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G = C5×S3order 30 = 2·3·5

Direct product of C5 and S3

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C3 — C5×S3
C1C3C15 — C5×S3
C3 — C5×S3
C1C5

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

3C2
3C10

Character table of C5×S3

 class 1235A5B5C5D10A10B10C10D15A15B15C15D
 size 132111133332222
ρ1111111111111111    trivial
ρ21-111111-1-1-1-11111    linear of order 2
ρ31-11ζ5ζ54ζ52ζ535255453ζ54ζ5ζ52ζ53    linear of order 10
ρ41-11ζ52ζ53ζ54ζ55452535ζ53ζ52ζ54ζ5    linear of order 10
ρ5111ζ52ζ53ζ54ζ5ζ54ζ52ζ53ζ5ζ53ζ52ζ54ζ5    linear of order 5
ρ6111ζ5ζ54ζ52ζ53ζ52ζ5ζ54ζ53ζ54ζ5ζ52ζ53    linear of order 5
ρ71-11ζ53ζ52ζ5ζ545535254ζ52ζ53ζ5ζ54    linear of order 10
ρ8111ζ53ζ52ζ5ζ54ζ5ζ53ζ52ζ54ζ52ζ53ζ5ζ54    linear of order 5
ρ9111ζ54ζ5ζ53ζ52ζ53ζ54ζ5ζ52ζ5ζ54ζ53ζ52    linear of order 5
ρ101-11ζ54ζ5ζ53ζ525354552ζ5ζ54ζ53ζ52    linear of order 10
ρ1120-122220000-1-1-1-1    orthogonal lifted from S3
ρ1220-1545535200005545352    complex faithful
ρ1320-1535255400005253554    complex faithful
ρ1420-1525354500005352545    complex faithful
ρ1520-1554525300005455253    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 9 12)(2 10 13)(3 6 14)(4 7 15)(5 8 11)
(6 14)(7 15)(8 11)(9 12)(10 13)

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

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

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

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);

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

50
05
,
08
410
,
18
010
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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