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## G = C5×S3order 30 = 2·3·5

### Direct product of C5 and S3

Aliases: C5×S3, C3⋊C10, C153C2, SmallGroup(30,1)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×S3
G = < a,b,c | a5=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Character table of C5×S3

 class 1 2 3 5A 5B 5C 5D 10A 10B 10C 10D 15A 15B 15C 15D size 1 3 2 1 1 1 1 3 3 3 3 2 2 2 2 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 ζ5 ζ54 ζ52 ζ53 -ζ52 -ζ5 -ζ54 -ζ53 ζ54 ζ5 ζ52 ζ53 linear of order 10 ρ4 1 -1 1 ζ52 ζ53 ζ54 ζ5 -ζ54 -ζ52 -ζ53 -ζ5 ζ53 ζ52 ζ54 ζ5 linear of order 10 ρ5 1 1 1 ζ52 ζ53 ζ54 ζ5 ζ54 ζ52 ζ53 ζ5 ζ53 ζ52 ζ54 ζ5 linear of order 5 ρ6 1 1 1 ζ5 ζ54 ζ52 ζ53 ζ52 ζ5 ζ54 ζ53 ζ54 ζ5 ζ52 ζ53 linear of order 5 ρ7 1 -1 1 ζ53 ζ52 ζ5 ζ54 -ζ5 -ζ53 -ζ52 -ζ54 ζ52 ζ53 ζ5 ζ54 linear of order 10 ρ8 1 1 1 ζ53 ζ52 ζ5 ζ54 ζ5 ζ53 ζ52 ζ54 ζ52 ζ53 ζ5 ζ54 linear of order 5 ρ9 1 1 1 ζ54 ζ5 ζ53 ζ52 ζ53 ζ54 ζ5 ζ52 ζ5 ζ54 ζ53 ζ52 linear of order 5 ρ10 1 -1 1 ζ54 ζ5 ζ53 ζ52 -ζ53 -ζ54 -ζ5 -ζ52 ζ5 ζ54 ζ53 ζ52 linear of order 10 ρ11 2 0 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 0 -1 2ζ54 2ζ5 2ζ53 2ζ52 0 0 0 0 -ζ5 -ζ54 -ζ53 -ζ52 complex faithful ρ13 2 0 -1 2ζ53 2ζ52 2ζ5 2ζ54 0 0 0 0 -ζ52 -ζ53 -ζ5 -ζ54 complex faithful ρ14 2 0 -1 2ζ52 2ζ53 2ζ54 2ζ5 0 0 0 0 -ζ53 -ζ52 -ζ54 -ζ5 complex faithful ρ15 2 0 -1 2ζ5 2ζ54 2ζ52 2ζ53 0 0 0 0 -ζ54 -ζ5 -ζ52 -ζ53 complex faithful

Permutation representations of C5×S3
On 15 points - transitive group 15T4
Generators in S15
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(1 8 12)(2 9 13)(3 10 14)(4 6 15)(5 7 11)
(6 15)(7 11)(8 12)(9 13)(10 14)

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

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

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

G:=TransitiveGroup(15,4);

Regular action on 30 points - transitive group 30T2
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 23 17)(2 24 18)(3 25 19)(4 21 20)(5 22 16)(6 15 29)(7 11 30)(8 12 26)(9 13 27)(10 14 28)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 16)(12 17)(13 18)(14 19)(15 20)(21 29)(22 30)(23 26)(24 27)(25 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), (1,23,17)(2,24,18)(3,25,19)(4,21,20)(5,22,16)(6,15,29)(7,11,30)(8,12,26)(9,13,27)(10,14,28), (1,8)(2,9)(3,10)(4,6)(5,7)(11,16)(12,17)(13,18)(14,19)(15,20)(21,29)(22,30)(23,26)(24,27)(25,28)>;

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

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

G:=TransitiveGroup(30,2);

C5×S3 is a maximal subgroup of   C3⋊F11
C5×S3 is a maximal quotient of   C3⋊F11

Polynomial with Galois group C5×S3 over ℚ
actionf(x)Disc(f)
15T4x15+x14-4x13+x12-18x11-24x10-5x9-58x8+79x7-32x6+97x5+35x4+42x3+15x2+1-1112·315·672·1312·11872·4676992

Matrix representation of C5×S3 in GL2(𝔽11) generated by

 5 0 0 5
,
 0 8 4 10
,
 1 8 0 10
G:=sub<GL(2,GF(11))| [5,0,0,5],[0,4,8,10],[1,0,8,10] >;

C5×S3 in GAP, Magma, Sage, TeX

C_5\times S_3
% in TeX

G:=Group("C5xS3");
// GroupNames label

G:=SmallGroup(30,1);
// by ID

G=gap.SmallGroup(30,1);
# by ID

G:=PCGroup([3,-2,-5,-3,182]);
// Polycyclic

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

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