S3xC2^4: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

Subgroups: 1464 in 164 conjugacy classes, 9 normal (all characteristic)
C₁, C₂^[×7], C₃, C₂²^[×31], C₅, S₃, S₃^[×3], C₆^[×3], C₂³^[×31], C₁₀, D₆^[×24], C₂×C₆^[×7], C₁₅, C₂⁴, C₂⁴^[×6], C₂²×S₃^[×28], C₂²×C₆^[×3], C₅×S₃, C₂⁵, S₃×C₂³^[×6], 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

Generators and relations
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 >

Permutation representations
►On 30 points - transitive group 30T111

Generators in S₃₀

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

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

(3 28)(5 30)(6 21)(8 23)(13 20)(15 17)

(1 26)(2 27)(3 28)(4 29)(6 21)(7 22)(9 24)(10 25)(11 18)(12 19)(13 20)(14 16)

(1 26)(5 30)(8 23)(9 24)(11 18)(15 17)

(1 26)(4 29)(7 22)(9 24)(11 18)(14 16)

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

(6 13)(7 14)(8 15)(9 11)(10 12)(16 22)(17 23)(18 24)(19 25)(20 21)

(3 28)(5 30)(6 21)(8 23)(13 20)(15 17)

(1 26)(2 27)(3 28)(4 29)(6 21)(7 22)(9 24)(10 25)(11 18)(12 19)(13 20)(14 16)

(1 26)(5 30)(8 23)(9 24)(11 18)(15 17)

(1 26)(4 29)(7 22)(9 24)(11 18)(14 16)

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

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

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

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

G = S₃×C₂⁴⋊C₅ order 480 = 2⁵·3·5

Direct product of S₃ and C₂⁴⋊C₅