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G = C3⋊F5order 60 = 22·3·5

The semidirect product of C3 and F5 acting via F5/D5=C2

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

Aliases: C3⋊F5, C5⋊Dic3, C151C4, D5.S3, (C3×D5).1C2, SmallGroup(60,7)

Series: Derived Chief Lower central Upper central

C1C15 — C3⋊F5
C1C5C15C3×D5 — C3⋊F5
C15 — C3⋊F5
C1

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

5C2
15C4
5C6
5Dic3
3F5

Character table of C3⋊F5

 class 1234A4B5615A15B
 size 152151541044
ρ1111111111    trivial
ρ2111-1-11111    linear of order 2
ρ31-11i-i1-111    linear of order 4
ρ41-11-ii1-111    linear of order 4
ρ522-1002-1-1-1    orthogonal lifted from S3
ρ62-2-10021-1-1    symplectic lifted from Dic3, Schur index 2
ρ740400-10-1-1    orthogonal lifted from F5
ρ840-200-101--15/21+-15/2    complex faithful
ρ940-200-101+-15/21--15/2    complex faithful

Permutation representations of C3⋊F5
On 15 points - transitive group 15T6
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)(6 11)(7 13 10 14)(8 15 9 12)

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)(6,11)(7,13,10,14)(8,15,9,12)>;

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)(6,11)(7,13,10,14)(8,15,9,12) );

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),(6,11),(7,13,10,14),(8,15,9,12)])

G:=TransitiveGroup(15,6);

On 30 points - transitive group 30T6
Generators in S30
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(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 17)(2 19 5 20)(3 16 4 18)(6 22)(7 24 10 25)(8 21 9 23)(11 27)(12 29 15 30)(13 26 14 28)

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

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

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

G:=TransitiveGroup(30,6);

Polynomial with Galois group C3⋊F5 over ℚ
actionf(x)Disc(f)
15T6x15-30x10-3708x5-2214·320·530·231110

Matrix representation of C3⋊F5 in GL4(𝔽2) generated by

0110
1011
0101
1100
,
0111
0011
0001
1111
,
1011
0001
0101
0011
G:=sub<GL(4,GF(2))| [0,1,0,1,1,0,1,1,1,1,0,0,0,1,1,0],[0,0,0,1,1,0,0,1,1,1,0,1,1,1,1,1],[1,0,0,0,0,0,1,0,1,0,0,1,1,1,1,1] >;

C3⋊F5 in GAP, Magma, Sage, TeX

C_3\rtimes F_5
% in TeX

G:=Group("C3:F5");
// GroupNames label

G:=SmallGroup(60,7);
// by ID

G=gap.SmallGroup(60,7);
# by ID

G:=PCGroup([4,-2,-2,-3,-5,8,98,579,391]);
// Polycyclic

G:=Group<a,b,c|a^3=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations

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