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G = C3×F5order 60 = 22·3·5

Direct product of C3 and F5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C5 — C3×F5
C1C5D5C3×D5 — C3×F5
C5 — C3×F5
C1C3

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

5C2
5C4
5C6
5C12

Character table of C3×F5

 class 123A3B4A4B56A6B12A12B12C12D15A15B
 size 151155455555544
ρ1111111111111111    trivial
ρ21111-1-1111-1-1-1-111    linear of order 2
ρ311ζ32ζ3111ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ32    linear of order 3
ρ411ζ3ζ32-1-11ζ3ζ32ζ6ζ65ζ6ζ65ζ32ζ3    linear of order 6
ρ511ζ32ζ3-1-11ζ32ζ3ζ65ζ6ζ65ζ6ζ3ζ32    linear of order 6
ρ611ζ3ζ32111ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ3    linear of order 3
ρ71-111i-i1-1-1ii-i-i11    linear of order 4
ρ81-111-ii1-1-1-i-iii11    linear of order 4
ρ91-1ζ32ζ62ζ2ζ21ζ6ζ65ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32ζ62ζ32    linear of order 12
ρ101-1ζ62ζ32ζ2ζ21ζ65ζ6ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3ζ32ζ62    linear of order 12
ρ111-1ζ32ζ62ζ2ζ21ζ6ζ65ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32ζ62ζ32    linear of order 12
ρ121-1ζ62ζ32ζ2ζ21ζ65ζ6ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3ζ32ζ62    linear of order 12
ρ13404400-1000000-1-1    orthogonal lifted from F5
ρ1440-2-2-3-2+2-300-1000000ζ65ζ6    complex faithful
ρ1540-2+2-3-2-2-300-1000000ζ6ζ65    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);

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

4000
0400
0040
0004
,
0323
2032
5650
2341
,
1030
0512
0243
0414
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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