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## G = S3×D5order 60 = 22·3·5

### Direct product of S3 and D5

Aliases: S3×D5, C51D6, D15⋊C2, C31D10, C15⋊C22, (C5×S3)⋊C2, (C3×D5)⋊C2, SmallGroup(60,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — S3×D5
 Chief series C1 — C5 — C15 — C3×D5 — S3×D5
 Lower central C15 — S3×D5
 Upper central C1

Generators and relations for S3×D5
G = < a,b,c,d | a3=b2=c5=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of S3×D5

 class 1 2A 2B 2C 3 5A 5B 6 10A 10B 15A 15B size 1 3 5 15 2 2 2 10 6 6 4 4 ρ1 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 linear of order 2 ρ3 1 -1 1 -1 1 1 1 1 -1 -1 1 1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 -1 1 1 1 1 linear of order 2 ρ5 2 0 2 0 -1 2 2 -1 0 0 -1 -1 orthogonal lifted from S3 ρ6 2 0 -2 0 -1 2 2 1 0 0 -1 -1 orthogonal lifted from D6 ρ7 2 -2 0 0 2 -1-√5/2 -1+√5/2 0 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ8 2 2 0 0 2 -1+√5/2 -1-√5/2 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ9 2 -2 0 0 2 -1+√5/2 -1-√5/2 0 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ10 2 2 0 0 2 -1-√5/2 -1+√5/2 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ11 4 0 0 0 -2 -1+√5 -1-√5 0 0 0 1-√5/2 1+√5/2 orthogonal faithful ρ12 4 0 0 0 -2 -1-√5 -1+√5 0 0 0 1+√5/2 1-√5/2 orthogonal faithful

Permutation representations of S3×D5
On 15 points - transitive group 15T7
Generators in S15
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)
(6 11)(7 12)(8 13)(9 14)(10 15)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)

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

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

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

G:=TransitiveGroup(15,7);

On 30 points - transitive group 30T8
Generators in S30
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)

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

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

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

G:=TransitiveGroup(30,8);

On 30 points - transitive group 30T10
Generators in S30
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(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 20)(2 19)(3 18)(4 17)(5 16)(6 23)(7 22)(8 21)(9 25)(10 24)(11 28)(12 27)(13 26)(14 30)(15 29)

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

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

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

G:=TransitiveGroup(30,10);

On 30 points - transitive group 30T13
Generators in S30
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 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)(2 17)(3 16)(4 20)(5 19)(6 26)(7 30)(8 29)(9 28)(10 27)(11 21)(12 25)(13 24)(14 23)(15 22)

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

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

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

G:=TransitiveGroup(30,13);

S3×D5 is a maximal subgroup of
D15⋊S3  D15⋊D5  D15⋊D7
S3×D5 is a maximal quotient of
D30.C2  C15⋊D4  C3⋊D20  C5⋊D12  C15⋊Q8  D15⋊S3  D15⋊D5  D15⋊D7

Polynomial with Galois group S3×D5 over ℚ
actionf(x)Disc(f)
15T7x15-2x14+2x13-2x11+6x10+7x9+2x8+x7+5x6-3x5-3x4+6x3+2x2+1-132·292·315·476·1992

Matrix representation of S3×D5 in GL4(𝔽31) generated by

 1 0 0 0 0 1 0 0 0 0 1 10 0 0 9 29
,
 1 0 0 0 0 1 0 0 0 0 30 21 0 0 0 1
,
 0 1 0 0 30 18 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(31))| [1,0,0,0,0,1,0,0,0,0,1,9,0,0,10,29],[1,0,0,0,0,1,0,0,0,0,30,0,0,0,21,1],[0,30,0,0,1,18,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

S3×D5 in GAP, Magma, Sage, TeX

S_3\times D_5
% in TeX

G:=Group("S3xD5");
// GroupNames label

G:=SmallGroup(60,8);
// by ID

G=gap.SmallGroup(60,8);
# by ID

G:=PCGroup([4,-2,-2,-3,-5,54,771]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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