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G = C4×D5order 40 = 23·5

Direct product of C4 and D5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D5, C202C2, C4Dic5, C2.1D10, Dic52C2, D10.2C2, C10.2C22, C52(C2×C4), SmallGroup(40,5)

Series: Derived Chief Lower central Upper central

C1C5 — C4×D5
C1C5C10D10 — C4×D5
C5 — C4×D5
C1C4

Generators and relations for C4×D5
 G = < a,b,c | a4=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
5C22
5C4
5C2×C4

Character table of C4×D5

 class 12A2B2C4A4B4C4D5A5B10A10B20A20B20C20D
 size 1155115522222222
ρ11111111111111111    trivial
ρ21111-1-1-1-11111-1-1-1-1    linear of order 2
ρ311-1-1-1-1111111-1-1-1-1    linear of order 2
ρ411-1-111-1-111111111    linear of order 2
ρ51-1-11i-i-ii11-1-1i-i-ii    linear of order 4
ρ61-11-1i-ii-i11-1-1i-i-ii    linear of order 4
ρ71-11-1-ii-ii11-1-1-iii-i    linear of order 4
ρ81-1-11-iii-i11-1-1-iii-i    linear of order 4
ρ922002200-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ102200-2-200-1-5/2-1+5/2-1+5/2-1-5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ112200-2-200-1+5/2-1-5/2-1-5/2-1+5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ1222002200-1+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
ρ132-2002i-2i00-1+5/2-1-5/21+5/21-5/2ζ4ζ544ζ5ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ534ζ52    complex faithful
ρ142-2002i-2i00-1-5/2-1+5/21-5/21+5/2ζ4ζ534ζ52ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ544ζ5    complex faithful
ρ152-200-2i2i00-1+5/2-1-5/21+5/21-5/2ζ43ζ5443ζ5ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5343ζ52    complex faithful
ρ162-200-2i2i00-1-5/2-1+5/21-5/21+5/2ζ43ζ5343ζ52ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5443ζ5    complex faithful

Permutation representations of C4×D5
On 20 points - transitive group 20T6
Generators in S20
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)

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

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

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

G:=TransitiveGroup(20,6);

Matrix representation of C4×D5 in GL2(𝔽29) generated by

170
017
,
2815
156
,
612
1423
G:=sub<GL(2,GF(29))| [17,0,0,17],[28,15,15,6],[6,14,12,23] >;

C4×D5 in GAP, Magma, Sage, TeX

C_4\times D_5
% in TeX

G:=Group("C4xD5");
// GroupNames label

G:=SmallGroup(40,5);
// by ID

G=gap.SmallGroup(40,5);
# by ID

G:=PCGroup([4,-2,-2,-2,-5,21,515]);
// Polycyclic

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

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