9 18 25 33)(8 34 26 19 10)
(1 35 27 20 11)(2 28 12 36 21)(3 13 22 29 37)(4 23 38 14 30)(5 39 31 24 15)(6 32 16 40 17)(7 9 18 25 33)(8 19 34 10 26)
(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)| (1,11,20,27,35)(2,36,28,21,12)(3,13,22,29,37)(4,38,30,23,14)(5,15,24,31,39)(6,40,32,17,16)(7,9,18,25,33)(8,34,26,19,10), (1,35,27,20,11)(2,28,12,36,21)(3,13,22,29,37)(4,23,38,14,30)(5,39,31,24,15)(6,32,16,40,17)(7,9,18,25,33)(8,19,34,10,26), (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( (1,11,20,27,35)(2,36,28,21,12)(3,13,22,29,37)(4,38,30,23,14)(5,15,24,31,39)(6,40,32,17,16)(7,9,18,25,33)(8,34,26,19,10), (1,35,27,20,11)(2,28,12,36,21)(3,13,22,29,37)(4,23,38,14,30)(5,39,31,24,15)(6,32,16,40,17)(7,9,18,25,33)(8,19,34,10,26), (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([[(1,11,20,27,35),(2,36,28,21,12),(3,13,22,29,37),(4,38,30,23,14),(5,15,24,31,39),(6,40,32,17,16),(7,9,18,25,33),(8,34,26,19,10)], [(1,35,27,20,11),(2,28,12,36,21),(3,13,22,29,37),(4,23,38,14,30),(5,39,31,24,15),(6,32,16,40,17),(7,9,18,25,33),(8,19,34,10,26)], [(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⋊3C8 is a maximal subgroup of
D5×C5⋊C8 Dic5.4F5 D10.F5 Dic5.F5 C20.14F5 C20.12F5 C102.C4
C52⋊3C8 is a maximal quotient of C52⋊3C16
Matrix representation of C52⋊3C8 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 21 | 0 | 16 | 0 |
| 16 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 7 | 18 | 0 | 0 |
| 5 | 0 | 37 | 0 |
| 7 | 0 | 0 | 10 |
| 27 | 0 | 15 | 0 |
| 0 | 0 | 9 | 1 |
| 0 | 1 | 14 | 0 |
| 0 | 0 | 38 | 0 |
G:=sub<GL(4,GF(41))| [18,0,21,16,0,18,0,0,0,0,16,0,0,0,0,16],[16,7,5,7,0,18,0,0,0,0,37,0,0,0,0,10],[27,0,0,0,0,0,1,0,15,9,14,38,0,1,0,0] >;
C52⋊3C8 in GAP, Magma, Sage, TeX
C_5^2\rtimes_3C_8
% in TeX
G:=Group("C5^2:3C8"); // GroupNames label
G:=SmallGroup(200,19);
// by ID
G=gap.SmallGroup(200,19);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,10,26,643,3004,2009]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^8=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^2>;
// generators/relations
Export
Subgroup lattice of C52⋊3C8 in TeX
Character table of C52⋊3C8 in TeX