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## G = C4×D5order 40 = 23·5

### Direct product of C4 and D5

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

 Derived series C1 — C5 — C4×D5
 Chief series C1 — C5 — C10 — D10 — C4×D5
 Lower central C5 — C4×D5
 Upper central C1 — C4

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

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);

C4×D5 is a maximal subgroup of
C8⋊D5  D5⋊C8  C4.F5  C4⋊F5  C4○D20  D42D5  Q82D5  D30.C2  Dic52D5  C4.A5  D70.C2  D552C4
C4×D5 is a maximal quotient of
C8⋊D5  C10.D4  D10⋊C4  D30.C2  Dic52D5  D70.C2  D552C4

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

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