C3xS3xS4 - GroupNames

G = C₃xS₃xS₄ order 432 = 2⁴·3³

Direct product of C₃, S₃ and S₄

direct product, non-abelian, soluble, monomial

Aliases: C₃xS₃xS₄, C₆²:₇D₆, (C₃xS₄):C₆, (S₃xA₄):C₆, C₃:S₄:₂C₆, C₃:₁(C₆xS₄), A₄:₁(S₃xC₆), (C₃xA₄):₇D₆, C₃²:₆(C₂xS₄), (C₃²xS₄):₁C₂, (C₃²xA₄):₁C₂², (C₂xC₆):₃S₃², (C₂xC₆):(S₃xC₆), C₂²:(C₃xS₃²), (C₃xS₃xA₄):₁C₂, (S₃xC₂xC₆):₁S₃, (C₃xC₃:S₄):₃C₂, (C₃xA₄):₂(C₂xC₆), (C₂²xS₃):(C₃xS₃), SmallGroup(432,745)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃xA₄ — C₃xS₃xS₄

Chief series

C₁ — C₂² — C₂xC₆ — C₃xA₄ — C₃²xA₄ — C₃xS₃xA₄ — C₃xS₃xS₄

Lower central

C₃xA₄ — C₃xS₃xS₄

Upper central

C₁ — C₃

Generators and relations for C₃xS₃xS₄
G = < a,b,c,d,e,f,g | a³=b³=c²=d²=e²=f³=g²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, cbc=b^-1, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fdf^-1=gdg=de=ed, fef^-1=d, eg=ge, gfg=f^-1 >

Subgroups: 904 in 163 conjugacy classes, 26 normal (all characteristic)
C₁, C₂, C₃, C₃, C₄, C₂², C₂², S₃, S₃, C₆, C₂xC₄, D₄, C₂³, C₃², C₃², Dic₃, C₁₂, A₄, A₄, D₆, C₂xC₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃xS₃, C₃:S₃, C₃xC₆, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₃xD₄, S₄, S₄, C₂xA₄, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₃³, C₃xDic₃, C₃xC₁₂, S₃², C₃xA₄, C₃xA₄, S₃xC₆, C₆², C₆², S₃xD₄, C₆xD₄, C₂xS₄, S₃xC₃², C₃xC₃:S₃, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, D₄xC₃², C₃xS₄, C₃xS₄, C₃:S₄, S₃xA₄, S₃xA₄, C₆xA₄, S₃xC₂xC₆, S₃xC₂xC₆, C₃xS₃², C₃²xA₄, C₃xS₃xD₄, S₃xS₄, C₆xS₄, C₃²xS₄, C₃xC₃:S₄, C₃xS₃xA₄, C₃xS₃xS₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₆, C₂xC₆, C₃xS₃, S₄, S₃², S₃xC₆, C₂xS₄, C₃xS₄, C₃xS₃², S₃xS₄, C₆xS₄, C₃xS₃xS₄

Permutation representations of C₃xS₃xS₄
►On 24 points - transitive group 24T1328

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

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

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

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

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

(10 13)(11 14)(12 15)(16 23)(17 24)(18 22)

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

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

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

G:=TransitiveGroup(24,1328);

45 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	3E	3F	3G	3H	3I	3J	3K	4A	4B	6A	6B	6C	6D	6E	···	6I	6J	6K	6L	6M	6N	6O	6P	6Q	6R	6S	12A	12B	12C	12D	12E	12F	12G
order	1	2	2	2	2	2	3	3	3	3	3	3	3	3	3	3	3	4	4	6	6	6	6	6	···	6	6	6	6	6	6	6	6	6	6	6	12	12	12	12	12	12	12
size	1	3	3	6	9	18	1	1	2	2	2	8	8	8	16	16	16	6	18	3	3	3	3	6	···	6	9	9	12	12	12	18	18	24	24	24	6	6	12	12	12	18	18

45 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	3	3	3	3	4	4	6	6
type	+	+	+	+					+	+	+	+					+	+			+		+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	S₃	D₆	D₆	C₃xS₃	C₃xS₃	S₃xC₆	S₃xC₆	S₄	C₂xS₄	C₃xS₄	C₆xS₄	S₃²	C₃xS₃²	S₃xS₄	C₃xS₃xS₄
kernel	C₃xS₃xS₄	C₃²xS₄	C₃xC₃:S₄	C₃xS₃xA₄	S₃xS₄	C₃xS₄	C₃:S₄	S₃xA₄	C₃xS₄	S₃xC₂xC₆	C₃xA₄	C₆²	S₄	C₂²xS₃	A₄	C₂xC₆	C₃xS₃	C₃²	S₃	C₃	C₂xC₆	C₂²	C₃	C₁
# reps	1	1	1	1	2	2	2	2	1	1	1	1	2	2	2	2	2	2	4	4	1	2	2	4

Matrix representation of C₃xS₃xS₄ ►in GL₇(F₁₃)

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	3	0	0	0	0
0	0	0	3	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

9	0	0	0	0	0	0
3	3	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

12	2	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

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

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

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

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

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[9,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,1,12,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,1,0,0,0,0,12,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C₃xS₃xS₄ in GAP, Magma, Sage, TeX

C_3\times S_3\times S_4

% in TeX

G:=Group("C3xS3xS4");

// GroupNames label

G:=SmallGroup(432,745);

// by ID

G=gap.SmallGroup(432,745);

# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,346,2524,4548,782,2659,1350]);

// Polycyclic

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

// generators/relations

G = C3xS3xS4 order 432 = 24·33

Direct product of C3, S3 and S4

G = C₃xS₃xS₄ order 432 = 2⁴·3³

Direct product of C₃, S₃ and S₄