metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊2D5, C8⋊2D10, D4⋊2D10, C40⋊4C22, D10.14D4, C20.2C23, Dic5.16D4, D20.1C22, Dic10⋊1C22, D4⋊D5⋊2C2, (D4×D5)⋊2C2, (C5×D8)⋊4C2, C8⋊D5⋊3C2, C40⋊C2⋊3C2, C5⋊2(C8⋊C22), D4.D5⋊1C2, C2.16(D4×D5), D4⋊2D5⋊1C2, C5⋊2C8⋊1C22, C10.28(C2×D4), (C5×D4)⋊2C22, C4.2(C22×D5), (C4×D5).1C22, SmallGroup(160,132)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊D5
G = < a,b,c,d | a8=b2=c5=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >
Subgroups: 272 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, M4(2), D8, D8, SD16, C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C8⋊C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C22×D5, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D4⋊2D5, D8⋊D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C8⋊C22, C22×D5, D4×D5, D8⋊D5
Character table of D8⋊D5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 4 | 4 | 10 | 20 | 2 | 10 | 20 | 2 | 2 | 4 | 20 | 2 | 2 | 8 | 8 | 8 | 8 | 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 | 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 | 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 | 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 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 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 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | 0 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 0 | 0 | -2 | 0 | -2 | 2 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ14 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ15 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ16 | 2 | 2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ17 | 2 | 2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ20 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | orthogonal lifted from D4×D5 |
| ρ21 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | orthogonal lifted from D4×D5 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ87ζ54+ζ87ζ5+ζ85ζ54-ζ85ζ5 | -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 | -ζ83ζ54+ζ83ζ5+ζ8ζ54-ζ8ζ5 | ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ54+ζ83ζ5+ζ8ζ54-ζ8ζ5 | ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 | -ζ87ζ54+ζ87ζ5+ζ85ζ54-ζ85ζ5 | -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 | -ζ87ζ54+ζ87ζ5+ζ85ζ54-ζ85ζ5 | -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 | -ζ83ζ54+ζ83ζ5+ζ8ζ54-ζ8ζ5 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 | -ζ83ζ54+ζ83ζ5+ζ8ζ54-ζ8ζ5 | ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 | -ζ87ζ54+ζ87ζ5+ζ85ζ54-ζ85ζ5 | complex faithful |
►On 40 points
(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)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 19)(20 24)(21 23)(25 31)(26 30)(27 29)(33 37)(34 36)(38 40)
(1 18 10 39 32)(2 19 11 40 25)(3 20 12 33 26)(4 21 13 34 27)(5 22 14 35 28)(6 23 15 36 29)(7 24 16 37 30)(8 17 9 38 31)
(1 32)(2 29)(3 26)(4 31)(5 28)(6 25)(7 30)(8 27)(9 13)(11 15)(17 34)(18 39)(19 36)(20 33)(21 38)(22 35)(23 40)(24 37)
G:=sub<Sym(40)| (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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,19)(20,24)(21,23)(25,31)(26,30)(27,29)(33,37)(34,36)(38,40), (1,18,10,39,32)(2,19,11,40,25)(3,20,12,33,26)(4,21,13,34,27)(5,22,14,35,28)(6,23,15,36,29)(7,24,16,37,30)(8,17,9,38,31), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,13)(11,15)(17,34)(18,39)(19,36)(20,33)(21,38)(22,35)(23,40)(24,37)>;
G:=Group( (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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,19)(20,24)(21,23)(25,31)(26,30)(27,29)(33,37)(34,36)(38,40), (1,18,10,39,32)(2,19,11,40,25)(3,20,12,33,26)(4,21,13,34,27)(5,22,14,35,28)(6,23,15,36,29)(7,24,16,37,30)(8,17,9,38,31), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,13)(11,15)(17,34)(18,39)(19,36)(20,33)(21,38)(22,35)(23,40)(24,37) );
G=PermutationGroup([[(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)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(17,19),(20,24),(21,23),(25,31),(26,30),(27,29),(33,37),(34,36),(38,40)], [(1,18,10,39,32),(2,19,11,40,25),(3,20,12,33,26),(4,21,13,34,27),(5,22,14,35,28),(6,23,15,36,29),(7,24,16,37,30),(8,17,9,38,31)], [(1,32),(2,29),(3,26),(4,31),(5,28),(6,25),(7,30),(8,27),(9,13),(11,15),(17,34),(18,39),(19,36),(20,33),(21,38),(22,35),(23,40),(24,37)]])
D8⋊D5 is a maximal subgroup of
D8⋊13D10 Q16⋊D10 D8⋊11D10 D5×C8⋊C22 SD16⋊D10 D8⋊5D10 D8⋊6D10 D24⋊D5 D24⋊6D5 Dic10⋊3D6 D30.8D4 D12⋊10D10 D12⋊5D10 D8⋊D15
D8⋊D5 is a maximal quotient of
D4.D5⋊5C4 D4⋊Dic10 Dic10⋊2D4 D4.Dic10 C4⋊C4.D10 C20⋊Q8⋊C2 (D4×D5)⋊C4 D4⋊(C4×D5) D20.8D4 D10.16SD16 C40⋊6C4⋊C2 C5⋊2C8⋊D4 D4⋊3D20 C5⋊(C8⋊2D4) D4⋊D5⋊6C4 D20.D4 Dic10⋊2Q8 C40⋊4Q8 C40⋊20(C2×C4) D10.8Q16 C2.D8⋊D5 C8⋊3D20 C40⋊21(C2×C4) D20.2Q8 Dic5⋊D8 D8⋊Dic5 (C2×D8).D5 C40⋊11D4 D20⋊D4 Dic10⋊D4 C40⋊12D4 D24⋊D5 D24⋊6D5 Dic10⋊3D6 D30.8D4 D12⋊10D10 D12⋊5D10 D8⋊D15
Matrix representation of D8⋊D5 ►in GL4(𝔽41) generated by
| 0 | 0 | 11 | 31 |
| 0 | 0 | 10 | 30 |
| 19 | 20 | 22 | 21 |
| 21 | 22 | 20 | 19 |
| 1 | 0 | 40 | 0 |
| 0 | 1 | 0 | 40 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 6 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 |
| 0 | 0 | 40 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [0,0,19,21,0,0,20,22,11,10,22,20,31,30,21,19],[1,0,0,0,0,1,0,0,40,0,40,0,0,40,0,40],[6,40,0,0,1,0,0,0,0,0,6,40,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
D8⋊D5 in GAP, Magma, Sage, TeX
D_8\rtimes D_5
% in TeX
G:=Group("D8:D5"); // GroupNames label
G:=SmallGroup(160,132);
// by ID
G=gap.SmallGroup(160,132);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,362,116,297,159,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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