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## G = C22×D5order 40 = 23·5

### Direct product of C22 and D5

Aliases: C22×D5, C5⋊C23, C10⋊C22, (C2×C10)⋊3C2, SmallGroup(40,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C22×D5
 Chief series C1 — C5 — D5 — D10 — C22×D5
 Lower central C5 — C22×D5
 Upper central C1 — C22

Generators and relations for C22×D5
G = < a,b,c,d | a2=b2=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of C22×D5

 class 1 2A 2B 2C 2D 2E 2F 2G 5A 5B 10A 10B 10C 10D 10E 10F size 1 1 1 1 5 5 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 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ6 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ7 1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 -1 1 -1 linear of order 2 ρ8 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ9 2 -2 2 -2 0 0 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 ρ10 2 -2 -2 2 0 0 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 -2 -2 0 0 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 2 2 0 0 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 2 2 0 0 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 ρ14 2 -2 -2 2 0 0 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 ρ15 2 -2 2 -2 0 0 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 ρ16 2 2 -2 -2 0 0 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

Permutation representations of C22×D5
On 20 points - transitive group 20T8
Generators in S20
(1 19)(2 20)(3 16)(4 17)(5 18)(6 11)(7 12)(8 13)(9 14)(10 15)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 13)(2 12)(3 11)(4 15)(5 14)(6 16)(7 20)(8 19)(9 18)(10 17)

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

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

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

G:=TransitiveGroup(20,8);

C22×D5 is a maximal subgroup of   D10⋊C4  C22⋊F5
C22×D5 is a maximal quotient of   C4○D20  D42D5  Q82D5

Matrix representation of C22×D5 in GL3(𝔽11) generated by

 10 0 0 0 10 0 0 0 10
,
 10 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 1 0 10 7
,
 1 0 0 0 0 10 0 10 0
G:=sub<GL(3,GF(11))| [10,0,0,0,10,0,0,0,10],[10,0,0,0,1,0,0,0,1],[1,0,0,0,0,10,0,1,7],[1,0,0,0,0,10,0,10,0] >;

C22×D5 in GAP, Magma, Sage, TeX

C_2^2\times D_5
% in TeX

G:=Group("C2^2xD5");
// GroupNames label

G:=SmallGroup(40,13);
// by ID

G=gap.SmallGroup(40,13);
# by ID

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

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

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