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G = D5×D8order 160 = 25·5

Direct product of D5 and D8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5×D8, C84D10, D404C2, D41D10, C402C22, D10.23D4, D201C22, C20.1C23, Dic5.7D4, C52(C2×D8), D4⋊D51C2, (C5×D8)⋊2C2, (D4×D5)⋊1C2, (C8×D5)⋊1C2, C2.15(D4×D5), C52C85C22, C10.27(C2×D4), (C5×D4)⋊1C22, C4.1(C22×D5), (C4×D5).15C22, SmallGroup(160,131)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D5×D8
 G = < a,b,c,d | a5=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 344 in 76 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2 [×6], C4, C4, C22 [×9], C5, C8, C8, C2×C4, D4 [×2], D4 [×4], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C2×C8, D8, D8 [×3], C2×D4 [×2], Dic5, C20, D10, D10 [×6], C2×C10 [×2], C2×D8, C52C8, C40, C4×D5, D20 [×2], C5⋊D4 [×2], C5×D4 [×2], C22×D5 [×2], C8×D5, D40, D4⋊D5 [×2], C5×D8, D4×D5 [×2], D5×D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, C22×D5, D4×D5, D5×D8

Character table of D5×D8

 class 12A2B2C2D2E2F2G4A4B5A5B8A8B8C8D10A10B10C10D10E10F20A20B40A40B40C40D
 size 114455202021022221010228888444444
ρ11111111111111111111111111111    trivial
ρ211-1-1-1-1111-11111-1-111-1-1-1-1111111    linear of order 2
ρ311-1111-111111-1-1-1-111-111-111-1-1-1-1    linear of order 2
ρ4111-1-1-1-111-111-1-111111-1-1111-1-1-1-1    linear of order 2
ρ511-11-1-11-11-111-1-11111-111-111-1-1-1-1    linear of order 2
ρ6111-1111-11111-1-1-1-1111-1-1111-1-1-1-1    linear of order 2
ρ711-1-111-1-11111111111-1-1-1-1111111    linear of order 2
ρ81111-1-1-1-11-11111-1-1111111111111    linear of order 2
ρ92200-2-200-22220000220000-2-20000    orthogonal lifted from D4
ρ1022002200-2-2220000220000-2-20000    orthogonal lifted from D4
ρ1122-22000020-1+5/2-1-5/2-2-200-1-5/2-1+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
ρ1222-22000020-1-5/2-1+5/2-2-200-1+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
ρ1322-2-2000020-1+5/2-1-5/22200-1-5/2-1+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
ρ142222000020-1-5/2-1+5/22200-1+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
ρ152222000020-1+5/2-1-5/22200-1-5/2-1+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
ρ1622-2-2000020-1-5/2-1+5/22200-1+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
ρ17222-2000020-1+5/2-1-5/2-2-200-1-5/2-1+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
ρ18222-2000020-1-5/2-1+5/2-2-200-1+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
ρ192-200-22000022-222-2-2-20000002-2-22    orthogonal lifted from D8
ρ202-200-220000222-2-22-2-2000000-222-2    orthogonal lifted from D8
ρ212-2002-20000222-22-2-2-2000000-222-2    orthogonal lifted from D8
ρ222-2002-2000022-22-22-2-20000002-2-22    orthogonal lifted from D8
ρ2344000000-40-1+5-1-50000-1-5-1+500001+51-50000    orthogonal lifted from D4×D5
ρ2444000000-40-1-5-1+50000-1+5-1-500001-51+50000    orthogonal lifted from D4×D5
ρ254-400000000-1+5-1-5-2222001+51-5000000ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ583ζ5383ζ528ζ538ζ52    orthogonal faithful
ρ264-400000000-1-5-1+522-22001-51+5000000ζ83ζ5383ζ528ζ538ζ52ζ87ζ5487ζ585ζ5485ζ583ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ5    orthogonal faithful
ρ274-400000000-1-5-1+5-2222001-51+500000083ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ5ζ83ζ5383ζ528ζ538ζ52ζ87ζ5487ζ585ζ5485ζ5    orthogonal faithful
ρ284-400000000-1+5-1-522-22001+51-5000000ζ83ζ5483ζ58ζ548ζ583ζ5383ζ528ζ538ζ52ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5383ζ528ζ538ζ52    orthogonal faithful

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

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

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

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

D5×D8 is a maximal subgroup of
D5.D16  D40.C4  D16⋊D5  C16⋊D10  D40⋊C4  D813D10  D815D10  D85D10  C405D6  D15⋊D8
D5×D8 is a maximal quotient of
Dic54D8  Dic5.14D8  Dic5.5D8  D4⋊D20  D10.12D8  D10⋊D8  D203D4  D4012C4  C402Q8  D10.13D8  C87D20  D202Q8  D16⋊D5  D163D5  C16⋊D10  SD32⋊D5  SD323D5  Q32⋊D5  D805C2  Dic5⋊D8  C405D4  D20⋊D4  C406D4  C405D6  D15⋊D8

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

40100
53500
0010
0001
,
40000
5100
0010
0001
,
1000
0100
001229
001212
,
1000
0100
0010
00040
G:=sub<GL(4,GF(41))| [40,5,0,0,1,35,0,0,0,0,1,0,0,0,0,1],[40,5,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,29,12],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40] >;

D5×D8 in GAP, Magma, Sage, TeX

D_5\times D_8
% in TeX

G:=Group("D5xD8");
// GroupNames label

G:=SmallGroup(160,131);
// by ID

G=gap.SmallGroup(160,131);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,116,297,159,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of D5×D8 in TeX

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