Copied to
clipboard

G = D5×SD16  order 160 = 25·5

Direct product of D5 and SD16

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

Aliases: D5×SD16, C8⋊5D10, Q8⋊1D10, C40⋊5C22, D4.2D10, D10.24D4, C20.4C23, Dic5.8D4, D20.2C22, Dic10⋊2C22, (C8×D5)⋊4C2, Q8⋊D5⋊1C2, (Q8×D5)⋊1C2, C5⋊2(C2×SD16), C40⋊C2⋊5C2, D4.D5⋊3C2, (D4×D5).1C2, C2.18(D4×D5), C5⋊2C8⋊6C22, (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

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, C4, C4, C22, C5, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C2×C8, SD16, SD16, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×SD16, C5⋊2C8, 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, C22, D4, C23, D5, SD16, C2×D4, D10, 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-√5000000ζ87ζ54+ζ87ζ5+ζ85ζ54+ζ85ζ5ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52ζ87ζ53+ζ87ζ52+ζ85ζ53+ζ85ζ52    complex faithful
ρ264-400000000-1+√5-1-√5-2√-22√-2001+√51-√5000000ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5ζ87ζ54+ζ87ζ5+ζ85ζ54+ζ85ζ5ζ87ζ53+ζ87ζ52+ζ85ζ53+ζ85ζ52ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52    complex faithful
ρ274-400000000-1-√5-1+√52√-2-2√-2001-√51+√5000000ζ87ζ53+ζ87ζ52+ζ85ζ53+ζ85ζ52ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5ζ87ζ54+ζ87ζ5+ζ85ζ54+ζ85ζ5    complex faithful
ρ284-400000000-1-√5-1+√5-2√-22√-2001-√51+√5000000ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52ζ87ζ53+ζ87ζ52+ζ85ζ53+ζ85ζ52ζ87ζ54+ζ87ζ5+ζ85ζ54+ζ85ζ5ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5    complex faithful

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

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

D5×SD16 is a maximal subgroup of
 SD16⋊F5  D20.29D4  D8⋊11D10  D8⋊6D10  C40.C23  C40⋊14D6  Dic10⋊D6  D15⋊SD16
D5×SD16 is a maximal quotient of
 Dic5⋊6SD16  Dic5.5D8  D4⋊Dic10  Dic10⋊2D4  D20.8D4  D10.16SD16  D10⋊SD16  Dic5⋊7SD16  Q8⋊Dic10  Dic5.3Q16  D10.11SD16  Q8⋊2D20  D10⋊2SD16  Dic5⋊SD16  Dic5⋊8SD16  Dic10⋊Q8  C40⋊5Q8  D10.12SD16  D10.17SD16  C8⋊8D20  D20⋊Q8  Dic5⋊3SD16  Dic5⋊5SD16  D10⋊6SD16  D10⋊8SD16  C40⋊14D4  C40⋊15D4  C40⋊14D6  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

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁