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## G = C2×C10order 20 = 22·5

### Abelian group of type [2,10]

Aliases: C2×C10, SmallGroup(20,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10
 Chief series C1 — C5 — C10 — C2×C10
 Lower central C1 — C2×C10
 Upper central C1 — C2×C10

Generators and relations for C2×C10
G = < a,b | a2=b10=1, ab=ba >

Character table of C2×C10

 class 1 2A 2B 2C 5A 5B 5C 5D 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 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 -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 -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 1 1 1 -1 linear of order 2 ρ5 1 1 1 1 ζ54 ζ5 ζ52 ζ53 ζ54 ζ54 ζ5 ζ5 ζ52 ζ53 ζ54 ζ52 ζ5 ζ52 ζ53 ζ53 linear of order 5 ρ6 1 -1 1 -1 ζ54 ζ5 ζ52 ζ53 -ζ54 ζ54 ζ5 -ζ5 -ζ52 -ζ53 -ζ54 ζ52 -ζ5 -ζ52 -ζ53 ζ53 linear of order 10 ρ7 1 1 -1 -1 ζ54 ζ5 ζ52 ζ53 -ζ54 -ζ54 -ζ5 ζ5 ζ52 ζ53 ζ54 -ζ52 -ζ5 -ζ52 -ζ53 -ζ53 linear of order 10 ρ8 1 -1 -1 1 ζ54 ζ5 ζ52 ζ53 ζ54 -ζ54 -ζ5 -ζ5 -ζ52 -ζ53 -ζ54 -ζ52 ζ5 ζ52 ζ53 -ζ53 linear of order 10 ρ9 1 1 1 1 ζ53 ζ52 ζ54 ζ5 ζ53 ζ53 ζ52 ζ52 ζ54 ζ5 ζ53 ζ54 ζ52 ζ54 ζ5 ζ5 linear of order 5 ρ10 1 -1 1 -1 ζ53 ζ52 ζ54 ζ5 -ζ53 ζ53 ζ52 -ζ52 -ζ54 -ζ5 -ζ53 ζ54 -ζ52 -ζ54 -ζ5 ζ5 linear of order 10 ρ11 1 1 -1 -1 ζ53 ζ52 ζ54 ζ5 -ζ53 -ζ53 -ζ52 ζ52 ζ54 ζ5 ζ53 -ζ54 -ζ52 -ζ54 -ζ5 -ζ5 linear of order 10 ρ12 1 -1 -1 1 ζ53 ζ52 ζ54 ζ5 ζ53 -ζ53 -ζ52 -ζ52 -ζ54 -ζ5 -ζ53 -ζ54 ζ52 ζ54 ζ5 -ζ5 linear of order 10 ρ13 1 1 1 1 ζ52 ζ53 ζ5 ζ54 ζ52 ζ52 ζ53 ζ53 ζ5 ζ54 ζ52 ζ5 ζ53 ζ5 ζ54 ζ54 linear of order 5 ρ14 1 -1 1 -1 ζ52 ζ53 ζ5 ζ54 -ζ52 ζ52 ζ53 -ζ53 -ζ5 -ζ54 -ζ52 ζ5 -ζ53 -ζ5 -ζ54 ζ54 linear of order 10 ρ15 1 1 -1 -1 ζ52 ζ53 ζ5 ζ54 -ζ52 -ζ52 -ζ53 ζ53 ζ5 ζ54 ζ52 -ζ5 -ζ53 -ζ5 -ζ54 -ζ54 linear of order 10 ρ16 1 -1 -1 1 ζ52 ζ53 ζ5 ζ54 ζ52 -ζ52 -ζ53 -ζ53 -ζ5 -ζ54 -ζ52 -ζ5 ζ53 ζ5 ζ54 -ζ54 linear of order 10 ρ17 1 1 1 1 ζ5 ζ54 ζ53 ζ52 ζ5 ζ5 ζ54 ζ54 ζ53 ζ52 ζ5 ζ53 ζ54 ζ53 ζ52 ζ52 linear of order 5 ρ18 1 -1 1 -1 ζ5 ζ54 ζ53 ζ52 -ζ5 ζ5 ζ54 -ζ54 -ζ53 -ζ52 -ζ5 ζ53 -ζ54 -ζ53 -ζ52 ζ52 linear of order 10 ρ19 1 1 -1 -1 ζ5 ζ54 ζ53 ζ52 -ζ5 -ζ5 -ζ54 ζ54 ζ53 ζ52 ζ5 -ζ53 -ζ54 -ζ53 -ζ52 -ζ52 linear of order 10 ρ20 1 -1 -1 1 ζ5 ζ54 ζ53 ζ52 ζ5 -ζ5 -ζ54 -ζ54 -ζ53 -ζ52 -ζ5 -ζ53 ζ54 ζ53 ζ52 -ζ52 linear of order 10

Permutation representations of C2×C10
Regular action on 20 points - transitive group 20T3
Generators in S20
(1 17)(2 18)(3 19)(4 20)(5 11)(6 12)(7 13)(8 14)(9 15)(10 16)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)

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

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

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

G:=TransitiveGroup(20,3);

C2×C10 is a maximal subgroup of   C5⋊D4  C22.58C24⋊C5

Polynomial with Galois group C2×C10 over ℚ
actionf(x)Disc(f)
20T3x20-x19+x17-x16+x14-x13+x11-x10+x9-x7+x6-x4+x3-x+1310·1118

Matrix representation of C2×C10 in GL2(𝔽11) generated by

 10 0 0 10
,
 10 0 0 9
G:=sub<GL(2,GF(11))| [10,0,0,10],[10,0,0,9] >;

C2×C10 in GAP, Magma, Sage, TeX

C_2\times C_{10}
% in TeX

G:=Group("C2xC10");
// GroupNames label

G:=SmallGroup(20,5);
// by ID

G=gap.SmallGroup(20,5);
# by ID

G:=PCGroup([3,-2,-2,-5]);
// Polycyclic

G:=Group<a,b|a^2=b^10=1,a*b=b*a>;
// generators/relations

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