C2^2xS3xD4 - GroupNames

G = C₂²×S₃×D₄ order 192 = 2⁶·3

Direct product of C₂², S₃ and D₄

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

Aliases: C₂²×S₃×D₄, C₁₂⋊C₂⁴, D₆⋊₂C₂⁴, C₂⁴⋊₁₅D₆, C₆.₅C₂⁵, D₁₂⋊₈C₂³, Dic₃⋊₁C₂⁴, (C₂×C₆)⋊C₂⁴, C₃⋊₂(D₄×C₂³), C₄⋊₁(S₃×C₂³), C₆⋊₂(C₂²×D₄), (S₃×C₂⁴)⋊₅C₂, (C₄×S₃)⋊₄C₂³, (C₂×C₁₂)⋊₄C₂³, (C₃×D₄)⋊₆C₂³, (C₂²×C₄)⋊₄₂D₆, C₃⋊D₄⋊₁C₂³, C₂.₆(S₃×C₂⁴), (C₆×D₄)⋊₄₉C₂², (C₂²×C₆)⋊₆C₂³, C₂³⋊₆(C₂²×S₃), C₂²⋊₂(S₃×C₂³), (C₂²×D₁₂)⋊₂₂C₂, (C₂×D₁₂)⋊₆₀C₂², (C₂²×S₃)⋊₈C₂³, (C₂³×C₆)⋊₁₅C₂², (S₃×C₂³)⋊₂₃C₂², (C₂²×C₁₂)⋊₂₅C₂², (C₂×Dic₃)⋊₁₀C₂³, (C₂²×Dic₃)⋊₅₁C₂², (D₄×C₂×C₆)⋊₉C₂, (C₂×C₆)⋊₁₄(C₂×D₄), (S₃×C₂²×C₄)⋊₈C₂, (S₃×C₂×C₄)⋊₅₈C₂², (C₂×C₄)⋊₈(C₂²×S₃), (C₂²×C₃⋊D₄)⋊₁₉C₂, (C₂×C₃⋊D₄)⋊₅₀C₂², SmallGroup(192,1514)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₂²×S₃×D₄

Chief series

C₁ — C₃ — C₆ — D₆ — C₂²×S₃ — S₃×C₂³ — S₃×C₂⁴ — C₂²×S₃×D₄

Lower central

C₃ — C₆ — C₂²×S₃×D₄

Upper central

C₁ — C₂³ — C₂²×D₄

Generators and relations for C₂²×S₃×D₄
G = < a,b,c,d,e,f | a²=b²=c³=d²=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 3432 in 1362 conjugacy classes, 503 normal (15 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₆, C₂×C₄, C₂×C₄, D₄, D₄, C₂³, C₂³, C₂³, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, 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×C₆, C₂²×C₆, C₂²×C₆, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, S₃×C₂×C₄, C₂×D₁₂, S₃×D₄, C₂²×Dic₃, C₂×C₃⋊D₄, C₂²×C₁₂, C₆×D₄, S₃×C₂³, S₃×C₂³, S₃×C₂³, C₂³×C₆, D₄×C₂³, S₃×C₂²×C₄, C₂²×D₁₂, C₂×S₃×D₄, C₂²×C₃⋊D₄, D₄×C₂×C₆, S₃×C₂⁴, C₂²×S₃×D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₂⁴, C₂²×S₃, C₂²×D₄, C₂⁵, S₃×D₄, S₃×C₂³, D₄×C₂³, C₂×S₃×D₄, S₃×C₂⁴, C₂²×S₃×D₄

Smallest permutation representation of C₂²×S₃×D₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

60 conjugacy classes

class	1	2A	···	2G	2H	···	2O	2P	···	2W	2X	···	2AE	3	4A	4B	4C	4D	4E	4F	4G	4H	6A	···	6G	6H	···	6O	12A	12B	12C	12D
order	1	2	···	2	2	···	2	2	···	2	2	···	2	3	4	4	4	4	4	4	4	4	6	···	6	6	···	6	12	12	12	12
size	1	1	···	1	2	···	2	3	···	3	6	···	6	2	2	2	2	2	6	6	6	6	2	···	2	4	···	4	4	4	4	4

60 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

S₃×D₄

kernel

C₂²×S₃×D₄

S₃×C₂²×C₄

C₂²×D₁₂

C₂×S₃×D₄

C₂²×C₃⋊D₄

D₄×C₂×C₆

S₃×C₂⁴

C₂²×D₄

C₂²×S₃

C₂²×C₄

C₂×D₄

C₂⁴

C₂²

# reps

Matrix representation of C₂²×S₃×D₄ ►in GL₅(ℤ)

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

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

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

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

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

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

G:=sub<GL(5,Integers())| [1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,0],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1] >;

C₂²×S₃×D₄ in GAP, Magma, Sage, TeX

C_2^2\times S_3\times D_4

% in TeX

G:=Group("C2^2xS3xD4");

// GroupNames label

G:=SmallGroup(192,1514);

// by ID

G=gap.SmallGroup(192,1514);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,235,6278]);

// Polycyclic

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

// generators/relations

G = C22×S3×D4 order 192 = 26·3

Direct product of C22, S3 and D4

G = C₂²×S₃×D₄ order 192 = 2⁶·3

Direct product of C₂², S₃ and D₄