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## G = D5×SD16order 160 = 25·5

### Direct product of D5 and SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D5×SD16
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D4×D5 — D5×SD16
 Lower central C5 — C10 — C20 — D5×SD16
 Upper central C1 — C2 — C4 — SD16

Generators and relations for D5×SD16
G = < a,b,c,d | a5=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 264 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2 [×4], C4, C4 [×3], C22 [×5], C5, C8, C8, C2×C4 [×2], D4, D4 [×2], Q8, Q8 [×2], C23, D5 [×2], D5, C10, C10, C2×C8, SD16, SD16 [×3], C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10 [×3], C2×C10, C2×SD16, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, D5×SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, D10 [×3], C2×SD16, C22×D5, D4×D5, D5×SD16

Character table of D5×SD16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 20A 20B 20C 20D 40A 40B 40C 40D size 1 1 4 5 5 20 2 4 10 20 2 2 2 2 10 10 2 2 8 8 4 4 8 8 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 2 2 0 -2 0 -2 0 2 2 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 -2 -2 0 -2 0 2 0 2 2 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 0 0 2 2 0 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 0 0 0 2 -2 0 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 0 0 0 2 2 0 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 ρ14 2 2 2 0 0 0 2 -2 0 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 ρ15 2 2 2 0 0 0 2 2 0 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 0 0 0 2 -2 0 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 0 0 0 2 2 0 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 0 0 0 2 -2 0 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 2 -2 0 0 0 0 0 2 2 -√-2 √-2 √-2 -√-2 -2 -2 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ20 2 -2 0 -2 2 0 0 0 0 0 2 2 √-2 -√-2 √-2 -√-2 -2 -2 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ21 2 -2 0 -2 2 0 0 0 0 0 2 2 -√-2 √-2 -√-2 √-2 -2 -2 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ22 2 -2 0 2 -2 0 0 0 0 0 2 2 √-2 -√-2 -√-2 √-2 -2 -2 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ23 4 4 0 0 0 0 -4 0 0 0 -1-√5 -1+√5 0 0 0 0 -1+√5 -1-√5 0 0 1+√5 1-√5 0 0 0 0 0 0 orthogonal lifted from D4×D5 ρ24 4 4 0 0 0 0 -4 0 0 0 -1+√5 -1-√5 0 0 0 0 -1-√5 -1+√5 0 0 1-√5 1+√5 0 0 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 ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5 ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ87ζ53+ζ87ζ52+ζ85ζ53+ζ85ζ52 complex 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 ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5 ζ87ζ54+ζ87ζ5+ζ85ζ54+ζ85ζ5 ζ87ζ53+ζ87ζ52+ζ85ζ53+ζ85ζ52 ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52 complex 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 ζ87ζ53+ζ87ζ52+ζ85ζ53+ζ85ζ52 ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5 ζ87ζ54+ζ87ζ5+ζ85ζ54+ζ85ζ5 complex 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 ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ87ζ53+ζ87ζ52+ζ85ζ53+ζ85ζ52 ζ87ζ54+ζ87ζ5+ζ85ζ54+ζ85ζ5 ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5 complex faithful

Smallest permutation representation of D5×SD16
On 40 points
Generators in S40
(1 35 27 20 14)(2 36 28 21 15)(3 37 29 22 16)(4 38 30 23 9)(5 39 31 24 10)(6 40 32 17 11)(7 33 25 18 12)(8 34 26 19 13)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(25 29)(26 30)(27 31)(28 32)
(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 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 19)(18 22)(21 23)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)

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

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

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

Matrix representation of D5×SD16 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 6 1 0 0 40 0
,
 40 0 0 0 0 40 0 0 0 0 1 6 0 0 0 40
,
 26 26 0 0 15 26 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,6,40,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,6,40],[26,15,0,0,26,26,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1] >;

D5×SD16 in GAP, Magma, Sage, TeX

D_5\times {\rm SD}_{16}
% in TeX

G:=Group("D5xSD16");
// GroupNames label

G:=SmallGroup(160,134);
// by ID

G=gap.SmallGroup(160,134);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,116,86,297,159,69,4613]);
// 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^3>;
// generators/relations

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