Copied to
clipboard

G = D40⋊C2order 160 = 25·5

6th semidirect product of D40 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C83D10, D406C2, Q82D10, C403C22, SD161D5, D4.3D10, D10.15D4, D202C22, C20.5C23, Dic5.17D4, D4⋊D53C2, (D4×D5)⋊3C2, Q8⋊D52C2, C8⋊D51C2, C53(C8⋊C22), C2.19(D4×D5), C52C82C22, Q82D51C2, (C5×SD16)⋊1C2, C10.31(C2×D4), (C5×Q8)⋊2C22, C4.5(C22×D5), (C4×D5).2C22, (C5×D4).3C22, SmallGroup(160,135)

Series: Derived Chief Lower central Upper central

C1C20 — D40⋊C2
C1C5C10C20C4×D5D4×D5 — D40⋊C2
C5C10C20 — D40⋊C2
C1C2C4SD16

Generators and relations for D40⋊C2
 G = < a,b,c | a40=b2=c2=1, bab=a-1, cac=a11, bc=cb >

Subgroups: 288 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×4], C4, C4 [×2], C22 [×6], C5, C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×3], C10, C10, M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, D10 [×4], C2×C10, C8⋊C22, C52C8, C40, C4×D5, C4×D5, D20 [×2], D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, D40⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], C8⋊C22, C22×D5, D4×D5, D40⋊C2

Character table of D40⋊C2

 class 12A2B2C2D2E4A4B4C5A5B8A8B10A10B10C10D20A20B20C20D40A40B40C40D
 size 114102020241022420228844884444
ρ11111111111111111111111111    trivial
ρ2111-1-111-1-111-11111111-1-1-1-1-1-1    linear of order 2
ρ311-1-1111-1-1111-111-1-111-1-11111    linear of order 2
ρ411-11-1111111-1-111-1-11111-1-1-1-1    linear of order 2
ρ511-11-1-11-11111111-1-111-1-11111    linear of order 2
ρ611-1-11-111-111-1111-1-11111-1-1-1-1    linear of order 2
ρ7111-1-1-111-1111-1111111111111    linear of order 2
ρ811111-11-1111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ9220-200-20222002200-2-2000000    orthogonal lifted from D4
ρ10220200-20-222002200-2-2000000    orthogonal lifted from D4
ρ1122-20002-20-1+5/2-1-5/220-1+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
ρ12222000220-1+5/2-1-5/220-1+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
ρ1322-2000220-1-5/2-1+5/2-20-1-5/2-1+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
ρ142220002-20-1-5/2-1+5/2-20-1-5/2-1+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
ρ152220002-20-1+5/2-1-5/2-20-1+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
ρ1622-2000220-1+5/2-1-5/2-20-1+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
ρ17222000220-1-5/2-1+5/220-1-5/2-1+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
ρ1822-20002-20-1-5/2-1+5/220-1-5/2-1+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
ρ194-400000004400-4-40000000000    orthogonal lifted from C8⋊C22
ρ20440000-400-1-5-1+500-1-5-1+5001-51+5000000    orthogonal lifted from D4×D5
ρ21440000-400-1+5-1-500-1+5-1-5001+51-5000000    orthogonal lifted from D4×D5
ρ224-40000000-1+5-1-5001-51+500000083ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ5287ζ5487ζ585ζ5485ζ583ζ5483ζ58ζ548ζ5    orthogonal faithful
ρ234-40000000-1+5-1-5001-51+5000000ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ5283ζ5483ζ58ζ548ζ587ζ5487ζ585ζ5485ζ5    orthogonal faithful
ρ244-40000000-1-5-1+5001+51-500000087ζ5487ζ585ζ5485ζ583ζ5483ζ58ζ548ζ5ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52    orthogonal faithful
ρ254-40000000-1-5-1+5001+51-500000083ζ5483ζ58ζ548ζ587ζ5487ζ585ζ5485ζ583ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52    orthogonal faithful

Smallest permutation representation of D40⋊C2
On 40 points
Generators in S40
(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)
(1 5)(2 4)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)
(2 12)(3 23)(4 34)(6 16)(7 27)(8 38)(10 20)(11 31)(14 24)(15 35)(18 28)(19 39)(22 32)(26 36)(30 40)

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

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

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

D40⋊C2 is a maximal subgroup of
D20.29D4  Q16⋊D10  D815D10  D5×C8⋊C22  D85D10  D40⋊C22  C40.C23  C24⋊D10  C408D6  D20.9D6  Dic6⋊D10  D20⋊D6  D60⋊C22  Q83D30  GL2(𝔽3)⋊D5
D40⋊C2 is a maximal quotient of
C4⋊C4.D10  D4.2Dic10  (D4×D5)⋊C4  D4⋊D20  C5⋊(C82D4)  C405C4⋊C2  D4⋊D56C4  D203D4  Dic5.9Q16  Q8⋊C4⋊D5  Q8⋊(C4×D5)  D10.7Q16  D204D4  C5⋊(C8⋊D4)  Q8⋊D56C4  D20.12D4  C403Q8  C8⋊(C4×D5)  D10.17SD16  C82D20  C4.Q8⋊D5  D4015C4  D20⋊Q8  D20.Q8  Dic55SD16  SD16⋊Dic5  (C5×D4).D4  D106SD16  D207D4  C408D4  C409D4  C24⋊D10  C408D6  D20.9D6  Dic6⋊D10  D20⋊D6  D60⋊C22  Q83D30

Matrix representation of D40⋊C2 in GL6(𝔽41)

40350000
6350000
000001
000010
0040000
000100
,
100000
35400000
0040000
000100
000001
000010
,
4000000
0400000
001000
0004000
000001
000010

G:=sub<GL(6,GF(41))| [40,6,0,0,0,0,35,35,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D40⋊C2 in GAP, Magma, Sage, TeX

D_{40}\rtimes C_2
% in TeX

G:=Group("D40:C2");
// GroupNames label

G:=SmallGroup(160,135);
// by ID

G=gap.SmallGroup(160,135);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,362,116,86,297,159,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^11,b*c=c*b>;
// generators/relations

Export

Character table of D40⋊C2 in TeX

׿
×
𝔽