metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5⋊C8, C4.3F5, C20.3C4, D10.2C4, Dic5.4C22, C4○(C5⋊C8), C5⋊C8⋊3C2, C5⋊1(C2×C8), C2.1(C2×F5), C10.1(C2×C4), (C4×D5).5C2, SmallGroup(80,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5⋊C8 |
Generators and relations for D5⋊C8
G = < a,b,c | a5=b2=c8=1, bab=a-1, cac-1=a3, cbc-1=a2b >
Character table of D5⋊C8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 20A | 20B | |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 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 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | i | -i | -i | -i | -i | i | i | 1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | -i | i | i | i | i | -i | -i | 1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | i | -i | -i | i | i | -i | i | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | i | -i | i | i | -i | -i | i | -i | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | -1 | 1 | i | -i | i | -i | 1 | ζ8 | ζ85 | ζ83 | ζ87 | ζ83 | ζ87 | ζ85 | ζ8 | -1 | -i | i | linear of order 8 |
| ρ10 | 1 | -1 | 1 | -1 | -i | i | i | -i | 1 | ζ8 | ζ8 | ζ87 | ζ83 | ζ83 | ζ87 | ζ85 | ζ85 | -1 | i | -i | linear of order 8 |
| ρ11 | 1 | -1 | 1 | -1 | -i | i | i | -i | 1 | ζ85 | ζ85 | ζ83 | ζ87 | ζ87 | ζ83 | ζ8 | ζ8 | -1 | i | -i | linear of order 8 |
| ρ12 | 1 | -1 | -1 | 1 | i | -i | i | -i | 1 | ζ85 | ζ8 | ζ87 | ζ83 | ζ87 | ζ83 | ζ8 | ζ85 | -1 | -i | i | linear of order 8 |
| ρ13 | 1 | -1 | 1 | -1 | i | -i | -i | i | 1 | ζ87 | ζ87 | ζ8 | ζ85 | ζ85 | ζ8 | ζ83 | ζ83 | -1 | -i | i | linear of order 8 |
| ρ14 | 1 | -1 | 1 | -1 | i | -i | -i | i | 1 | ζ83 | ζ83 | ζ85 | ζ8 | ζ8 | ζ85 | ζ87 | ζ87 | -1 | -i | i | linear of order 8 |
| ρ15 | 1 | -1 | -1 | 1 | -i | i | -i | i | 1 | ζ83 | ζ87 | ζ8 | ζ85 | ζ8 | ζ85 | ζ87 | ζ83 | -1 | i | -i | linear of order 8 |
| ρ16 | 1 | -1 | -1 | 1 | -i | i | -i | i | 1 | ζ87 | ζ83 | ζ85 | ζ8 | ζ85 | ζ8 | ζ83 | ζ87 | -1 | i | -i | linear of order 8 |
| ρ17 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ18 | 4 | 4 | 0 | 0 | -4 | -4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ19 | 4 | -4 | 0 | 0 | -4i | 4i | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -i | i | complex faithful, Schur index 2 |
| ρ20 | 4 | -4 | 0 | 0 | 4i | -4i | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | i | -i | complex faithful, Schur index 2 |
►On 40 points
(1 35 9 22 26)(2 23 36 27 10)(3 28 24 11 37)(4 12 29 38 17)(5 39 13 18 30)(6 19 40 31 14)(7 32 20 15 33)(8 16 25 34 21)
(1 26)(2 10)(3 37)(4 17)(5 30)(6 14)(7 33)(8 21)(11 28)(12 38)(15 32)(16 34)(18 39)(19 31)(22 35)(23 27)
(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)| (1,35,9,22,26)(2,23,36,27,10)(3,28,24,11,37)(4,12,29,38,17)(5,39,13,18,30)(6,19,40,31,14)(7,32,20,15,33)(8,16,25,34,21), (1,26)(2,10)(3,37)(4,17)(5,30)(6,14)(7,33)(8,21)(11,28)(12,38)(15,32)(16,34)(18,39)(19,31)(22,35)(23,27), (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( (1,35,9,22,26)(2,23,36,27,10)(3,28,24,11,37)(4,12,29,38,17)(5,39,13,18,30)(6,19,40,31,14)(7,32,20,15,33)(8,16,25,34,21), (1,26)(2,10)(3,37)(4,17)(5,30)(6,14)(7,33)(8,21)(11,28)(12,38)(15,32)(16,34)(18,39)(19,31)(22,35)(23,27), (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([(1,35,9,22,26),(2,23,36,27,10),(3,28,24,11,37),(4,12,29,38,17),(5,39,13,18,30),(6,19,40,31,14),(7,32,20,15,33),(8,16,25,34,21)], [(1,26),(2,10),(3,37),(4,17),(5,30),(6,14),(7,33),(8,21),(11,28),(12,38),(15,32),(16,34),(18,39),(19,31),(22,35),(23,27)], [(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)])
D5⋊C8 is a maximal subgroup of
C8×F5 C8⋊F5 D20⋊C4 Q8⋊F5 D5⋊M4(2) D4.F5 Q8.F5 D15⋊C8 C60.C4 D25⋊C8 Dic5.4F5 D10.2F5 C20.14F5 C20.F5 C20.11F5 A5⋊C8 C4.6S5
D5⋊C8 is a maximal quotient of
D5⋊C16 C8.F5 C4×C5⋊C8 D10⋊C8 Dic5⋊C8 D15⋊C8 C60.C4 D25⋊C8 Dic5.4F5 D10.2F5 C20.14F5 C20.F5 C20.11F5
Matrix representation of D5⋊C8 ►in GL4(𝔽41) generated by
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 40 | 40 | 40 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 1 |
| 32 | 0 | 15 | 15 |
| 15 | 15 | 0 | 32 |
| 26 | 17 | 26 | 0 |
| 9 | 24 | 24 | 9 |
G:=sub<GL(4,GF(41))| [0,0,0,40,1,0,0,40,0,1,0,40,0,0,1,40],[0,1,40,0,1,0,40,0,0,0,40,0,0,0,40,1],[32,15,26,9,0,15,17,24,15,0,26,24,15,32,0,9] >;
D5⋊C8 in GAP, Magma, Sage, TeX
D_5\rtimes C_8
% in TeX
G:=Group("D5:C8"); // GroupNames label
G:=SmallGroup(80,28);
// by ID
G=gap.SmallGroup(80,28);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,20,46,42,804,414]);
// Polycyclic
G:=Group<a,b,c|a^5=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^2*b>;
// generators/relations
Export
Subgroup lattice of D5⋊C8 in TeX
Character table of D5⋊C8 in TeX