S3xC4.D4 - GroupNames

G = S₃xC₄.D₄ order 192 = 2⁶·3

Direct product of S₃ and C₄.D₄

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

Aliases: S₃xC₄.D₄, M₄(2):₁₄D₆, (C₄xS₃).₁D₄, C₄.₁₄₆(S₃xD₄), C₁₂.₈₉(C₂xD₄), (C₂xD₄).₁₂₁D₆, C₁₂.D₄:₃C₂, (S₃xM₄(2)):₅C₂, (S₃xC₂³).₂C₄, (C₂xC₁₂).₁C₂³, C₂³.₁₁(C₄xS₃), C₁₂.₄₆D₄:₉C₂, (C₆xD₄).₁₁C₂², C₄.Dic₃:₁C₂², D₆.₁₆(C₂²:C₄), (C₂xD₁₂).₃₈C₂², (C₃xM₄(2)):₁₆C₂², Dic₃.₄(C₂²:C₄), (C₂xS₃xD₄).₂C₂, C₃:₁(C₂xC₄.D₄), (C₂xC₃:D₄).₁C₄, (S₃xC₂xC₄).₁C₂², C₂².₁₄(S₃xC₂xC₄), (C₃xC₄.D₄):₉C₂, C₂.₁₃(S₃xC₂²:C₄), C₆.₁₂(C₂xC₂²:C₄), (C₂xC₆).₈(C₂²xC₄), (C₂²xC₆).₆(C₂xC₄), (C₂xC₄).₁(C₂²xS₃), (C₂²xS₃).₂(C₂xC₄), (C₂xDic₃).₁₉(C₂xC₄), SmallGroup(192,303)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — S₃xC₄.D₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — S₃xC₂xC₄ — C₂xS₃xD₄ — S₃xC₄.D₄

Lower central

C₃ — C₆ — C₂xC₆ — S₃xC₄.D₄

Upper central

C₁ — C₂ — C₂xC₄ — C₄.D₄

Generators and relations for S₃xC₄.D₄
G = < a,b,c,d,e | a³=b²=c⁴=1, d⁴=c², e²=c, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd^-1=c^-1, ce=ec, ede^-1=c^-1d³ >

Subgroups: 656 in 186 conjugacy classes, 53 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂xC₈, M₄(2), M₄(2), C₂²xC₄, C₂xD₄, C₂xD₄, C₂⁴, C₃:C₈, C₂₄, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₄.D₄, C₄.D₄, C₂xM₄(2), C₂²xD₄, S₃xC₈, C₈:S₃, C₄.Dic₃, C₃xM₄(2), S₃xC₂xC₄, C₂xD₁₂, S₃xD₄, C₂xC₃:D₄, C₆xD₄, S₃xC₂³, C₂xC₄.D₄, C₁₂.₄₆D₄, C₁₂.D₄, C₃xC₄.D₄, S₃xM₄(2), C₂xS₃xD₄, S₃xC₄.D₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, C₂³, D₆, C₂²:C₄, C₂²xC₄, C₂xD₄, C₄xS₃, C₂²xS₃, C₄.D₄, C₂xC₂²:C₄, S₃xC₂xC₄, S₃xD₄, C₂xC₄.D₄, S₃xC₂²:C₄, S₃xC₄.D₄

Permutation representations of S₃xC₄.D₄
►On 24 points - transitive group 24T339

Generators in S₂₄

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

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

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

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

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

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

G:=TransitiveGroup(24,339);

33 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	3	4A	4B	4C	4D	6A	6B	6C	6D	8A	8B	8C	8D	8E	8F	8G	8H	12A	12B	24A	24B	24C	24D
order	1	2	2	2	2	2	2	2	2	2	3	4	4	4	4	6	6	6	6	8	8	8	8	8	8	8	8	12	12	24	24	24	24
size	1	1	2	3	3	4	4	6	12	12	2	2	2	6	6	2	4	8	8	4	4	4	4	12	12	12	12	4	4	8	8	8	8

33 irreducible representations

dim

type

image

C₁

C₂

C₄

S₃

D₄

D₆

C₄xS₃

C₄.D₄

S₃xD₄

S₃xC₄.D₄

kernel

S₃xC₄.D₄

C₁₂.₄₆D₄

C₁₂.D₄

C₃xC₄.D₄

S₃xM₄(2)

C₂xS₃xD₄

C₂xC₃:D₄

S₃xC₂³

C₄.D₄

C₄xS₃

M₄(2)

C₂xD₄

C₂³

S₃

C₄

C₁

# reps

Matrix representation of S₃xC₄.D₄ ►in GL₆(F₇₃)

72	72	0	0	0	0
1	0	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
72	72	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	72	0
0	0	0	0	0	72

72	0	0	0	0	0
0	72	0	0	0	0
0	0	72	3	0	0
0	0	48	1	0	0
0	0	59	0	0	1
0	0	59	42	72	0

27	0	0	0	0	0
0	27	0	0	0	0
0	0	46	42	71	0
0	0	55	27	48	25
0	0	60	35	13	14
0	0	59	4	59	60

46	0	0	0	0	0
0	46	0	0	0	0
0	0	46	42	71	0
0	0	0	1	48	48
0	0	0	0	13	59
0	0	1	0	59	13

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,48,59,59,0,0,3,1,0,42,0,0,0,0,0,72,0,0,0,0,1,0],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,46,55,60,59,0,0,42,27,35,4,0,0,71,48,13,59,0,0,0,25,14,60],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,1,0,0,42,1,0,0,0,0,71,48,13,59,0,0,0,48,59,13] >;

S₃xC₄.D₄ in GAP, Magma, Sage, TeX

S_3\times C_4.D_4

% in TeX

G:=Group("S3xC4.D4");

// GroupNames label

G:=SmallGroup(192,303);

// by ID

G=gap.SmallGroup(192,303);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,58,570,136,438,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=1,d^4=c^2,e^2=c,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=c^-1*d^3>;

// generators/relations

G = S3xC4.D4 order 192 = 26·3

Direct product of S3 and C4.D4

G = S₃xC₄.D₄ order 192 = 2⁶·3

Direct product of S₃ and C₄.D₄