S3xC2^2wrC2 - GroupNames

G = S₃×C₂²≀C₂ order 192 = 2⁶·3

Direct product of S₃ and C₂²≀C₂

direct product, metabelian, supersoluble, monomial, rational, 2-hyperelementary

Aliases: S₃×C₂²≀C₂, C₂⁴⋊₁₄D₆, (C₂×D₄)⋊₁₇D₆, D₆⋊₁₁(C₂×D₄), C₂²⋊₆(S₃×D₄), C₂³⋊₂D₆⋊₂C₂, D₆⋊D₄⋊₈C₂, (S₃×C₂⁴)⋊₂C₂, (C₂×C₁₂)⋊₁C₂³, D₆⋊C₄⋊₉C₂², C₂²⋊C₄⋊₂₃D₆, (C₆×D₄)⋊₅C₂², C₂⁴⋊₄S₃⋊₅C₂, (C₂²×S₃)⋊₁₄D₄, (C₂²×C₆)⋊₁C₂³, (C₂³×C₆)⋊₈C₂², C₂³⋊₂(C₂²×S₃), C₆.₅₄(C₂²×D₄), (C₂×D₁₂)⋊₁₇C₂², (C₂×C₆).₁₃₂C₂⁴, (C₂²×S₃)⋊₂C₂³, (C₂×Dic₃)⋊₂C₂³, (S₃×C₂³)⋊₂₀C₂², C₆.D₄⋊₁₃C₂², C₂².₁₅₃(S₃×C₂³), (C₂×S₃×D₄)⋊₅C₂, (C₂×C₆)⋊₅(C₂×D₄), C₂.₂₇(C₂×S₃×D₄), C₃⋊₂(C₂×C₂²≀C₂), (S₃×C₂×C₄)⋊₅C₂², (S₃×C₂²⋊C₄)⋊₁C₂, (C₂×C₄)⋊₁(C₂²×S₃), (C₃×C₂²≀C₂)⋊₃C₂, (C₂×C₃⋊D₄)⋊₇C₂², (C₃×C₂²⋊C₄)⋊₃C₂², SmallGroup(192,1147)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — S₃×C₂²≀C₂

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂³ — S₃×C₂⁴ — S₃×C₂²≀C₂

Lower central

C₃ — C₂×C₆ — S₃×C₂²≀C₂

Upper central

C₁ — C₂² — C₂²≀C₂

Generators and relations for S₃×C₂²≀C₂
G = < a,b,c,d,e,f,g | a³=b²=c²=d²=e²=f²=g²=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, gcg=ce=ec, cf=fc, de=ed, gdg=df=fd, ef=fe, eg=ge, fg=gf >

Subgroups: 2064 in 662 conjugacy classes, 131 normal (16 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, S₃, C₆, C₆, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₂²≀C₂, C₂²≀C₂, C₂²×D₄, C₂⁵, D₆⋊C₄, C₆.D₄, C₃×C₂²⋊C₄, S₃×C₂×C₄, C₂×D₁₂, S₃×D₄, C₂×C₃⋊D₄, C₆×D₄, S₃×C₂³, S₃×C₂³, S₃×C₂³, C₂³×C₆, C₂×C₂²≀C₂, S₃×C₂²⋊C₄, D₆⋊D₄, C₂³⋊₂D₆, C₂⁴⋊₄S₃, C₃×C₂²≀C₂, C₂×S₃×D₄, S₃×C₂⁴, S₃×C₂²≀C₂
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₂⁴, C₂²×S₃, C₂²≀C₂, C₂²×D₄, S₃×D₄, S₃×C₂³, C₂×C₂²≀C₂, C₂×S₃×D₄, S₃×C₂²≀C₂

Permutation representations of S₃×C₂²≀C₂
►On 24 points - transitive group 24T360

Generators in S₂₄

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

(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)

(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

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

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

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

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

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

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

G:=TransitiveGroup(24,360);

42 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	2N	2O	···	2T	2U	3	4A	4B	4C	4D	4E	4F	6A	6B	6C	6D	···	6I	6J	12A	12B	12C
order	1	2	2	2	2	···	2	2	2	2	2	2	2	···	2	2	3	4	4	4	4	4	4	6	6	6	6	···	6	6	12	12	12
size	1	1	1	1	2	···	2	3	3	3	3	4	6	···	6	12	2	4	4	4	12	12	12	2	2	2	4	···	4	8	8	8	8

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

S₃×D₄

kernel

S₃×C₂²≀C₂

S₃×C₂²⋊C₄

D₆⋊D₄

C₂³⋊₂D₆

C₂⁴⋊₄S₃

C₃×C₂²≀C₂

C₂×S₃×D₄

S₃×C₂⁴

C₂²≀C₂

C₂²×S₃

C₂²⋊C₄

C₂×D₄

C₂⁴

C₂²

# reps

Matrix representation of S₃×C₂²≀C₂ ►in GL₆(𝔽₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	12	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	12	12	0	0
0	0	0	0	1	0
0	0	0	0	0	1

12	0	0	0	0	0
5	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	12

12	0	0	0	0	0
0	12	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	0
0	0	0	0	0	1

12	0	0	0	0	0
0	12	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	0
0	0	0	0	0	12

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	12	0
0	0	0	0	0	12

1	3	0	0	0	0
0	12	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,5,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,3,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

S₃×C₂²≀C₂ in GAP, Magma, Sage, TeX

S_3\times C_2^2\wr C_2

% in TeX

G:=Group("S3xC2^2wrC2");

// GroupNames label

G:=SmallGroup(192,1147);

// by ID

G=gap.SmallGroup(192,1147);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,570,185,6278]);

// Polycyclic

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

// generators/relations

G = S3×C22≀C2 order 192 = 26·3

Direct product of S3 and C22≀C2

G = S₃×C₂²≀C₂ order 192 = 2⁶·3

Direct product of S₃ and C₂²≀C₂