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G = C3xS5order 360 = 23·32·5

Direct product of C3 and S5

direct product, non-abelian, not soluble

Aliases: C3xS5, A5:C6, (C3xA5):3C2, SmallGroup(360,119)

Series: ChiefDerived Lower central Upper central

C1C3C3xA5 — C3xS5
A5 — C3xS5
A5 — C3xS5
C1C3

Subgroups: 362 in 41 conjugacy classes, 6 normal (all characteristic)
Quotients: C1, C2, C3, C6, S5, C3xS5
10C2
15C2
10C3
20C3
6C5
5C22
15C22
15C4
10S3
10C6
10C6
10S3
15C6
20C6
10C32
6D5
6C15
15D4
5A4
5C2xC6
10D6
10A4
15C2xC6
15C12
10C3xS3
10C3xC6
10C3xS3
6F5
6C3xD5
5S4
15C3xD4
5C3xA4
10S3xC6
6C3xF5
5C3xS4

Character table of C3xS5

 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 C3xS5
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 C3xS5 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 C3xS5 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] >;

C3xS5 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 C3xS5 in TeX
Character table of C3xS5 in TeX

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