direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D5, C20⋊2C2, C4○Dic5, C2.1D10, Dic5⋊2C2, D10.2C2, C10.2C22, C5⋊2(C2×C4), SmallGroup(40,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C4×D5 |
Generators and relations for C4×D5
G = < a,b,c | a4=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C4×D5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 10A | 10B | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | 1 | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | i | -i | i | -i | 1 | 1 | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | -i | i | -i | i | 1 | 1 | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | -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 |
| ρ10 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | -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 D10 |
| ρ11 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | -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 D10 |
| ρ12 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | -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 |
| ρ13 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | complex faithful |
| ρ14 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | complex faithful |
| ρ15 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | complex faithful |
| ρ16 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | complex faithful |
►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);
C4×D5 is a maximal subgroup of
C8⋊D5 D5⋊C8 C4.F5 C4⋊F5 C4○D20 D4⋊2D5 Q8⋊2D5 D30.C2 Dic5⋊2D5 C4.A5 D70.C2 D55⋊2C4
C4×D5 is a maximal quotient of
C8⋊D5 C10.D4 D10⋊C4 D30.C2 Dic5⋊2D5 D70.C2 D55⋊2C4
Matrix representation of C4×D5 ►in GL2(𝔽29) generated by
| 17 | 0 |
| 0 | 17 |
| 28 | 15 |
| 15 | 6 |
| 6 | 12 |
| 14 | 23 |
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
Export
Subgroup lattice of C4×D5 in TeX
Character table of C4×D5 in TeX