C2xS3xC3^2:C4 - GroupNames

G = C₂×S₃×C₃²⋊C₄ order 432 = 2⁴·3³

Direct product of C₂, S₃ and C₃²⋊C₄

direct product, metabelian, soluble, monomial, A-group

Aliases: C₂×S₃×C₃²⋊C₄, C₃³⋊₁(C₂²×C₄), C₃³⋊C₄⋊₂C₂², (S₃×C₃×C₆)⋊₂C₄, (S₃×C₃⋊S₃)⋊₂C₄, C₃⋊S₃⋊₅(C₄×S₃), (C₃×C₆)⋊₅(C₄×S₃), (C₃²×C₆)⋊(C₂×C₄), C₆⋊₁(C₂×C₃²⋊C₄), (C₆×C₃²⋊C₄)⋊₆C₂, C₃²⋊₁₀(S₃×C₂×C₄), (C₂×C₃⋊S₃).₃₈D₆, C₃⋊₁(C₂²×C₃²⋊C₄), (S₃×C₃²)⋊₁(C₂×C₄), (C₂×C₃³⋊C₂)⋊₂C₄, C₃³⋊C₂⋊₁(C₂×C₄), (C₂×C₃³⋊C₄)⋊₅C₂, C₃⋊S₃.₆(C₂²×S₃), (C₃×C₃⋊S₃).₅C₂³, (S₃×C₃⋊S₃).₃C₂², (C₃×C₃²⋊C₄)⋊₃C₂², (C₆×C₃⋊S₃).₃₂C₂², (C₂×S₃×C₃⋊S₃).₃C₂, (C₃×C₃⋊S₃)⋊₄(C₂×C₄), SmallGroup(432,753)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₂×S₃×C₃²⋊C₄

Chief series

C₁ — C₃ — C₃³ — C₃×C₃⋊S₃ — S₃×C₃⋊S₃ — S₃×C₃²⋊C₄ — C₂×S₃×C₃²⋊C₄

Lower central

C₃³ — C₂×S₃×C₃²⋊C₄

Upper central

C₁ — C₂

Generators and relations for C₂×S₃×C₃²⋊C₄
G = < a,b,c,d,e,f | a²=b³=c²=d³=e³=f⁴=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b^-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fef^-1=de=ed, fdf^-1=d^-1e >

Subgroups: 1632 in 192 conjugacy classes, 42 normal (22 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², S₃, S₃, C₆, C₆, C₂×C₄, C₂³, C₃², C₃², Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂²×C₄, C₃×S₃, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₂²×S₃, C₃³, C₃²⋊C₄, C₃²⋊C₄, S₃², S₃×C₆, C₂×C₃⋊S₃, C₂×C₃⋊S₃, C₆², S₃×C₂×C₄, S₃×C₃², C₃×C₃⋊S₃, C₃³⋊C₂, C₃²×C₆, C₂×C₃²⋊C₄, C₂×C₃²⋊C₄, C₂×S₃², C₂²×C₃⋊S₃, C₃×C₃²⋊C₄, C₃³⋊C₄, S₃×C₃⋊S₃, S₃×C₃×C₆, C₆×C₃⋊S₃, C₂×C₃³⋊C₂, C₂²×C₃²⋊C₄, S₃×C₃²⋊C₄, C₆×C₃²⋊C₄, C₂×C₃³⋊C₄, C₂×S₃×C₃⋊S₃, C₂×S₃×C₃²⋊C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, D₆, C₂²×C₄, C₄×S₃, C₂²×S₃, C₃²⋊C₄, S₃×C₂×C₄, C₂×C₃²⋊C₄, C₂²×C₃²⋊C₄, S₃×C₃²⋊C₄, C₂×S₃×C₃²⋊C₄

Permutation representations of C₂×S₃×C₃²⋊C₄
►On 24 points - transitive group 24T1310

Generators in S₂₄

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

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

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

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

(2 22 17)(4 19 24)(6 15 11)(8 9 13)

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

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

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

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

G:=TransitiveGroup(24,1310);

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	12A	12B	12C	12D
order	1	2	2	2	2	2	2	2	3	3	3	3	3	4	4	4	4	4	4	4	4	6	6	6	6	6	6	6	6	6	6	6	12	12	12	12
size	1	1	3	3	9	9	27	27	2	4	4	8	8	9	9	9	9	27	27	27	27	2	4	4	8	8	12	12	12	12	18	18	18	18	18	18

36 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	4	4	4	8	8
type	+	+	+	+	+				+	+	+			+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	S₃	D₆	D₆	C₄×S₃	C₄×S₃	C₃²⋊C₄	C₂×C₃²⋊C₄	C₂×C₃²⋊C₄	S₃×C₃²⋊C₄	C₂×S₃×C₃²⋊C₄
kernel	C₂×S₃×C₃²⋊C₄	S₃×C₃²⋊C₄	C₆×C₃²⋊C₄	C₂×C₃³⋊C₄	C₂×S₃×C₃⋊S₃	S₃×C₃⋊S₃	S₃×C₃×C₆	C₂×C₃³⋊C₂	C₂×C₃²⋊C₄	C₃²⋊C₄	C₂×C₃⋊S₃	C₃⋊S₃	C₃×C₆	D₆	S₃	C₆	C₂	C₁
# reps	1	4	1	1	1	4	2	2	1	2	1	2	2	2	4	2	2	2

Matrix representation of C₂×S₃×C₃²⋊C₄ ►in GL₆(𝔽₁₃)

12	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	0
0	0	0	0	0	12

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

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

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

5	0	0	0	0	0
0	5	0	0	0	0
0	0	8	0	0	0
0	0	8	0	5	0
0	0	0	8	5	5
0	0	8	5	0	0

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

C₂×S₃×C₃²⋊C₄ in GAP, Magma, Sage, TeX

C_2\times S_3\times C_3^2\rtimes C_4

% in TeX

G:=Group("C2xS3xC3^2:C4");

// GroupNames label

G:=SmallGroup(432,753);

// by ID

G=gap.SmallGroup(432,753);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,64,1411,165,1356,530,14118]);

// Polycyclic

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

// generators/relations

G = C2×S3×C32⋊C4 order 432 = 24·33

Direct product of C2, S3 and C32⋊C4

G = C₂×S₃×C₃²⋊C₄ order 432 = 2⁴·3³

Direct product of C₂, S₃ and C₃²⋊C₄