S3xC4^2:2C2 - GroupNames

G = S₃×C₄²⋊₂C₂ order 192 = 2⁶·3

Direct product of S₃ and C₄²⋊₂C₂

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

Aliases: S₃×C₄²⋊₂C₂, C₄²⋊₃₂D₆, C₄⋊C₄⋊₃₂D₆, (S₃×C₄²)⋊₁₉C₂, (C₄×C₁₂)⋊₃₁C₂², C₂²⋊C₄.₇₆D₆, C₄²⋊₃S₃⋊₁₃C₂, D₆.₄₂(C₄○D₄), (C₂×C₁₂).₉₄C₂³, (C₂×C₆).₂₄₇C₂⁴, D₆⋊C₄.₄₄C₂², C₄⋊Dic₃⋊₄₃C₂², Dic₃⋊C₄⋊₃₁C₂², (C₄×Dic₃)⋊₈₀C₂², C₂³.₈D₆⋊₄₄C₂, (C₂²×C₆).₆₁C₂³, C₂³.₆₃(C₂²×S₃), (S₃×C₂³).₆₇C₂², C₂².₂₆₈(S₃×C₂³), (C₂²×S₃).₂₅₈C₂³, (C₂×Dic₃).₁₂₈C₂³, C₆.D₄.₆₃C₂², (S₃×C₄⋊C₄)⋊₄₀C₂, C₃⋊₄(C₂×C₄²⋊₂C₂), C₄⋊C₄⋊S₃⋊₄₀C₂, C₂.₉₄(S₃×C₄○D₄), (C₃×C₄⋊C₄)⋊₃₁C₂², C₆.₂₀₅(C₂×C₄○D₄), (S₃×C₂²⋊C₄).₃C₂, (C₃×C₄²⋊₂C₂)⋊₂C₂, (S₃×C₂×C₄).₂₉₉C₂², (C₂×C₄).₈₄(C₂²×S₃), (C₃×C₂²⋊C₄).₇₂C₂², SmallGroup(192,1262)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — S₃×C₄²⋊₂C₂

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂³ — S₃×C₂²⋊C₄ — S₃×C₄²⋊₂C₂

Lower central

C₃ — C₂×C₆ — S₃×C₄²⋊₂C₂

Upper central

C₁ — C₂² — C₄²⋊₂C₂

Generators and relations for S₃×C₄²⋊₂C₂
G = < a,b,c,d,e | a³=b²=c⁴=d⁴=e²=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=cd², ede=c²d^-1 >

Subgroups: 624 in 246 conjugacy classes, 101 normal (16 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₂×C₄, C₂×C₄, C₂³, C₂³, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂⁴, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊₂C₂, C₄²⋊₂C₂, C₄×Dic₃, Dic₃⋊C₄, C₄⋊Dic₃, D₆⋊C₄, C₆.D₄, C₄×C₁₂, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, S₃×C₂×C₄, S₃×C₂³, C₂×C₄²⋊₂C₂, S₃×C₄², C₄²⋊₃S₃, C₂³.₈D₆, S₃×C₂²⋊C₄, S₃×C₄⋊C₄, C₄⋊C₄⋊S₃, C₃×C₄²⋊₂C₂, S₃×C₄²⋊₂C₂
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄○D₄, C₂⁴, C₂²×S₃, C₄²⋊₂C₂, C₂×C₄○D₄, S₃×C₂³, C₂×C₄²⋊₂C₂, S₃×C₄○D₄, S₃×C₄²⋊₂C₂

Smallest permutation representation of S₃×C₄²⋊₂C₂
►On 48 points

Generators in S₄₈

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

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

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

(2 10)(4 12)(5 20)(6 8)(7 18)(13 41)(14 16)(15 43)(17 19)(21 48)(23 46)(26 40)(28 38)(29 31)(30 33)(32 35)(34 36)(42 44)

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

C₄○D₄

S₃×C₄○D₄

kernel

S₃×C₄²⋊₂C₂

S₃×C₄²

C₄²⋊₃S₃

C₂³.₈D₆

S₃×C₂²⋊C₄

S₃×C₄⋊C₄

C₄⋊C₄⋊S₃

C₃×C₄²⋊₂C₂

C₄²⋊₂C₂

C₄²

C₂²⋊C₄

C₄⋊C₄

D₆

C₂

# reps

Matrix representation of S₃×C₄²⋊₂C₂ ►in GL₆(𝔽₁₃)

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	8	3	0	0
0	0	0	5	0	0
0	0	0	0	8	0
0	0	0	0	0	8

12	0	0	0	0	0
0	12	0	0	0	0
0	0	5	0	0	0
0	0	0	5	0	0
0	0	0	0	5	8
0	0	0	0	10	8

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	2	12

G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,3,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,10,0,0,0,0,8,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,2,0,0,0,0,0,12] >;

S₃×C₄²⋊₂C₂ in GAP, Magma, Sage, TeX

S_3\times C_4^2\rtimes_2C_2

% in TeX

G:=Group("S3xC4^2:2C2");

// GroupNames label

G:=SmallGroup(192,1262);

// by ID

G=gap.SmallGroup(192,1262);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,346,297,136,6278]);

// Polycyclic

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

// generators/relations

G = S3×C42⋊2C2 order 192 = 26·3

Direct product of S3 and C42⋊2C2

G = S₃×C₄²⋊₂C₂ order 192 = 2⁶·3

Direct product of S₃ and C₄²⋊₂C₂