Copied to
clipboard

## G = S3×F5order 120 = 23·3·5

### Direct product of S3 and F5

Aliases: S3×F5, D15⋊C4, D5.1D6, C5⋊(C4×S3), C3⋊F5⋊C2, C15⋊(C2×C4), (C5×S3)⋊C4, (C3×F5)⋊C2, C31(C2×F5), (S3×D5).C2, (C3×D5).C22, Aut(D15), Hol(C15), SmallGroup(120,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — S3×F5
 Chief series C1 — C5 — C15 — C3×D5 — C3×F5 — S3×F5
 Lower central C15 — S3×F5
 Upper central C1

Generators and relations for S3×F5
G = < a,b,c,d | a3=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Character table of S3×F5

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5 6 10 12A 12B 15 size 1 3 5 15 2 5 5 15 15 4 10 12 10 10 8 ρ1 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 linear of order 2 ρ3 1 -1 1 -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 1 1 1 linear of order 2 ρ5 1 -1 -1 1 1 i -i i -i 1 -1 -1 -i i 1 linear of order 4 ρ6 1 1 -1 -1 1 -i i i -i 1 -1 1 i -i 1 linear of order 4 ρ7 1 -1 -1 1 1 -i i -i i 1 -1 -1 i -i 1 linear of order 4 ρ8 1 1 -1 -1 1 i -i -i i 1 -1 1 -i i 1 linear of order 4 ρ9 2 0 2 0 -1 2 2 0 0 2 -1 0 -1 -1 -1 orthogonal lifted from S3 ρ10 2 0 2 0 -1 -2 -2 0 0 2 -1 0 1 1 -1 orthogonal lifted from D6 ρ11 2 0 -2 0 -1 -2i 2i 0 0 2 1 0 -i i -1 complex lifted from C4×S3 ρ12 2 0 -2 0 -1 2i -2i 0 0 2 1 0 i -i -1 complex lifted from C4×S3 ρ13 4 4 0 0 4 0 0 0 0 -1 0 -1 0 0 -1 orthogonal lifted from F5 ρ14 4 -4 0 0 4 0 0 0 0 -1 0 1 0 0 -1 orthogonal lifted from C2×F5 ρ15 8 0 0 0 -4 0 0 0 0 -2 0 0 0 0 1 orthogonal faithful

Permutation representations of S3×F5
On 15 points - transitive group 15T11
Generators in S15
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)
(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)
(2 3 5 4)(7 8 10 9)(12 13 15 14)

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

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

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

G:=TransitiveGroup(15,11);

On 30 points - transitive group 30T23
Generators in S30
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 16)(2 17)(3 18)(4 19)(5 20)(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 16)(2 18 5 19)(3 20 4 17)(6 26)(7 28 10 29)(8 30 9 27)(11 21)(12 23 15 24)(13 25 14 22)

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

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

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

G:=TransitiveGroup(30,23);

On 30 points - transitive group 30T24
Generators in S30
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(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 18)(2 20 5 16)(3 17 4 19)(6 23)(7 25 10 21)(8 22 9 24)(11 28)(12 30 15 26)(13 27 14 29)

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

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

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

G:=TransitiveGroup(30,24);

On 30 points - transitive group 30T32
Generators in S30
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 16)(2 17)(3 18)(4 19)(5 20)(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)
(2 3 5 4)(7 8 10 9)(12 13 15 14)(17 18 20 19)(22 23 25 24)(27 28 30 29)

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

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

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

G:=TransitiveGroup(30,32);

S3×F5 is a maximal subgroup of   C3⋊F5⋊S3
S3×F5 is a maximal quotient of   D6⋊F5  Dic3⋊F5  D15⋊C8  D6.F5  Dic3.F5  C3⋊F5⋊S3

Polynomial with Galois group S3×F5 over ℚ
actionf(x)Disc(f)
15T11x15+12-228·329·515

Matrix representation of S3×F5 in GL6(𝔽61)

 60 60 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 1 0 0 60 0 0 0 1 0 60 0 0 0 0 1 60
,
 50 0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(61))| [60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,60,60,60],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

S3×F5 in GAP, Magma, Sage, TeX

S_3\times F_5
% in TeX

G:=Group("S3xF5");
// GroupNames label

G:=SmallGroup(120,36);
// by ID

G=gap.SmallGroup(120,36);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,20,168,1204,614]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^5=d^4=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^-1=c^3>;
// generators/relations

Export

׿
×
𝔽