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G = C3×S5order 360 = 23·32·5

Direct product of C3 and S5

direct product, non-abelian, not soluble

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

Series: ChiefDerived Lower central Upper central

C1C3C3×A5 — C3×S5
A5 — C3×S5
A5 — C3×S5
C1C3

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

Character table of C3×S5

 class 12A2B3A3B3C3D3E456A6B6C6D6E6F6G12A12B15A15B
 size 110151120202030241010151520202030302424
ρ1111111111111111111111    trivial
ρ21-1111111-11-1-111-1-1-1-1-111    linear of order 2
ρ31-11ζ32ζ3ζ3ζ321-11ζ65ζ6ζ3ζ32ζ6ζ65-1ζ6ζ65ζ3ζ32    linear of order 6
ρ41-11ζ3ζ32ζ32ζ31-11ζ6ζ65ζ32ζ3ζ65ζ6-1ζ65ζ6ζ32ζ3    linear of order 6
ρ5111ζ32ζ3ζ3ζ32111ζ3ζ32ζ3ζ32ζ32ζ31ζ32ζ3ζ3ζ32    linear of order 3
ρ6111ζ3ζ32ζ32ζ3111ζ32ζ3ζ32ζ3ζ3ζ321ζ3ζ32ζ32ζ3    linear of order 3
ρ7420441110-12200-1-1-100-1-1    orthogonal lifted from S5
ρ84-20441110-1-2-20011100-1-1    orthogonal lifted from S5
ρ9420-2-2-3-2+2-3ζ3ζ3210-1-1+-3-1--300ζ6ζ65-100ζ65ζ6    complex faithful
ρ10420-2+2-3-2-2-3ζ32ζ310-1-1--3-1+-300ζ65ζ6-100ζ6ζ65    complex faithful
ρ114-20-2+2-3-2-2-3ζ32ζ310-11+-31--300ζ3ζ32100ζ6ζ65    complex faithful
ρ124-20-2-2-3-2+2-3ζ3ζ3210-11--31+-300ζ32ζ3100ζ65ζ6    complex faithful
ρ1351155-1-1-1-101111111-1-100    orthogonal lifted from S5
ρ145-1155-1-1-110-1-111-1-1-11100    orthogonal lifted from S5
ρ155-11-5-5-3/2-5+5-3/2ζ65ζ6-110ζ65ζ6ζ3ζ32ζ6ζ65-1ζ32ζ300    complex faithful
ρ16511-5-5-3/2-5+5-3/2ζ65ζ6-1-10ζ3ζ32ζ3ζ32ζ32ζ31ζ6ζ6500    complex faithful
ρ17511-5+5-3/2-5-5-3/2ζ6ζ65-1-10ζ32ζ3ζ32ζ3ζ3ζ321ζ65ζ600    complex faithful
ρ185-11-5+5-3/2-5-5-3/2ζ6ζ65-110ζ6ζ65ζ32ζ3ζ65ζ6-1ζ3ζ3200    complex faithful
ρ1960-2660000100-2-20000011    orthogonal lifted from S5
ρ2060-2-3+3-3-3-3-300001001+-31--300000ζ32ζ3    complex faithful
ρ2160-2-3-3-3-3+3-300001001--31+-300000ζ3ζ32    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 8 11)(3 15 5)(6 13 7)

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

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

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

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 4)(2 5 7)(3 11 8)(6 13 16)(9 12 17)(10 15 18)

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

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

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

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

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

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

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

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 18 10)(2 9 30)(3 14 28)(4 13 21)(5 25 17)(6 23 29)(7 12 22)(8 26 20)(11 24 16)

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), (1,18,10)(2,9,30)(3,14,28)(4,13,21)(5,25,17)(6,23,29)(7,12,22)(8,26,20)(11,24,16)>;

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

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

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 19 9)(2 27 13)(4 22 29)(5 7 16)(11 21 14)(17 25 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), (1,19,9)(2,27,13)(4,22,29)(5,7,16)(11,21,14)(17,25,24)>;

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), (1,19,9)(2,27,13)(4,22,29)(5,7,16)(11,21,14)(17,25,24) );

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)], [(1,19,9),(2,27,13),(4,22,29),(5,7,16),(11,21,14),(17,25,24)])

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

4325
0035
1320
4005
,
0304
2051
5354
2543
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

Export

Subgroup lattice of C3×S5 in TeX
Character table of C3×S5 in TeX

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