metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊1F5, C40⋊1C4, D5.1D8, D10.9D4, D5.1Q16, Dic5.3Q8, C5⋊(C2.D8), C5⋊2C8⋊6C4, C4⋊F5.4C2, C4.9(C2×F5), C20.9(C2×C4), (C8×D5).3C2, C2.5(C4⋊F5), C10.2(C4⋊C4), (C4×D5).26C22, SmallGroup(160,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5.D8
G = < a,b,c,d | a5=b2=c8=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=c-1 >
Character table of D5.D8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 10 | 20A | 20B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 5 | 5 | 2 | 10 | 20 | 20 | 20 | 20 | 4 | 2 | 2 | 10 | 10 | 4 | 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 | 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 | 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 | 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 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | -1 | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | -i | i | i | -i | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | i | -i | -i | i | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | √2 | -√2 | -2 | 0 | 0 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | -√2 | √2 | -2 | 0 | 0 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | -√2 | √2 | -2 | 0 | 0 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | √2 | -√2 | -2 | 0 | 0 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ15 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ16 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ17 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√-5 | √-5 | √-5 | -√-5 | complex lifted from C4⋊F5 |
| ρ18 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √-5 | -√-5 | -√-5 | √-5 | complex lifted from C4⋊F5 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2√2 | 2√2 | 0 | 0 | 1 | √-5 | -√-5 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2√2 | -2√2 | 0 | 0 | 1 | -√-5 | √-5 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2√2 | -2√2 | 0 | 0 | 1 | √-5 | -√-5 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2√2 | 2√2 | 0 | 0 | 1 | -√-5 | √-5 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | complex faithful |
►On 40 points
Generators in S40
(1 26 12 24 35)(2 27 13 17 36)(3 28 14 18 37)(4 29 15 19 38)(5 30 16 20 39)(6 31 9 21 40)(7 32 10 22 33)(8 25 11 23 34)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 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)
(1 8)(2 7)(3 6)(4 5)(9 37 21 28)(10 36 22 27)(11 35 23 26)(12 34 24 25)(13 33 17 32)(14 40 18 31)(15 39 19 30)(16 38 20 29)
G:=sub<Sym(40)| (1,26,12,24,35)(2,27,13,17,36)(3,28,14,18,37)(4,29,15,19,38)(5,30,16,20,39)(6,31,9,21,40)(7,32,10,22,33)(8,25,11,23,34), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26), (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), (1,8)(2,7)(3,6)(4,5)(9,37,21,28)(10,36,22,27)(11,35,23,26)(12,34,24,25)(13,33,17,32)(14,40,18,31)(15,39,19,30)(16,38,20,29)>;
G:=Group( (1,26,12,24,35)(2,27,13,17,36)(3,28,14,18,37)(4,29,15,19,38)(5,30,16,20,39)(6,31,9,21,40)(7,32,10,22,33)(8,25,11,23,34), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26), (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), (1,8)(2,7)(3,6)(4,5)(9,37,21,28)(10,36,22,27)(11,35,23,26)(12,34,24,25)(13,33,17,32)(14,40,18,31)(15,39,19,30)(16,38,20,29) );
G=PermutationGroup([[(1,26,12,24,35),(2,27,13,17,36),(3,28,14,18,37),(4,29,15,19,38),(5,30,16,20,39),(6,31,9,21,40),(7,32,10,22,33),(8,25,11,23,34)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,37,21,28),(10,36,22,27),(11,35,23,26),(12,34,24,25),(13,33,17,32),(14,40,18,31),(15,39,19,30),(16,38,20,29)]])
D5.D8 is a maximal subgroup of
C80⋊2C4 C80⋊3C4 D5.D16 D5.Q32 (C2×C8)⋊6F5 M4(2)⋊1F5 D8×F5 SD16⋊F5 Q16×F5 Dic5.4Dic6 D5.D24
D5.D8 is a maximal quotient of
C80⋊2C4 C80⋊3C4 C16.F5 C80.2C4 C40⋊1C8 Dic5.13D8 D10.10D8 Dic5.4Dic6 D5.D24
Matrix representation of D5.D8 ►in GL4(𝔽7) generated by
| 3 | 1 | 3 | 3 |
| 0 | 1 | 6 | 3 |
| 4 | 4 | 2 | 2 |
| 1 | 5 | 6 | 0 |
| 6 | 0 | 4 | 2 |
| 0 | 6 | 1 | 4 |
| 0 | 3 | 3 | 1 |
| 0 | 1 | 3 | 6 |
| 3 | 3 | 0 | 2 |
| 1 | 3 | 1 | 1 |
| 2 | 2 | 6 | 3 |
| 3 | 4 | 2 | 3 |
| 4 | 1 | 3 | 3 |
| 6 | 2 | 6 | 6 |
| 1 | 2 | 1 | 2 |
| 0 | 2 | 1 | 0 |
G:=sub<GL(4,GF(7))| [3,0,4,1,1,1,4,5,3,6,2,6,3,3,2,0],[6,0,0,0,0,6,3,1,4,1,3,3,2,4,1,6],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[4,6,1,0,1,2,2,2,3,6,1,1,3,6,2,0] >;
D5.D8 in GAP, Magma, Sage, TeX
D_5.D_8
% in TeX
G:=Group("D5.D8"); // GroupNames label
G:=SmallGroup(160,69);
// by ID
G=gap.SmallGroup(160,69);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,151,579,69,2309,1169]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^8=1,d^2=a^-1*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of D5.D8 in TeX
Character table of D5.D8 in TeX