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G = D5×SD16order 160 = 25·5

Direct product of D5 and SD16

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

Aliases: D5×SD16, C85D10, Q81D10, C405C22, D4.2D10, D10.24D4, C20.4C23, Dic5.8D4, D20.2C22, Dic102C22, (C8×D5)⋊4C2, Q8⋊D51C2, (Q8×D5)⋊1C2, C52(C2×SD16), C40⋊C25C2, D4.D53C2, (D4×D5).1C2, C2.18(D4×D5), C52C86C22, (C5×SD16)⋊3C2, C10.30(C2×D4), (C5×Q8)⋊1C22, C4.4(C22×D5), (C5×D4).2C22, (C4×D5).17C22, SmallGroup(160,134)

Series: Derived Chief Lower central Upper central

C1C20 — D5×SD16
C1C5C10C20C4×D5D4×D5 — D5×SD16
C5C10C20 — D5×SD16
C1C2C4SD16

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 12A2B2C2D2E4A4B4C4D5A5B8A8B8C8D10A10B10C10D20A20B20C20D40A40B40C40D
 size 114552024102022221010228844884444
ρ11111111111111111111111111111    trivial
ρ21111111-11-111-1-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ311-111-1111111-1-1-1-111-1-11111-1-1-1-1    linear of order 2
ρ411-111-11-11-111111111-1-111-1-11111    linear of order 2
ρ5111-1-1-111-1-11111-1-1111111111111    linear of order 2
ρ6111-1-1-11-1-1111-1-111111111-1-1-1-1-1-1    linear of order 2
ρ711-1-1-1111-1-111-1-11111-1-11111-1-1-1-1    linear of order 2
ρ811-1-1-111-1-111111-1-111-1-111-1-11111    linear of order 2
ρ9220220-20-202200002200-2-2000000    orthogonal lifted from D4
ρ10220-2-20-20202200002200-2-2000000    orthogonal lifted from D4
ρ1122-20002200-1+5/2-1-5/2-2-200-1-5/2-1+5/21-5/21+5/2-1+5/2-1-5/2-1-5/2-1+5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ1222-20002-200-1-5/2-1+5/22200-1+5/2-1-5/21+5/21-5/2-1-5/2-1+5/21-5/21+5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D10
ρ132220002200-1+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
ρ142220002-200-1-5/2-1+5/2-2-200-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1+5/21-5/21+5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ152220002200-1-5/2-1+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
ρ162220002-200-1+5/2-1-5/2-2-200-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/21+5/21-5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ1722-20002200-1-5/2-1+5/2-2-200-1+5/2-1-5/21+5/21-5/2-1-5/2-1+5/2-1+5/2-1-5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ1822-20002-200-1+5/2-1-5/22200-1-5/2-1+5/21-5/21+5/2-1+5/2-1-5/21+5/21-5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D10
ρ192-202-20000022--2-2-2--2-2-2000000-2--2--2-2    complex lifted from SD16
ρ202-20-220000022-2--2-2--2-2-2000000--2-2-2--2    complex lifted from SD16
ρ212-20-220000022--2-2--2-2-2-2000000-2--2--2-2    complex lifted from SD16
ρ222-202-20000022-2--2--2-2-2-2000000--2-2-2--2    complex lifted from SD16
ρ23440000-4000-1-5-1+50000-1+5-1-5001+51-5000000    orthogonal lifted from D4×D5
ρ24440000-4000-1+5-1-50000-1-5-1+5001-51+5000000    orthogonal lifted from D4×D5
ρ254-400000000-1+5-1-52-2-2-2001+51-5000000ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ5ζ83ζ5383ζ528ζ538ζ52ζ87ζ5387ζ5285ζ5385ζ52    complex faithful
ρ264-400000000-1+5-1-5-2-22-2001+51-5000000ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ5ζ87ζ5387ζ5285ζ5385ζ52ζ83ζ5383ζ528ζ538ζ52    complex faithful
ρ274-400000000-1-5-1+52-2-2-2001-51+5000000ζ87ζ5387ζ5285ζ5385ζ52ζ83ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ5    complex faithful
ρ284-400000000-1-5-1+5-2-22-2001-51+5000000ζ83ζ5383ζ528ζ538ζ52ζ87ζ5387ζ5285ζ5385ζ52ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ5    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)])

D5×SD16 is a maximal subgroup of
SD16⋊F5  D20.29D4  D811D10  D86D10  C40.C23  C4014D6  Dic10⋊D6  D15⋊SD16
D5×SD16 is a maximal quotient of
Dic56SD16  Dic5.5D8  D4⋊Dic10  Dic102D4  D20.8D4  D10.16SD16  D10⋊SD16  Dic57SD16  Q8⋊Dic10  Dic5.3Q16  D10.11SD16  Q82D20  D102SD16  Dic5⋊SD16  Dic58SD16  Dic10⋊Q8  C405Q8  D10.12SD16  D10.17SD16  C88D20  D20⋊Q8  Dic53SD16  Dic55SD16  D106SD16  D108SD16  C4014D4  C4015D4  C4014D6  Dic10⋊D6  D15⋊SD16

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

1000
0100
0061
00400
,
40000
04000
0016
00040
,
262600
152600
00400
00040
,
1000
04000
0010
0001
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

Export

Character table of D5×SD16 in TeX

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