S3xC2^4:C5 - GroupNames

G = S₃×C₂⁴⋊C₅ order 480 = 2⁵·3·5

Direct product of S₃ and C₂⁴⋊C₅

direct product, metabelian, soluble, monomial, A-group

Aliases: S₃×C₂⁴⋊C₅, (S₃×C₂⁴)⋊C₅, (C₂³×C₆)⋊C₁₀, C₂⁴⋊₄(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

Export

Character table of S₃×C₂⁴⋊C₅ in TeX

G = S3×C24⋊C5 order 480 = 25·3·5

Direct product of S3 and C24⋊C5

G = S₃×C₂⁴⋊C₅ order 480 = 2⁵·3·5

Direct product of S₃ and C₂⁴⋊C₅