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G = C5⋊D20  order 200 = 23·52

The semidirect product of C5 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial

Aliases: C5⋊2D20, Dic5⋊D5, C52⋊3D4, D10⋊2D5, C10.4D10, C2.4D52, (D5×C10)⋊2C2, C5⋊1(C5⋊D4), (C5×Dic5)⋊1C2, (C5×C10).4C22, (C2×C5⋊D5)⋊1C2, SmallGroup(200,25)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5×C10 — C5⋊D20
Chief series
C1 — C5 — C52 — C5×C10 — D5×C10 — C5⋊D20
Lower central
C52 — C5×C10 — C5⋊D20
Upper central
C1 — C2

Generators and relations for C5⋊D20
 G = < a,b,c | a5=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >

C1
C2
10C2
50C2
C5
C5
2C5
2C5
5C4
5C22
25C22
C10
C10
2D5
2C10
2C10
10D5
10D5
10D5
10D5
10C10
10D5
10D5
C52
25D4
Dic5
D10
5D10
5C20
5D10
5C2×C10
10D10
10D10
C5×C10
2C5×D5
2C5⋊D5
5D20
5C5⋊D4
C5×Dic5
D5×C10
C2×C5⋊D5
C5⋊D20

Character table of C5⋊D20

 class 12A2B2C45A5B5C5D5E5F5G5H10A10B10C10D10E10F10G10H10I10J10K10L20A20B20C20D
 size 1110501022224444222244441010101010101010
ρ111111111111111111111111111111    trivial
ρ2111-1-111111111111111111111-1-1-1-1    linear of order 2
ρ311-11-11111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411-1-111111111111111111-1-1-1-11111    linear of order 2
ρ52-200022222222-2-2-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ6220022-1+√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/222-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/20000-1-√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ7220022-1-√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/222-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/20000-1+√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ822-200-1-√5/22-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/222-1+√5/2-1+√5/2-1-√5/2-1-√5/21-√5/21-√5/21+√5/21+√5/20000    orthogonal lifted from D10
ρ92200-22-1-√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/222-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/200001-√5/21+√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ102-20002-1-√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-2-21+√5/21-√5/21+√5/21-√5/21+√5/21-√5/20000ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52-ζ43ζ54+ζ43ζ5    orthogonal lifted from D20
ρ1122-200-1+√5/22-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/222-1-√5/2-1-√5/2-1+√5/2-1+√5/21+√5/21+√5/21-√5/21-√5/20000    orthogonal lifted from D10
ρ1222200-1-√5/22-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/222-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/20000    orthogonal lifted from D5
ρ132-20002-1+√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-2-21-√5/21+√5/21-√5/21+√5/21-√5/21+√5/20000-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5ζ4ζ53-ζ4ζ52    orthogonal lifted from D20
ρ142200-22-1+√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/222-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/200001+√5/21-√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ1522200-1+√5/22-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/222-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/20000    orthogonal lifted from D5
ρ162-20002-1+√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-2-21-√5/21+√5/21-√5/21+√5/21-√5/21+√5/20000ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5-ζ4ζ53+ζ4ζ52    orthogonal lifted from D20
ρ172-20002-1-√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-2-21+√5/21-√5/21+√5/21-√5/21+√5/21-√5/20000-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52ζ43ζ54-ζ43ζ5    orthogonal lifted from D20
ρ182-2000-1+√5/22-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/21+√5/21-√5/2-2-21+√5/21+√5/21-√5/21-√5/2ζ53-ζ52-ζ53+ζ52ζ54-ζ5-ζ54+ζ50000    complex lifted from C5⋊D4
ρ192-2000-1-√5/22-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/21-√5/21+√5/2-2-21-√5/21-√5/21+√5/21+√5/2-ζ54+ζ5ζ54-ζ5ζ53-ζ52-ζ53+ζ520000    complex lifted from C5⋊D4
ρ202-2000-1+√5/22-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/21+√5/21-√5/2-2-21+√5/21+√5/21-√5/21-√5/2-ζ53+ζ52ζ53-ζ52-ζ54+ζ5ζ54-ζ50000    complex lifted from C5⋊D4
ρ212-2000-1-√5/22-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/21-√5/21+√5/2-2-21-√5/21-√5/21+√5/21+√5/2ζ54-ζ5-ζ54+ζ5-ζ53+ζ52ζ53-ζ520000    complex lifted from C5⋊D4
ρ2244000-1+√5-1+√5-1-√5-1-√5-13-√5/2-13+√5/2-1-√5-1+√5-1+√5-1-√5-13+√5/23-√5/2-100000000    orthogonal lifted from D52
ρ234-4000-1-√5-1-√5-1+√5-1+√5-13+√5/2-13-√5/21-√51+√51+√51-√51-3+√5/2-3-√5/2100000000    orthogonal faithful
ρ244-4000-1+√5-1+√5-1-√5-1-√5-13-√5/2-13+√5/21+√51-√51-√51+√51-3-√5/2-3+√5/2100000000    orthogonal faithful
ρ254-4000-1-√5-1+√5-1+√5-1-√53-√5/2-13+√5/2-11-√51+√51-√51+√5-3+√5/211-3-√5/200000000    orthogonal faithful
ρ2644000-1-√5-1-√5-1+√5-1+√5-13+√5/2-13-√5/2-1+√5-1-√5-1-√5-1+√5-13-√5/23+√5/2-100000000    orthogonal lifted from D52
ρ2744000-1+√5-1-√5-1-√5-1+√53+√5/2-13-√5/2-1-1-√5-1+√5-1-√5-1+√53+√5/2-1-13-√5/200000000    orthogonal lifted from D52
ρ284-4000-1+√5-1-√5-1-√5-1+√53+√5/2-13-√5/2-11+√51-√51+√51-√5-3-√5/211-3+√5/200000000    orthogonal faithful
ρ2944000-1-√5-1+√5-1+√5-1-√53-√5/2-13+√5/2-1-1+√5-1-√5-1+√5-1-√53-√5/2-1-13+√5/200000000    orthogonal lifted from D52

Permutation representations of C5⋊D20
►On 20 points - transitive group 20T60
Generators in S20
(1 9 17 5 13)(2 14 6 18 10)(3 11 19 7 15)(4 16 8 20 12)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)

(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)
 
G:=sub<Sym(20)| (1,9,17,5,13)(2,14,6,18,10)(3,11,19,7,15)(4,16,8,20,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)>;
 
G:=Group( (1,9,17,5,13)(2,14,6,18,10)(3,11,19,7,15)(4,16,8,20,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19) );
 
G=PermutationGroup([[(1,9,17,5,13),(2,14,6,18,10),(3,11,19,7,15),(4,16,8,20,12)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19)]])
 
G:=TransitiveGroup(20,60);
 

C5⋊D20 is a maximal subgroup of   D20⋊D5  D10.9D10  Dic10⋊5D5  D5×D20  Dic5.D10  D5×C5⋊D4  D10⋊D10
C5⋊D20 is a maximal quotient of   C5⋊D40  C52⋊3SD16  C52⋊4SD16  C52⋊3Q16  D10⋊Dic5  C10.D20  Dic5⋊Dic5

Matrix representation of C5⋊D20 ►in GL4(𝔽41) generated by

64000
1000
0010
0001
,
183500
202300
00040
0016
,
40000
35100
0016
00040
G:=sub<GL(4,GF(41))| [6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[18,20,0,0,35,23,0,0,0,0,0,1,0,0,40,6],[40,35,0,0,0,1,0,0,0,0,1,0,0,0,6,40] >;
 

C5⋊D20 in GAP, Magma, Sage, TeX 

C_5\rtimes D_{20}
 
% in TeX
 
G:=Group("C5:D20");
 
// GroupNames label
 
G:=SmallGroup(200,25);
 
// by ID
 
G=gap.SmallGroup(200,25);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-5,-5,61,26,328,4004]);
 
// Polycyclic
 
G:=Group<a,b,c|a^5=b^20=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C5⋊D20 in TeX
Character table of C5⋊D20 in TeX

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