C2xS3^3 - GroupNames

G = C₂×S₃³ order 432 = 2⁴·3³

Direct product of C₂, S₃, S₃ and S₃

direct product, metabelian, supersoluble, monomial, A-group, rational

Aliases: C₂×S₃³, C₃³⋊C₂⁴, (S₃×C₆)⋊₁₄D₆, (C₃²×C₆)⋊C₂³, (S₃×C₃²)⋊C₂³, C₃³⋊C₂⋊C₂³, C₃²⋊₄(S₃×C₂³), C₃²⋊₄D₆⋊₄C₂², C₆⋊₁(C₂×S₃²), (S₃²×C₆)⋊₁₁C₂, C₃⋊₁(C₂²×S₃²), (S₃×C₃⋊S₃)⋊C₂², (C₃×C₃⋊S₃)⋊C₂³, (C₂×C₃⋊S₃)⋊₁₈D₆, (C₃×S₃²)⋊₄C₂², C₃⋊S₃⋊₂(C₂²×S₃), (S₃×C₃×C₆)⋊₁₅C₂², (C₃×C₆)⋊₄(C₂²×S₃), (C₆×C₃⋊S₃)⋊₁₃C₂², (C₃×S₃)⋊₁(C₂²×S₃), (C₂×C₃²⋊₄D₆)⋊₇C₂, (C₂×C₃³⋊C₂)⋊₈C₂², (C₂×S₃×C₃⋊S₃)⋊₁₁C₂, SmallGroup(432,759)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₂×S₃³

Chief series

C₁ — C₃ — C₃² — C₃³ — S₃×C₃² — C₃×S₃² — S₃³ — C₂×S₃³

Lower central

C₃³ — C₂×S₃³

Upper central

C₁ — C₂

Generators and relations for C₂×S₃³
G = < a,b,c,d,e,f,g | a²=b³=c²=d³=e²=f³=g²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, cbc=b^-1, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, ede=d^-1, df=fd, dg=gd, ef=fe, eg=ge, gfg=f^-1 >

Subgroups: 3916 in 642 conjugacy classes, 132 normal (7 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², S₃, S₃, C₆, C₆, C₂³, C₃², C₃², D₆, D₆, C₂×C₆, C₂⁴, C₃×S₃, C₃×S₃, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₂²×S₃, C₂²×C₆, C₃³, S₃², S₃², S₃×C₆, S₃×C₆, C₂×C₃⋊S₃, C₂×C₃⋊S₃, C₆², S₃×C₂³, S₃×C₃², C₃×C₃⋊S₃, C₃³⋊C₂, C₃²×C₆, C₂×S₃², C₂×S₃², S₃×C₂×C₆, C₂²×C₃⋊S₃, C₃×S₃², S₃×C₃⋊S₃, C₃²⋊₄D₆, S₃×C₃×C₆, C₆×C₃⋊S₃, C₂×C₃³⋊C₂, C₂²×S₃², S₃³, S₃²×C₆, C₂×S₃×C₃⋊S₃, C₂×C₃²⋊₄D₆, C₂×S₃³
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, C₂²×S₃, S₃², S₃×C₂³, C₂×S₃², C₂²×S₃², S₃³, C₂×S₃³

Permutation representations of C₂×S₃³
►On 24 points - transitive group 24T1296

Generators in S₂₄

(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 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)

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

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

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

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

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

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

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

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

G:=TransitiveGroup(24,1296);

54 conjugacy classes

class	1	2A	2B	···	2G	2H	···	2M	2N	2O	3A	3B	3C	3D	3E	3F	3G	6A	6B	6C	6D	6E	6F	6G	···	6R	6S	6T	···	6Y	6Z	···	6AE
order	1	2	2	···	2	2	···	2	2	2	3	3	3	3	3	3	3	6	6	6	6	6	6	6	···	6	6	6	···	6	6	···	6
size	1	1	3	···	3	9	···	9	27	27	2	2	2	4	4	4	8	2	2	2	4	4	4	6	···	6	8	12	···	12	18	···	18

54 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

S₃²

C₂×S₃²

S₃³

C₂×S₃³

kernel

C₂×S₃³

S₃³

S₃²×C₆

C₂×S₃×C₃⋊S₃

C₂×C₃²⋊₄D₆

C₂×S₃²

S₃²

S₃×C₆

C₂×C₃⋊S₃

D₆

S₃

C₆

C₂

C₁

# reps

Matrix representation of C₂×S₃³ ►in GL₆(ℤ)

-1	0	0	0	0	0
0	-1	0	0	0	0
0	0	-1	0	0	0
0	0	0	-1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	-1	1	0	0
0	0	-1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	-1	0	0
0	0	-1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	-1	1
0	0	0	0	-1	0

-1	0	0	0	0	0
0	-1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

0	1	0	0	0	0
-1	-1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

-1	0	0	0	0	0
1	1	0	0	0	0
0	0	-1	0	0	0
0	0	0	-1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂×S₃³ in GAP, Magma, Sage, TeX

C_2\times S_3^3

% in TeX

G:=Group("C2xS3^3");

// GroupNames label

G:=SmallGroup(432,759);

// by ID

G=gap.SmallGroup(432,759);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,298,2028,14118]);

// Polycyclic

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

// generators/relations

G = C2×S33 order 432 = 24·33

Direct product of C2, S3, S3 and S3

G = C₂×S₃³ order 432 = 2⁴·3³

Direct product of C₂, S₃, S₃ and S₃