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 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 11)(2 28)(3 37)(4 23)(5 15)(6 32)(7 33)(8 19)(9 25)(12 21)(13 29)(16 17)(20 35)(24 39)(26 34)(30 38)
(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)| (1,35,27,20,11)(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), (1,11)(2,28)(3,37)(4,23)(5,15)(6,32)(7,33)(8,19)(9,25)(12,21)(13,29)(16,17)(20,35)(24,39)(26,34)(30,38), (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( (1,35,27,20,11)(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), (1,11)(2,28)(3,37)(4,23)(5,15)(6,32)(7,33)(8,19)(9,25)(12,21)(13,29)(16,17)(20,35)(24,39)(26,34)(30,38), (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([[(1,35,27,20,11),(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)], [(1,11),(2,28),(3,37),(4,23),(5,15),(6,32),(7,33),(8,19),(9,25),(12,21),(13,29),(16,17),(20,35),(24,39),(26,34),(30,38)], [(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