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G = C5⋊D4order 40 = 23·5

The semidirect product of C5 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52D4, C22⋊D5, Dic5⋊C2, D102C2, C2.5D10, C10.5C22, (C2×C10)⋊2C2, SmallGroup(40,8)

Series: Derived Chief Lower central Upper central

C1C10 — C5⋊D4
C1C5C10D10 — C5⋊D4
C5C10 — C5⋊D4
C1C2C22

Generators and relations for C5⋊D4
 G = < a,b,c | a5=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
10C2
5C4
5C22
2D5
2C10
5D4

Character table of C5⋊D4

 class 12A2B2C45A5B10A10B10C10D10E10F
 size 112101022222222
ρ11111111111111    trivial
ρ2111-1-111111111    linear of order 2
ρ311-11-111-1-1-1-111    linear of order 2
ρ411-1-1111-1-1-1-111    linear of order 2
ρ52-2000220000-2-2    orthogonal lifted from D4
ρ622200-1+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
ρ722-200-1+5/2-1-5/21-5/21+5/21+5/21-5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ822-200-1-5/2-1+5/21+5/21-5/21-5/21+5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ922200-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ102-2000-1-5/2-1+5/2ζ5352545ζ54553521-5/21+5/2    complex faithful
ρ112-2000-1-5/2-1+5/25352ζ545545ζ53521-5/21+5/2    complex faithful
ρ122-2000-1+5/2-1-5/2ζ545ζ535253525451+5/21-5/2    complex faithful
ρ132-2000-1+5/2-1-5/25455352ζ5352ζ5451+5/21-5/2    complex faithful

Permutation representations of C5⋊D4
On 20 points - transitive group 20T7
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 14 9 19)(2 13 10 18)(3 12 6 17)(4 11 7 16)(5 15 8 20)
(2 5)(3 4)(6 7)(8 10)(11 17)(12 16)(13 20)(14 19)(15 18)

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

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,14,9,19)(2,13,10,18)(3,12,6,17)(4,11,7,16)(5,15,8,20), (2,5)(3,4)(6,7)(8,10)(11,17)(12,16)(13,20)(14,19)(15,18) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,14,9,19),(2,13,10,18),(3,12,6,17),(4,11,7,16),(5,15,8,20)], [(2,5),(3,4),(6,7),(8,10),(11,17),(12,16),(13,20),(14,19),(15,18)])

G:=TransitiveGroup(20,7);

On 20 points - transitive group 20T11
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 16 9 11)(2 20 10 15)(3 19 6 14)(4 18 7 13)(5 17 8 12)
(1 11)(2 15)(3 14)(4 13)(5 12)(6 19)(7 18)(8 17)(9 16)(10 20)

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

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16,9,11)(2,20,10,15)(3,19,6,14)(4,18,7,13)(5,17,8,12), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,16,9,11),(2,20,10,15),(3,19,6,14),(4,18,7,13),(5,17,8,12)], [(1,11),(2,15),(3,14),(4,13),(5,12),(6,19),(7,18),(8,17),(9,16),(10,20)])

G:=TransitiveGroup(20,11);

Matrix representation of C5⋊D4 in GL2(𝔽11) generated by

58
82
,
82
63
,
11
010
G:=sub<GL(2,GF(11))| [5,8,8,2],[8,6,2,3],[1,0,1,10] >;

C5⋊D4 in GAP, Magma, Sage, TeX

C_5\rtimes D_4
% in TeX

G:=Group("C5:D4");
// GroupNames label

G:=SmallGroup(40,8);
// by ID

G=gap.SmallGroup(40,8);
# by ID

G:=PCGroup([4,-2,-2,-2,-5,49,515]);
// Polycyclic

G:=Group<a,b,c|a^5=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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