C3xS3xS4 - GroupNames

G = C₃×S₃×S₄ order 432 = 2⁴·3³

Direct product of C₃, S₃ and S₄

direct product, non-abelian, soluble, monomial

Aliases: C₃×S₃×S₄, C₆²⋊₇D₆, (C₃×S₄)⋊C₆, (S₃×A₄)⋊C₆, C₃⋊S₄⋊₂C₆, C₃⋊₁(C₆×S₄), A₄⋊₁(S₃×C₆), (C₃×A₄)⋊₇D₆, C₃²⋊₆(C₂×S₄), (C₃²×S₄)⋊₁C₂, (C₃²×A₄)⋊₁C₂², (C₂×C₆)⋊₃S₃², (C₂×C₆)⋊(S₃×C₆), C₂²⋊(C₃×S₃²), (C₃×S₃×A₄)⋊₁C₂, (S₃×C₂×C₆)⋊₁S₃, (C₃×C₃⋊S₄)⋊₃C₂, (C₃×A₄)⋊₂(C₂×C₆), (C₂²×S₃)⋊(C₃×S₃), SmallGroup(432,745)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃×A₄ — C₃×S₃×S₄

Chief series

C₁ — C₂² — C₂×C₆ — C₃×A₄ — C₃²×A₄ — C₃×S₃×A₄ — C₃×S₃×S₄

Lower central

C₃×A₄ — C₃×S₃×S₄

Upper central

C₁ — C₃

Generators and relations for C₃×S₃×S₄
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, C₂×C₄, D₄, C₂³, C₃², C₃², Dic₃, C₁₂, A₄, A₄, D₆, C₂×C₆, C₂×C₆, C₂×D₄, C₃×S₃, C₃×S₃, C₃⋊S₃, C₃×C₆, C₄×S₃, D₁₂, C₃⋊D₄, C₂×C₁₂, C₃×D₄, S₄, S₄, C₂×A₄, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₃³, C₃×Dic₃, C₃×C₁₂, S₃², C₃×A₄, C₃×A₄, S₃×C₆, C₆², C₆², S₃×D₄, C₆×D₄, C₂×S₄, S₃×C₃², C₃×C₃⋊S₃, S₃×C₁₂, C₃×D₁₂, C₃×C₃⋊D₄, D₄×C₃², C₃×S₄, C₃×S₄, C₃⋊S₄, S₃×A₄, S₃×A₄, C₆×A₄, S₃×C₂×C₆, S₃×C₂×C₆, C₃×S₃², C₃²×A₄, C₃×S₃×D₄, S₃×S₄, C₆×S₄, C₃²×S₄, C₃×C₃⋊S₄, C₃×S₃×A₄, C₃×S₃×S₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₆, C₂×C₆, C₃×S₃, S₄, S₃², S₃×C₆, C₂×S₄, C₃×S₄, C₃×S₃², S₃×S₄, C₆×S₄, C₃×S₃×S₄

Permutation representations of C₃×S₃×S₄
►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	C₃×S₃	C₃×S₃	S₃×C₆	S₃×C₆	S₄	C₂×S₄	C₃×S₄	C₆×S₄	S₃²	C₃×S₃²	S₃×S₄	C₃×S₃×S₄
kernel	C₃×S₃×S₄	C₃²×S₄	C₃×C₃⋊S₄	C₃×S₃×A₄	S₃×S₄	C₃×S₄	C₃⋊S₄	S₃×A₄	C₃×S₄	S₃×C₂×C₆	C₃×A₄	C₆²	S₄	C₂²×S₃	A₄	C₂×C₆	C₃×S₃	C₃²	S₃	C₃	C₂×C₆	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 C₃×S₃×S₄ ►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] >;

C₃×S₃×S₄ 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 = C3×S3×S4 order 432 = 24·33

Direct product of C3, S3 and S4

G = C₃×S₃×S₄ order 432 = 2⁴·3³

Direct product of C₃, S₃ and S₄