direct product, abelian, monomial, 2-elementary
Aliases: C2×C10, SmallGroup(20,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C10 |
| C1 — C2×C10 |
| 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 |
►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
| action | f(x) | Disc(f) |
|---|---|---|
| 20T3 | x20-x19+x17-x16+x14-x13+x11-x10+x9-x7+x6-x4+x3-x+1 | 310·1118 |
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
Export
Subgroup lattice of C2×C10 in TeX
Character table of C2×C10 in TeX