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## G = C3×S5order 360 = 23·32·5

### Direct product of C3 and S5

Aliases: C3×S5, A5⋊C6, (C3×A5)⋊3C2, SmallGroup(360,119)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C3 — C3×A5 — C3×S5
 Derived series A5 — C3×S5
 Lower central A5 — C3×S5
 Upper central C1 — C3

10C2
15C2
10C3
20C3
6C5
5C22
15C22
15C4
10S3
10C6
10C6
10S3
15C6
20C6
10C32
6D5
6C15
15D4
5A4
10D6
10A4
15C2×C6
15C12
10C3×S3
10C3×C6
10C3×S3
6F5
5S4
15C3×D4
10S3×C6

Character table of C3×S5

 class 1 2A 2B 3A 3B 3C 3D 3E 4 5 6A 6B 6C 6D 6E 6F 6G 12A 12B 15A 15B size 1 10 15 1 1 20 20 20 30 24 10 10 15 15 20 20 20 30 30 24 24 ρ1 1 1 1 1 1 1 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 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 -1 1 ζ32 ζ3 ζ3 ζ32 1 -1 1 ζ65 ζ6 ζ3 ζ32 ζ6 ζ65 -1 ζ6 ζ65 ζ3 ζ32 linear of order 6 ρ4 1 -1 1 ζ3 ζ32 ζ32 ζ3 1 -1 1 ζ6 ζ65 ζ32 ζ3 ζ65 ζ6 -1 ζ65 ζ6 ζ32 ζ3 linear of order 6 ρ5 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ6 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ7 4 2 0 4 4 1 1 1 0 -1 2 2 0 0 -1 -1 -1 0 0 -1 -1 orthogonal lifted from S5 ρ8 4 -2 0 4 4 1 1 1 0 -1 -2 -2 0 0 1 1 1 0 0 -1 -1 orthogonal lifted from S5 ρ9 4 2 0 -2-2√-3 -2+2√-3 ζ3 ζ32 1 0 -1 -1+√-3 -1-√-3 0 0 ζ6 ζ65 -1 0 0 ζ65 ζ6 complex faithful ρ10 4 2 0 -2+2√-3 -2-2√-3 ζ32 ζ3 1 0 -1 -1-√-3 -1+√-3 0 0 ζ65 ζ6 -1 0 0 ζ6 ζ65 complex faithful ρ11 4 -2 0 -2+2√-3 -2-2√-3 ζ32 ζ3 1 0 -1 1+√-3 1-√-3 0 0 ζ3 ζ32 1 0 0 ζ6 ζ65 complex faithful ρ12 4 -2 0 -2-2√-3 -2+2√-3 ζ3 ζ32 1 0 -1 1-√-3 1+√-3 0 0 ζ32 ζ3 1 0 0 ζ65 ζ6 complex faithful ρ13 5 1 1 5 5 -1 -1 -1 -1 0 1 1 1 1 1 1 1 -1 -1 0 0 orthogonal lifted from S5 ρ14 5 -1 1 5 5 -1 -1 -1 1 0 -1 -1 1 1 -1 -1 -1 1 1 0 0 orthogonal lifted from S5 ρ15 5 -1 1 -5-5√-3/2 -5+5√-3/2 ζ65 ζ6 -1 1 0 ζ65 ζ6 ζ3 ζ32 ζ6 ζ65 -1 ζ32 ζ3 0 0 complex faithful ρ16 5 1 1 -5-5√-3/2 -5+5√-3/2 ζ65 ζ6 -1 -1 0 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 ζ6 ζ65 0 0 complex faithful ρ17 5 1 1 -5+5√-3/2 -5-5√-3/2 ζ6 ζ65 -1 -1 0 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 ζ65 ζ6 0 0 complex faithful ρ18 5 -1 1 -5+5√-3/2 -5-5√-3/2 ζ6 ζ65 -1 1 0 ζ6 ζ65 ζ32 ζ3 ζ65 ζ6 -1 ζ3 ζ32 0 0 complex faithful ρ19 6 0 -2 6 6 0 0 0 0 1 0 0 -2 -2 0 0 0 0 0 1 1 orthogonal lifted from S5 ρ20 6 0 -2 -3+3√-3 -3-3√-3 0 0 0 0 1 0 0 1+√-3 1-√-3 0 0 0 0 0 ζ32 ζ3 complex faithful ρ21 6 0 -2 -3-3√-3 -3+3√-3 0 0 0 0 1 0 0 1-√-3 1+√-3 0 0 0 0 0 ζ3 ζ32 complex faithful

Permutation representations of C3×S5
On 15 points - transitive group 15T24
Generators in S15
(1 2 3)(4 5 6)(7 8 9)(10 11 12 13 14 15)
(2 5 12)(3 10 8)(4 9 14)

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

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

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

G:=TransitiveGroup(15,24);

On 18 points - transitive group 18T144
Generators in S18
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 14 17)(2 15 5)(3 6 10)(4 8 11)(7 16 13)(9 18 12)

G:=sub<Sym(18)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,14,17)(2,15,5)(3,6,10)(4,8,11)(7,16,13)(9,18,12)>;

G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,14,17)(2,15,5)(3,6,10)(4,8,11)(7,16,13)(9,18,12) );

G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,14,17),(2,15,5),(3,6,10),(4,8,11),(7,16,13),(9,18,12)]])

G:=TransitiveGroup(18,144);

On 30 points - transitive group 30T90
Generators in S30
(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 4 5)(2 23 7)(3 9 29)(6 25 21)(8 22 30)(10 17 12)(11 27 19)(13 28 26)(14 18 16)(15 20 24)

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

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

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

G:=TransitiveGroup(30,90);

On 30 points - transitive group 30T98
Generators in S30
(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 21 18)(2 27 19)(3 14 25)(4 28 17)(5 26 9)(6 11 13)(7 22 30)(8 20 12)(10 15 24)

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

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

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

G:=TransitiveGroup(30,98);

On 30 points - transitive group 30T103
Generators in S30
(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 26 14)(2 20 9)(4 29 22)(5 18 12)(7 24 25)(10 16 28)

G:=sub<Sym(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), (1,26,14)(2,20,9)(4,29,22)(5,18,12)(7,24,25)(10,16,28)>;

G:=Group( (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), (1,26,14)(2,20,9)(4,29,22)(5,18,12)(7,24,25)(10,16,28) );

G=PermutationGroup([[(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)], [(1,26,14),(2,20,9),(4,29,22),(5,18,12),(7,24,25),(10,16,28)]])

G:=TransitiveGroup(30,103);

Polynomial with Galois group C3×S5 over ℚ
actionf(x)Disc(f)
15T24x15-x14-2x13+x12+16x10+22x9-13x8-57x5+26x4-1212·710·193·1132·1513·19732·124332

Matrix representation of C3×S5 in GL4(𝔽7) generated by

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

C3×S5 in GAP, Magma, Sage, TeX

C_3\times S_5
% in TeX

G:=Group("C3xS5");
// GroupNames label

G:=SmallGroup(360,119);
// by ID

G=gap.SmallGroup(360,119);
# by ID

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