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## G = C3×F5order 60 = 22·3·5

### Direct product of C3 and F5

Aliases: C3×F5, C5⋊C12, C152C4, D5.C6, (C3×D5).2C2, SmallGroup(60,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C3×F5
 Chief series C1 — C5 — D5 — C3×D5 — C3×F5
 Lower central C5 — C3×F5
 Upper central C1 — C3

Generators and relations for C3×F5
G = < a,b,c | a3=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Character table of C3×F5

 class 1 2 3A 3B 4A 4B 5 6A 6B 12A 12B 12C 12D 15A 15B size 1 5 1 1 5 5 4 5 5 5 5 5 5 4 4 ρ1 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 linear of order 2 ρ3 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ4 1 1 ζ3 ζ32 -1 -1 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 ζ32 ζ3 linear of order 6 ρ5 1 1 ζ32 ζ3 -1 -1 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 ζ3 ζ32 linear of order 6 ρ6 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ7 1 -1 1 1 i -i 1 -1 -1 i i -i -i 1 1 linear of order 4 ρ8 1 -1 1 1 -i i 1 -1 -1 -i -i i i 1 1 linear of order 4 ρ9 1 -1 ζ32 ζ3 -i i 1 ζ6 ζ65 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 ζ3 ζ32 linear of order 12 ρ10 1 -1 ζ3 ζ32 -i i 1 ζ65 ζ6 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 ζ32 ζ3 linear of order 12 ρ11 1 -1 ζ32 ζ3 i -i 1 ζ6 ζ65 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 ζ3 ζ32 linear of order 12 ρ12 1 -1 ζ3 ζ32 i -i 1 ζ65 ζ6 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 ζ32 ζ3 linear of order 12 ρ13 4 0 4 4 0 0 -1 0 0 0 0 0 0 -1 -1 orthogonal lifted from F5 ρ14 4 0 -2-2√-3 -2+2√-3 0 0 -1 0 0 0 0 0 0 ζ65 ζ6 complex faithful ρ15 4 0 -2+2√-3 -2-2√-3 0 0 -1 0 0 0 0 0 0 ζ6 ζ65 complex faithful

Permutation representations of C3×F5
On 15 points - transitive group 15T8
Generators in S15
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(2 3 5 4)(7 8 10 9)(12 13 15 14)

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

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

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

G:=TransitiveGroup(15,8);

On 30 points - transitive group 30T7
Generators in S30
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 18)(2 20 5 16)(3 17 4 19)(6 23)(7 25 10 21)(8 22 9 24)(11 28)(12 30 15 26)(13 27 14 29)

G:=sub<Sym(30)| (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,18)(2,20,5,16)(3,17,4,19)(6,23)(7,25,10,21)(8,22,9,24)(11,28)(12,30,15,26)(13,27,14,29)>;

G:=Group( (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,18)(2,20,5,16)(3,17,4,19)(6,23)(7,25,10,21)(8,22,9,24)(11,28)(12,30,15,26)(13,27,14,29) );

G=PermutationGroup([(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,18),(2,20,5,16),(3,17,4,19),(6,23),(7,25,10,21),(8,22,9,24),(11,28),(12,30,15,26),(13,27,14,29)])

G:=TransitiveGroup(30,7);

C3×F5 is a maximal subgroup of   C35⋊C12
C3×F5 is a maximal quotient of   C35⋊C12

Polynomial with Galois group C3×F5 over ℚ
actionf(x)Disc(f)
15T8x15+32x10-228x5-8242·515·710·18110

Matrix representation of C3×F5 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 0 3 2 3 2 0 3 2 5 6 5 0 2 3 4 1
,
 1 0 3 0 0 5 1 2 0 2 4 3 0 4 1 4
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,2,5,2,3,0,6,3,2,3,5,4,3,2,0,1],[1,0,0,0,0,5,2,4,3,1,4,1,0,2,3,4] >;

C3×F5 in GAP, Magma, Sage, TeX

C_3\times F_5
% in TeX

G:=Group("C3xF5");
// GroupNames label

G:=SmallGroup(60,6);
// by ID

G=gap.SmallGroup(60,6);
# by ID

G:=PCGroup([4,-2,-3,-2,-5,24,387,139]);
// Polycyclic

G:=Group<a,b,c|a^3=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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