direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×F5, C10⋊C4, D5⋊C4, D10.C2, D5.C22, C5⋊(C2×C4), Aut(D10), Hol(C10), SmallGroup(40,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C2×F5 |
Generators and relations for C2×F5
G = < a,b,c | a2=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Character table of C2×F5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 10 | |
| size | 1 | 1 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 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 | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | -i | -i | i | i | 1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -i | i | -i | i | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | i | -i | i | -i | 1 | 1 | linear of order 4 |
| ρ9 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | orthogonal faithful |
| ρ10 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from F5 |
►On 10 points - transitive group 10T5
Generators in S10
(1 6)(2 7)(3 8)(4 9)(5 10)
(1 2 3 4 5)(6 7 8 9 10)
(1 6)(2 8 5 9)(3 10 4 7)
G:=sub<Sym(10)| (1,6)(2,7)(3,8)(4,9)(5,10), (1,2,3,4,5)(6,7,8,9,10), (1,6)(2,8,5,9)(3,10,4,7)>;
G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10), (1,2,3,4,5)(6,7,8,9,10), (1,6)(2,8,5,9)(3,10,4,7) );
G=PermutationGroup([[(1,6),(2,7),(3,8),(4,9),(5,10)], [(1,2,3,4,5),(6,7,8,9,10)], [(1,6),(2,8,5,9),(3,10,4,7)]])
G:=TransitiveGroup(10,5);
►On 20 points - transitive group 20T9
Generators in S20
(1 9)(2 10)(3 6)(4 7)(5 8)(11 20)(12 16)(13 17)(14 18)(15 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 13 9 17)(2 15 8 20)(3 12 7 18)(4 14 6 16)(5 11 10 19)
G:=sub<Sym(20)| (1,9)(2,10)(3,6)(4,7)(5,8)(11,20)(12,16)(13,17)(14,18)(15,19), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,13,9,17)(2,15,8,20)(3,12,7,18)(4,14,6,16)(5,11,10,19)>;
G:=Group( (1,9)(2,10)(3,6)(4,7)(5,8)(11,20)(12,16)(13,17)(14,18)(15,19), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,13,9,17)(2,15,8,20)(3,12,7,18)(4,14,6,16)(5,11,10,19) );
G=PermutationGroup([[(1,9),(2,10),(3,6),(4,7),(5,8),(11,20),(12,16),(13,17),(14,18),(15,19)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,13,9,17),(2,15,8,20),(3,12,7,18),(4,14,6,16),(5,11,10,19)]])
G:=TransitiveGroup(20,9);
►On 20 points - transitive group 20T13
Generators in S20
(1 6)(2 7)(3 8)(4 9)(5 10)(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 18)(2 20 5 16)(3 17 4 19)(6 13)(7 15 10 11)(8 12 9 14)
G:=sub<Sym(20)| (1,6)(2,7)(3,8)(4,9)(5,10)(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), (1,18)(2,20,5,16)(3,17,4,19)(6,13)(7,15,10,11)(8,12,9,14)>;
G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(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), (1,18)(2,20,5,16)(3,17,4,19)(6,13)(7,15,10,11)(8,12,9,14) );
G=PermutationGroup([[(1,6),(2,7),(3,8),(4,9),(5,10),(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)], [(1,18),(2,20,5,16),(3,17,4,19),(6,13),(7,15,10,11),(8,12,9,14)]])
G:=TransitiveGroup(20,13);
C2×F5 is a maximal subgroup of
C4⋊F5 C22⋊F5 D5⋊F5 A5⋊C4 C32⋊F5⋊C2
C2×F5 is a maximal quotient of D5⋊C8 C4.F5 C4⋊F5 C22.F5 C22⋊F5 D5⋊F5 C32⋊F5⋊C2
| action | f(x) | Disc(f) |
|---|---|---|
| 10T5 | x10+2x9-23x8-74x7+62x6+542x5+871x4+608x3+192x2+24x+1 | 215·76·136·232 |
Matrix representation of C2×F5 ►in GL4(ℤ) generated by
| -1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| -1 | -1 | -1 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| -1 | -1 | -1 | -1 |
G:=sub<GL(4,Integers())| [-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[-1,1,0,0,-1,0,1,0,-1,0,0,1,-1,0,0,0],[1,0,0,-1,0,0,1,-1,0,0,0,-1,0,1,0,-1] >;
C2×F5 in GAP, Magma, Sage, TeX
C_2\times F_5
% in TeX
G:=Group("C2xF5"); // GroupNames label
G:=SmallGroup(40,12);
// by ID
G=gap.SmallGroup(40,12);
# by ID
G:=PCGroup([4,-2,-2,-2,-5,16,259,139]);
// Polycyclic
G:=Group<a,b,c|a^2=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C2×F5 in TeX
Character table of C2×F5 in TeX