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G = D5×D8order 160 = 25·5

Direct product of D5 and D8

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

 Derived series C1 — C20 — D5×D8
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D4×D5 — D5×D8
 Lower central C5 — C10 — C20 — D5×D8
 Upper central C1 — C2 — C4 — D8

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, C4, C4, C22, C5, C8, C8, C2×C4, D4, D4, C23, D5, D5, C10, C10, C2×C8, D8, D8, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×D8, C52C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C8×D5, D40, D4⋊D5, C5×D8, D4×D5, D5×D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C22×D5, D4×D5, D5×D8

Character table of D5×D8

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 20A 20B 40A 40B 40C 40D size 1 1 4 4 5 5 20 20 2 10 2 2 2 2 10 10 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 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 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 -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 -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 -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 -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 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 1 1 1 linear of order 2 ρ9 2 2 0 0 -2 -2 0 0 -2 2 2 2 0 0 0 0 2 2 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 0 2 2 0 0 -2 -2 2 2 0 0 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 0 0 2 0 -1+√5/2 -1-√5/2 -2 -2 0 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 ρ12 2 2 -2 2 0 0 0 0 2 0 -1-√5/2 -1+√5/2 -2 -2 0 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 0 0 2 0 -1+√5/2 -1-√5/2 2 2 0 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 0 0 2 0 -1-√5/2 -1+√5/2 2 2 0 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 ρ15 2 2 2 2 0 0 0 0 2 0 -1+√5/2 -1-√5/2 2 2 0 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 ρ16 2 2 -2 -2 0 0 0 0 2 0 -1-√5/2 -1+√5/2 2 2 0 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 0 0 2 0 -1+√5/2 -1-√5/2 -2 -2 0 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 0 0 2 0 -1-√5/2 -1+√5/2 -2 -2 0 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 ρ19 2 -2 0 0 -2 2 0 0 0 0 2 2 -√2 √2 √2 -√2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ20 2 -2 0 0 -2 2 0 0 0 0 2 2 √2 -√2 -√2 √2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ21 2 -2 0 0 2 -2 0 0 0 0 2 2 √2 -√2 √2 -√2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ22 2 -2 0 0 2 -2 0 0 0 0 2 2 -√2 √2 -√2 √2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ23 4 4 0 0 0 0 0 0 -4 0 -1+√5 -1-√5 0 0 0 0 -1-√5 -1+√5 0 0 0 0 1+√5 1-√5 0 0 0 0 orthogonal lifted from D4×D5 ρ24 4 4 0 0 0 0 0 0 -4 0 -1-√5 -1+√5 0 0 0 0 -1+√5 -1-√5 0 0 0 0 1-√5 1+√5 0 0 0 0 orthogonal lifted from D4×D5 ρ25 4 -4 0 0 0 0 0 0 0 0 -1+√5 -1-√5 -2√2 2√2 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 orthogonal faithful ρ26 4 -4 0 0 0 0 0 0 0 0 -1-√5 -1+√5 2√2 -2√2 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 orthogonal faithful ρ27 4 -4 0 0 0 0 0 0 0 0 -1-√5 -1+√5 -2√2 2√2 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 orthogonal faithful ρ28 4 -4 0 0 0 0 0 0 0 0 -1+√5 -1-√5 2√2 -2√2 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 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)]])

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

 40 1 0 0 5 35 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 5 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 12 29 0 0 12 12
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 40
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

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