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## G = S3×C24⋊C5order 480 = 25·3·5

### Direct product of S3 and C24⋊C5

Aliases: S3×C24⋊C5, (S3×C24)⋊C5, (C23×C6)⋊C10, C244(C5×S3), C3⋊(C2×C24⋊C5), (C3×C24⋊C5)⋊3C2, SmallGroup(480,1196)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23×C6 — S3×C24⋊C5
 Chief series C1 — C3 — C23×C6 — C3×C24⋊C5 — S3×C24⋊C5
 Lower central C23×C6 — S3×C24⋊C5
 Upper central C1

Generators and relations for S3×C24⋊C5
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f2=g5=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg-1=cde, de=ed, df=fd, gdg-1=def, geg-1=ef=fe, gfg-1=c >

Subgroups: 1464 in 164 conjugacy classes, 9 normal (all characteristic)
C1, C2, C3, C22, C5, S3, S3, C6, C23, C10, D6, C2×C6, C15, C24, C24, C22×S3, C22×C6, C5×S3, C25, S3×C23, C23×C6, C24⋊C5, S3×C24, C2×C24⋊C5, C3×C24⋊C5, S3×C24⋊C5
Quotients: C1, C2, C5, S3, C10, C5×S3, C24⋊C5, C2×C24⋊C5, S3×C24⋊C5

Character table of S3×C24⋊C5

 class 1 2A 2B 2C 2D 2E 2F 2G 3 5A 5B 5C 5D 6A 6B 6C 10A 10B 10C 10D 15A 15B 15C 15D size 1 3 5 5 5 15 15 15 2 16 16 16 16 10 10 10 48 48 48 48 32 32 32 32 ρ1 1 1 1 1 1 1 1 1 1 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 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 ζ52 ζ54 ζ5 ζ53 1 1 1 ζ5 ζ54 ζ53 ζ52 ζ53 ζ52 ζ54 ζ5 linear of order 5 ρ4 1 -1 1 1 1 -1 -1 -1 1 ζ5 ζ52 ζ53 ζ54 1 1 1 -ζ53 -ζ52 -ζ54 -ζ5 ζ54 ζ5 ζ52 ζ53 linear of order 10 ρ5 1 -1 1 1 1 -1 -1 -1 1 ζ52 ζ54 ζ5 ζ53 1 1 1 -ζ5 -ζ54 -ζ53 -ζ52 ζ53 ζ52 ζ54 ζ5 linear of order 10 ρ6 1 1 1 1 1 1 1 1 1 ζ53 ζ5 ζ54 ζ52 1 1 1 ζ54 ζ5 ζ52 ζ53 ζ52 ζ53 ζ5 ζ54 linear of order 5 ρ7 1 1 1 1 1 1 1 1 1 ζ5 ζ52 ζ53 ζ54 1 1 1 ζ53 ζ52 ζ54 ζ5 ζ54 ζ5 ζ52 ζ53 linear of order 5 ρ8 1 1 1 1 1 1 1 1 1 ζ54 ζ53 ζ52 ζ5 1 1 1 ζ52 ζ53 ζ5 ζ54 ζ5 ζ54 ζ53 ζ52 linear of order 5 ρ9 1 -1 1 1 1 -1 -1 -1 1 ζ54 ζ53 ζ52 ζ5 1 1 1 -ζ52 -ζ53 -ζ5 -ζ54 ζ5 ζ54 ζ53 ζ52 linear of order 10 ρ10 1 -1 1 1 1 -1 -1 -1 1 ζ53 ζ5 ζ54 ζ52 1 1 1 -ζ54 -ζ5 -ζ52 -ζ53 ζ52 ζ53 ζ5 ζ54 linear of order 10 ρ11 2 0 2 2 2 0 0 0 -1 2 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 0 2 2 2 0 0 0 -1 2ζ5 2ζ52 2ζ53 2ζ54 -1 -1 -1 0 0 0 0 -ζ54 -ζ5 -ζ52 -ζ53 complex lifted from C5×S3 ρ13 2 0 2 2 2 0 0 0 -1 2ζ52 2ζ54 2ζ5 2ζ53 -1 -1 -1 0 0 0 0 -ζ53 -ζ52 -ζ54 -ζ5 complex lifted from C5×S3 ρ14 2 0 2 2 2 0 0 0 -1 2ζ53 2ζ5 2ζ54 2ζ52 -1 -1 -1 0 0 0 0 -ζ52 -ζ53 -ζ5 -ζ54 complex lifted from C5×S3 ρ15 2 0 2 2 2 0 0 0 -1 2ζ54 2ζ53 2ζ52 2ζ5 -1 -1 -1 0 0 0 0 -ζ5 -ζ54 -ζ53 -ζ52 complex lifted from C5×S3 ρ16 5 -5 1 1 -3 -1 3 -1 5 0 0 0 0 1 1 -3 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C24⋊C5 ρ17 5 -5 1 -3 1 3 -1 -1 5 0 0 0 0 -3 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C24⋊C5 ρ18 5 -5 -3 1 1 -1 -1 3 5 0 0 0 0 1 -3 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C24⋊C5 ρ19 5 5 -3 1 1 1 1 -3 5 0 0 0 0 1 -3 1 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ20 5 5 1 -3 1 -3 1 1 5 0 0 0 0 -3 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ21 5 5 1 1 -3 1 -3 1 5 0 0 0 0 1 1 -3 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ22 10 0 -6 2 2 0 0 0 -5 0 0 0 0 -1 3 -1 0 0 0 0 0 0 0 0 orthogonal faithful ρ23 10 0 2 -6 2 0 0 0 -5 0 0 0 0 3 -1 -1 0 0 0 0 0 0 0 0 orthogonal faithful ρ24 10 0 2 2 -6 0 0 0 -5 0 0 0 0 -1 -1 3 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of S3×C24⋊C5
On 30 points - transitive group 30T111
Generators in S30
(1 17 15)(2 18 11)(3 19 12)(4 20 13)(5 16 14)(6 24 26)(7 25 27)(8 21 28)(9 22 29)(10 23 30)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 15)(7 11)(8 12)(9 13)(10 14)(16 23)(17 24)(18 25)(19 21)(20 22)
(3 28)(5 30)(8 19)(10 16)(12 21)(14 23)
(1 26)(2 27)(3 28)(4 29)(6 17)(7 18)(8 19)(9 20)(11 25)(12 21)(13 22)(15 24)
(1 26)(5 30)(6 17)(10 16)(14 23)(15 24)
(1 26)(4 29)(6 17)(9 20)(13 22)(15 24)
(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)

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

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

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

G:=TransitiveGroup(30,111);

On 30 points - transitive group 30T120
Generators in S30
(1 17 15)(2 18 11)(3 19 12)(4 20 13)(5 16 14)(6 24 26)(7 25 27)(8 21 28)(9 22 29)(10 23 30)
(6 24)(7 25)(8 21)(9 22)(10 23)(11 18)(12 19)(13 20)(14 16)(15 17)
(3 28)(5 30)(8 19)(10 16)(12 21)(14 23)
(1 26)(2 27)(3 28)(4 29)(6 17)(7 18)(8 19)(9 20)(11 25)(12 21)(13 22)(15 24)
(1 26)(5 30)(6 17)(10 16)(14 23)(15 24)
(1 26)(4 29)(6 17)(9 20)(13 22)(15 24)
(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)

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

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

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

G:=TransitiveGroup(30,120);

Matrix representation of S3×C24⋊C5 in GL7(𝔽31)

 30 30 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 30 30 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 30 0 0 0 0 0 0 19 1 0 0 0 0 19 0 0 30 0 0 0 0 1 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 30 0 0 0 0 0 19 0 30 0 0 0 0 30 0 0 30
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 30 0 0 0 0 0 1 19 1 0 0 0 0 12 12 0 1 0 0 0 19 1 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 30 0 0 0 0 12 0 0 1 0 0 0 19 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 30 12 29 0 0 0 0 0 0 19 1 0 0 0 0 0 12 0 1 0 0 0 0 1 0 0

G:=sub<GL(7,GF(31))| [30,1,0,0,0,0,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,19,0,0,0,0,30,19,0,1,0,0,0,0,1,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,12,19,30,0,0,0,0,30,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,30],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30,0,1,12,19,0,0,0,30,19,12,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30,0,0,12,19,0,0,0,1,12,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,30,0,0,0,0,0,1,12,0,0,0,0,0,0,29,19,12,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;

S3×C24⋊C5 in GAP, Magma, Sage, TeX

S_3\times C_2^4\rtimes C_5
% in TeX

G:=Group("S3xC2^4:C5");
// GroupNames label

G:=SmallGroup(480,1196);
// by ID

G=gap.SmallGroup(480,1196);
# by ID

G:=PCGroup([7,-2,-5,-2,2,2,2,-3,324,850,2111,222,15686]);
// Polycyclic

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

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