S3xC4.4D4 - GroupNames

G = S₃xC₄.₄D₄ order 192 = 2⁶·3

Direct product of S₃ and C₄.₄D₄

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

Aliases: S₃xC₄.₄D₄, C₄²:₃₅D₆, (C₂xQ₈):₂₀D₆, C₄.₃₁(S₃xD₄), (S₃xC₄²):₉C₂, C₂²:C₄:₃₂D₆, (C₄xS₃).₂₄D₄, D₆.₆₀(C₂xD₄), C₁₂.₆₀(C₂xD₄), (C₄xC₁₂):₂₁C₂², D₆:C₄:₂₉C₂², (C₂xD₄).₁₇₀D₆, Dic₃.₉(C₂xD₄), (C₆xQ₈):₁₁C₂², C₆.₈₇(C₂²xD₄), C₄²:₇S₃:₂₃C₂, D₆.₄₀(C₄oD₄), (C₂xC₆).₂₁₇C₂⁴, (C₂xC₁₂).₇₉C₂³, C₂³.₁₂D₆:₂₃C₂, C₁₂.₂₃D₄:₂₀C₂, (C₂xDic₆):₃₂C₂², (C₄xDic₃):₇₉C₂², (C₆xD₄).₁₅₂C₂², (C₂²xC₆).₄₇C₂³, C₂³.₄₉(C₂²xS₃), C₂³.₁₁D₆:₃₉C₂, (C₂xD₁₂).₁₆₂C₂², C₆.D₄:₃₂C₂², (C₂²xS₃).₉₅C₂³, (S₃xC₂³).₆₂C₂², C₂².₂₃₈(S₃xC₂³), (C₂xDic₃).₁₁₂C₂³, (C₂xS₃xQ₈):₉C₂, (C₂xS₃xD₄).₉C₂, C₂.₆₀(C₂xS₃xD₄), C₃:₃(C₂xC₄.₄D₄), C₂.₇₅(S₃xC₄oD₄), (S₃xC₂²:C₄):₁₆C₂, C₆.₁₈₆(C₂xC₄oD₄), (C₃xC₄.₄D₄):₉C₂, (S₃xC₂xC₄).₂₉₇C₂², (C₃xC₂²:C₄):₂₇C₂², (C₂xC₄).₃₀₀(C₂²xS₃), (C₂xC₃:D₄).₅₈C₂², SmallGroup(192,1232)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — S₃xC₄.₄D₄

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xS₃ — S₃xC₂³ — S₃xC₂²:C₄ — S₃xC₄.₄D₄

Lower central

C₃ — C₂xC₆ — S₃xC₄.₄D₄

Upper central

C₁ — C₂² — C₄.₄D₄

Generators and relations for S₃xC₄.₄D₄
G = < a,b,c,d,e | a³=b²=c⁴=d⁴=1, e²=c², bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=c^-1, ede^-1=c²d^-1 >

Subgroups: 944 in 330 conjugacy classes, 111 normal (29 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₄², C₄², C₂²:C₄, C₂²:C₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₂xQ₈, C₂⁴, Dic₆, C₄xS₃, C₄xS₃, D₁₂, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂xC₄², C₂xC₂²:C₄, C₄.₄D₄, C₄.₄D₄, C₂²xD₄, C₂²xQ₈, C₄xDic₃, C₄xDic₃, D₆:C₄, C₆.D₄, C₄xC₁₂, C₃xC₂²:C₄, C₂xDic₆, C₂xDic₆, S₃xC₂xC₄, S₃xC₂xC₄, C₂xD₁₂, S₃xD₄, S₃xQ₈, C₂xC₃:D₄, C₆xD₄, C₆xQ₈, S₃xC₂³, C₂xC₄.₄D₄, S₃xC₄², C₄²:₇S₃, S₃xC₂²:C₄, C₂³.₁₁D₆, C₂³.₁₂D₆, C₁₂.₂₃D₄, C₃xC₄.₄D₄, C₂xS₃xD₄, C₂xS₃xQ₈, S₃xC₄.₄D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₄oD₄, C₂⁴, C₂²xS₃, C₄.₄D₄, C₂²xD₄, C₂xC₄oD₄, S₃xD₄, S₃xC₂³, C₂xC₄.₄D₄, C₂xS₃xD₄, S₃xC₄oD₄, S₃xC₄.₄D₄

Smallest permutation representation of S₃xC₄.₄D₄
►On 48 points

Generators in S₄₈

(1 13 45)(2 14 46)(3 15 47)(4 16 48)(5 21 35)(6 22 36)(7 23 33)(8 24 34)(9 38 32)(10 39 29)(11 40 30)(12 37 31)(17 44 28)(18 41 25)(19 42 26)(20 43 27)

(1 9)(2 10)(3 11)(4 12)(5 43)(6 44)(7 41)(8 42)(13 32)(14 29)(15 30)(16 31)(17 22)(18 23)(19 24)(20 21)(25 33)(26 34)(27 35)(28 36)(37 48)(38 45)(39 46)(40 47)

(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 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)

(1 26 9 34)(2 27 10 35)(3 28 11 36)(4 25 12 33)(5 14 20 39)(6 15 17 40)(7 16 18 37)(8 13 19 38)(21 46 43 29)(22 47 44 30)(23 48 41 31)(24 45 42 32)

(1 35 3 33)(2 34 4 36)(5 15 7 13)(6 14 8 16)(9 27 11 25)(10 26 12 28)(17 39 19 37)(18 38 20 40)(21 47 23 45)(22 46 24 48)(29 42 31 44)(30 41 32 43)

G:=sub<Sym(48)| (1,13,45)(2,14,46)(3,15,47)(4,16,48)(5,21,35)(6,22,36)(7,23,33)(8,24,34)(9,38,32)(10,39,29)(11,40,30)(12,37,31)(17,44,28)(18,41,25)(19,42,26)(20,43,27), (1,9)(2,10)(3,11)(4,12)(5,43)(6,44)(7,41)(8,42)(13,32)(14,29)(15,30)(16,31)(17,22)(18,23)(19,24)(20,21)(25,33)(26,34)(27,35)(28,36)(37,48)(38,45)(39,46)(40,47), (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)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,26,9,34)(2,27,10,35)(3,28,11,36)(4,25,12,33)(5,14,20,39)(6,15,17,40)(7,16,18,37)(8,13,19,38)(21,46,43,29)(22,47,44,30)(23,48,41,31)(24,45,42,32), (1,35,3,33)(2,34,4,36)(5,15,7,13)(6,14,8,16)(9,27,11,25)(10,26,12,28)(17,39,19,37)(18,38,20,40)(21,47,23,45)(22,46,24,48)(29,42,31,44)(30,41,32,43)>;

G:=Group( (1,13,45)(2,14,46)(3,15,47)(4,16,48)(5,21,35)(6,22,36)(7,23,33)(8,24,34)(9,38,32)(10,39,29)(11,40,30)(12,37,31)(17,44,28)(18,41,25)(19,42,26)(20,43,27), (1,9)(2,10)(3,11)(4,12)(5,43)(6,44)(7,41)(8,42)(13,32)(14,29)(15,30)(16,31)(17,22)(18,23)(19,24)(20,21)(25,33)(26,34)(27,35)(28,36)(37,48)(38,45)(39,46)(40,47), (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)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,26,9,34)(2,27,10,35)(3,28,11,36)(4,25,12,33)(5,14,20,39)(6,15,17,40)(7,16,18,37)(8,13,19,38)(21,46,43,29)(22,47,44,30)(23,48,41,31)(24,45,42,32), (1,35,3,33)(2,34,4,36)(5,15,7,13)(6,14,8,16)(9,27,11,25)(10,26,12,28)(17,39,19,37)(18,38,20,40)(21,47,23,45)(22,46,24,48)(29,42,31,44)(30,41,32,43) );

G=PermutationGroup([[(1,13,45),(2,14,46),(3,15,47),(4,16,48),(5,21,35),(6,22,36),(7,23,33),(8,24,34),(9,38,32),(10,39,29),(11,40,30),(12,37,31),(17,44,28),(18,41,25),(19,42,26),(20,43,27)], [(1,9),(2,10),(3,11),(4,12),(5,43),(6,44),(7,41),(8,42),(13,32),(14,29),(15,30),(16,31),(17,22),(18,23),(19,24),(20,21),(25,33),(26,34),(27,35),(28,36),(37,48),(38,45),(39,46),(40,47)], [(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),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,26,9,34),(2,27,10,35),(3,28,11,36),(4,25,12,33),(5,14,20,39),(6,15,17,40),(7,16,18,37),(8,13,19,38),(21,46,43,29),(22,47,44,30),(23,48,41,31),(24,45,42,32)], [(1,35,3,33),(2,34,4,36),(5,15,7,13),(6,14,8,16),(9,27,11,25),(10,26,12,28),(17,39,19,37),(18,38,20,40),(21,47,23,45),(22,46,24,48),(29,42,31,44),(30,41,32,43)]])

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	3	4A	···	4F	4G	4H	4I	···	4N	4O	4P	6A	6B	6C	6D	6E	12A	···	12F	12G	12H
order	1	2	2	2	2	2	2	2	2	2	2	2	3	4	···	4	4	4	4	···	4	4	4	6	6	6	6	6	12	···	12	12	12
size	1	1	1	1	3	3	3	3	4	4	12	12	2	2	···	2	4	4	6	···	6	12	12	2	2	2	8	8	4	···	4	8	8

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

C₄oD₄

S₃xD₄

S₃xC₄oD₄

kernel

S₃xC₄.₄D₄

S₃xC₄²

C₄²:₇S₃

S₃xC₂²:C₄

C₂³.₁₁D₆

C₂³.₁₂D₆

C₁₂.₂₃D₄

C₃xC₄.₄D₄

C₂xS₃xD₄

C₂xS₃xQ₈

C₄.₄D₄

C₄xS₃

C₄²

C₂²:C₄

C₂xD₄

C₂xQ₈

D₆

C₄

C₂

# reps

Matrix representation of S₃xC₄.₄D₄ ►in GL₆(F₁₃)

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

0	8	0	0	0	0
8	0	0	0	0	0
0	0	1	3	0	0
0	0	8	12	0	0
0	0	0	0	1	0
0	0	0	0	0	1

5	0	0	0	0	0
0	5	0	0	0	0
0	0	5	2	0	0
0	0	1	8	0	0
0	0	0	0	12	0
0	0	0	0	0	12

0	1	0	0	0	0
12	0	0	0	0	0
0	0	8	0	0	0
0	0	12	5	0	0
0	0	0	0	12	0
0	0	0	0	0	12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,8,0,0,0,0,3,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,1,0,0,0,0,2,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,12,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

S₃xC₄.₄D₄ in GAP, Magma, Sage, TeX

S_3\times C_4._4D_4

% in TeX

G:=Group("S3xC4.4D4");

// GroupNames label

G:=SmallGroup(192,1232);

// by ID

G=gap.SmallGroup(192,1232);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,346,136,6278]);

// Polycyclic

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

// generators/relations

G = S3xC4.4D4 order 192 = 26·3

Direct product of S3 and C4.4D4

G = S₃xC₄.₄D₄ order 192 = 2⁶·3

Direct product of S₃ and C₄.₄D₄