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## G = C2×F5order 40 = 23·5

### Direct product of C2 and F5

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

 Derived series C1 — C5 — C2×F5
 Chief series C1 — C5 — D5 — F5 — C2×F5
 Lower central C5 — C2×F5
 Upper central C1 — C2

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

Permutation representations of C2×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

Polynomial with Galois group C2×F5 over ℚ
actionf(x)Disc(f)
10T5x10+2x9-23x8-74x7+62x6+542x5+871x4+608x3+192x2+24x+1215·76·136·232

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

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