Copied to
clipboard

G = D8⋊D5order 160 = 25·5

2nd semidirect product of D8 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D82D5, C82D10, D42D10, C404C22, D10.14D4, C20.2C23, Dic5.16D4, D20.1C22, Dic101C22, D4⋊D52C2, (D4×D5)⋊2C2, (C5×D8)⋊4C2, C8⋊D53C2, C40⋊C23C2, C52(C8⋊C22), D4.D51C2, C2.16(D4×D5), D42D51C2, C52C81C22, C10.28(C2×D4), (C5×D4)⋊2C22, C4.2(C22×D5), (C4×D5).1C22, SmallGroup(160,132)

Series: Derived Chief Lower central Upper central

C1C20 — D8⋊D5
C1C5C10C20C4×D5D4×D5 — D8⋊D5
C5C10C20 — D8⋊D5
C1C2C4D8

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 [×4], C4, C4 [×2], C22 [×6], C5, C8, C8, C2×C4 [×2], D4 [×2], D4 [×3], Q8, C23, D5 [×2], C10, C10 [×2], M4(2), D8, D8, SD16 [×2], C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10 [×3], C2×C10 [×2], C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4 [×2], C22×D5, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, D8⋊D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], C8⋊C22, C22×D5, D4×D5, D8⋊D5

Character table of D8⋊D5

 class 12A2B2C2D2E4A4B4C5A5B8A8B10A10B10C10D10E10F20A20B40A40B40C40D
 size 114410202102022420228888444444
ρ11111111111111111111111111    trivial
ρ2111-1-1-11-1111-11111-1-1111-1-1-1-1    linear of order 2
ρ311-1-11-111-1111111-1-1-1-1111111    linear of order 2
ρ411-11-111-1-111-1111-111-111-1-1-1-1    linear of order 2
ρ511-111-111111-1-111-111-111-1-1-1-1    linear of order 2
ρ611-1-1-111-11111-111-1-1-1-1111111    linear of order 2
ρ7111-11111-111-1-1111-1-1111-1-1-1-1    linear of order 2
ρ81111-1-11-1-1111-1111111111111    linear of order 2
ρ9220020-2-202200220000-2-20000    orthogonal lifted from D4
ρ102200-20-2202200220000-2-20000    orthogonal lifted from D4
ρ11222200200-1+5/2-1-5/220-1+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
ρ1222-2-200200-1+5/2-1-5/220-1+5/2-1-5/21+5/21-5/21+5/21-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ1322-2-200200-1-5/2-1+5/220-1-5/2-1+5/21-5/21+5/21-5/21+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ14222-200200-1-5/2-1+5/2-20-1-5/2-1+5/2-1+5/21+5/21-5/2-1-5/2-1-5/2-1+5/21-5/21+5/21-5/21+5/2    orthogonal lifted from D10
ρ15222-200200-1+5/2-1-5/2-20-1+5/2-1-5/2-1-5/21-5/21+5/2-1+5/2-1+5/2-1-5/21+5/21-5/21+5/21-5/2    orthogonal lifted from D10
ρ1622-2200200-1+5/2-1-5/2-20-1+5/2-1-5/21+5/2-1+5/2-1-5/21-5/2-1+5/2-1-5/21+5/21-5/21+5/21-5/2    orthogonal lifted from D10
ρ1722-2200200-1-5/2-1+5/2-20-1-5/2-1+5/21-5/2-1-5/2-1+5/21+5/2-1-5/2-1+5/21-5/21+5/21-5/21+5/2    orthogonal lifted from D10
ρ18222200200-1-5/2-1+5/220-1-5/2-1+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
ρ194-400000004400-4-40000000000    orthogonal lifted from C8⋊C22
ρ20440000-400-1-5-1+500-1-5-1+500001+51-50000    orthogonal lifted from D4×D5
ρ21440000-400-1+5-1-500-1+5-1-500001-51+50000    orthogonal lifted from D4×D5
ρ224-40000000-1-5-1+5001+51-500000087ζ5487ζ585ζ5485ζ583ζ5383ζ528ζ538ζ5283ζ5483ζ58ζ548ζ5ζ83ζ5383ζ528ζ538ζ52    complex faithful
ρ234-40000000-1-5-1+5001+51-500000083ζ5483ζ58ζ548ζ5ζ83ζ5383ζ528ζ538ζ5287ζ5487ζ585ζ5485ζ583ζ5383ζ528ζ538ζ52    complex faithful
ρ244-40000000-1+5-1-5001-51+5000000ζ83ζ5383ζ528ζ538ζ5287ζ5487ζ585ζ5485ζ583ζ5383ζ528ζ538ζ5283ζ5483ζ58ζ548ζ5    complex faithful
ρ254-40000000-1+5-1-5001-51+500000083ζ5383ζ528ζ538ζ5283ζ5483ζ58ζ548ζ5ζ83ζ5383ζ528ζ538ζ5287ζ5487ζ585ζ5485ζ5    complex faithful

Smallest permutation representation of D8⋊D5
On 40 points
Generators in S40
(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 15)(10 14)(11 13)(17 19)(20 24)(21 23)(26 32)(27 31)(28 30)(34 40)(35 39)(36 38)
(1 18 25 37 12)(2 19 26 38 13)(3 20 27 39 14)(4 21 28 40 15)(5 22 29 33 16)(6 23 30 34 9)(7 24 31 35 10)(8 17 32 36 11)
(1 12)(2 9)(3 14)(4 11)(5 16)(6 13)(7 10)(8 15)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(26 30)(28 32)

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)(9,15)(10,14)(11,13)(17,19)(20,24)(21,23)(26,32)(27,31)(28,30)(34,40)(35,39)(36,38), (1,18,25,37,12)(2,19,26,38,13)(3,20,27,39,14)(4,21,28,40,15)(5,22,29,33,16)(6,23,30,34,9)(7,24,31,35,10)(8,17,32,36,11), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(26,30)(28,32)>;

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)(9,15)(10,14)(11,13)(17,19)(20,24)(21,23)(26,32)(27,31)(28,30)(34,40)(35,39)(36,38), (1,18,25,37,12)(2,19,26,38,13)(3,20,27,39,14)(4,21,28,40,15)(5,22,29,33,16)(6,23,30,34,9)(7,24,31,35,10)(8,17,32,36,11), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(26,30)(28,32) );

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),(9,15),(10,14),(11,13),(17,19),(20,24),(21,23),(26,32),(27,31),(28,30),(34,40),(35,39),(36,38)], [(1,18,25,37,12),(2,19,26,38,13),(3,20,27,39,14),(4,21,28,40,15),(5,22,29,33,16),(6,23,30,34,9),(7,24,31,35,10),(8,17,32,36,11)], [(1,12),(2,9),(3,14),(4,11),(5,16),(6,13),(7,10),(8,15),(17,40),(18,37),(19,34),(20,39),(21,36),(22,33),(23,38),(24,35),(26,30),(28,32)])

D8⋊D5 is a maximal subgroup of
D813D10  Q16⋊D10  D811D10  D5×C8⋊C22  SD16⋊D10  D85D10  D86D10  D24⋊D5  D246D5  Dic103D6  D30.8D4  D1210D10  D125D10  D8⋊D15
D8⋊D5 is a maximal quotient of
D4.D55C4  D4⋊Dic10  Dic102D4  D4.Dic10  C4⋊C4.D10  C20⋊Q8⋊C2  (D4×D5)⋊C4  D4⋊(C4×D5)  D20.8D4  D10.16SD16  C406C4⋊C2  C52C8⋊D4  D43D20  C5⋊(C82D4)  D4⋊D56C4  D20.D4  Dic102Q8  C404Q8  C4020(C2×C4)  D10.8Q16  C2.D8⋊D5  C83D20  C4021(C2×C4)  D20.2Q8  Dic5⋊D8  D8⋊Dic5  (C2×D8).D5  C4011D4  D20⋊D4  Dic10⋊D4  C4012D4  D24⋊D5  D246D5  Dic103D6  D30.8D4  D1210D10  D125D10  D8⋊D15

Matrix representation of D8⋊D5 in GL4(𝔽41) generated by

001131
001030
19202221
21222019
,
10400
01040
00400
00040
,
6100
40000
0061
00400
,
0100
1000
0001
0010
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

Export

Character table of D8⋊D5 in TeX

׿
×
𝔽