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G = S3×F5order 120 = 23·3·5

Direct product of S3 and F5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C15 — S3×F5
C1C5C15C3×D5C3×F5 — S3×F5
C15 — S3×F5
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 >

3C2
5C2
15C2
5C4
15C4
15C22
5S3
5C6
3C10
3D5
15C2×C4
5C12
5Dic3
5D6
3F5
3D10
5C4×S3
3C2×F5

Character table of S3×F5

 class 12A2B2C34A4B4C4D561012A12B15
 size 1351525515154101210108
ρ1111111111111111    trivial
ρ211111-1-1-1-1111-1-11    linear of order 2
ρ31-11-11-1-11111-1-1-11    linear of order 2
ρ41-11-1111-1-111-1111    linear of order 2
ρ51-1-111i-ii-i1-1-1-ii1    linear of order 4
ρ611-1-11-iii-i1-11i-i1    linear of order 4
ρ71-1-111-ii-ii1-1-1i-i1    linear of order 4
ρ811-1-11i-i-ii1-11-ii1    linear of order 4
ρ92020-122002-10-1-1-1    orthogonal lifted from S3
ρ102020-1-2-2002-1011-1    orthogonal lifted from D6
ρ1120-20-1-2i2i00210-ii-1    complex lifted from C4×S3
ρ1220-20-12i-2i00210i-i-1    complex lifted from C4×S3
ρ13440040000-10-100-1    orthogonal lifted from F5
ρ144-40040000-10100-1    orthogonal lifted from C2×F5
ρ158000-40000-200001    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)

60600000
100000
001000
000100
000010
000001
,
010000
100000
0060000
0006000
0000600
0000060
,
100000
010000
0000060
0010060
0001060
0000160
,
5000000
0500000
000010
001000
000001
000100

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

Subgroup lattice of S3×F5 in TeX
Character table of S3×F5 in TeX

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