C6xC3:S4 - GroupNames

G = C₆×C₃⋊S₄ order 432 = 2⁴·3³

Direct product of C₆ and C₃⋊S₄

direct product, non-abelian, soluble, monomial

Aliases: C₆×C₃⋊S₄, C₆²⋊₁₈D₆, C₆⋊(C₃×S₄), C₃⋊₂(C₆×S₄), (C₃×C₆)⋊₃S₄, (C₆×A₄)⋊₃S₃, (C₆×A₄)⋊₄C₆, A₄⋊₂(S₃×C₆), (C₃×A₄)⋊₉D₆, (C₂×C₆²)⋊₈S₃, C₃²⋊₈(C₂×S₄), (C₃²×A₄)⋊₇C₂², (C₂×A₄)⋊(C₃×S₃), (A₄×C₃×C₆)⋊₂C₂, C₂²⋊(C₆×C₃⋊S₃), C₂³⋊(C₃×C₃⋊S₃), (C₂×C₆)⋊₄(S₃×C₆), (C₃×A₄)⋊₅(C₂×C₆), (C₂²×C₆)⋊₂(C₃×S₃), (C₂²×C₆)⋊₁(C₃⋊S₃), (C₂×C₆)⋊₂(C₂×C₃⋊S₃), SmallGroup(432,761)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃×A₄ — C₆×C₃⋊S₄

Chief series

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

Lower central

C₃×A₄ — C₆×C₃⋊S₄

Upper central

C₁ — C₆

Generators and relations for C₆×C₃⋊S₄
G = < a,b,c,d,e,f | a⁶=b³=c²=d²=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b^-1, ece^-1=fcf=cd=dc, ede^-1=c, df=fd, fef=e^-1 >

Subgroups: 1072 in 208 conjugacy classes, 38 normal (22 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, D₄, C₂³, C₂³, C₃², C₃², Dic₃, C₁₂, A₄, A₄, D₆, C₂×C₆, C₂×C₆, C₂×D₄, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₃×D₄, S₄, C₂×A₄, C₂×A₄, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₃³, C₃×Dic₃, C₃×A₄, C₃×A₄, C₃×A₄, S₃×C₆, C₂×C₃⋊S₃, C₆², C₆², C₂×C₃⋊D₄, C₆×D₄, C₂×S₄, C₃×C₃⋊S₃, C₃²×C₆, C₆×Dic₃, C₃×C₃⋊D₄, C₃×S₄, C₃⋊S₄, C₆×A₄, C₆×A₄, C₆×A₄, S₃×C₂×C₆, C₂×C₆², C₃²×A₄, C₆×C₃⋊S₃, C₆×C₃⋊D₄, C₆×S₄, C₂×C₃⋊S₄, C₃×C₃⋊S₄, A₄×C₃×C₆, C₆×C₃⋊S₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₆, C₂×C₆, C₃×S₃, C₃⋊S₃, S₄, S₃×C₆, C₂×C₃⋊S₃, C₂×S₄, C₃×C₃⋊S₃, C₃×S₄, C₃⋊S₄, C₆×C₃⋊S₃, C₆×S₄, C₂×C₃⋊S₄, C₃×C₃⋊S₄, C₆×C₃⋊S₄

Smallest permutation representation of C₆×C₃⋊S₄
►On 36 points

Generators in S₃₆

(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)

(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)

(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)

(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)

(1 11 17)(2 12 18)(3 7 13)(4 8 14)(5 9 15)(6 10 16)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)

(1 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 28)(8 29)(9 30)(10 25)(11 26)(12 27)(13 34)(14 35)(15 36)(16 31)(17 32)(18 33)

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

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)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(31,34)(32,35)(33,36), (1,11,17)(2,12,18)(3,7,13)(4,8,14)(5,9,15)(6,10,16)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30), (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,28)(8,29)(9,30)(10,25)(11,26)(12,27)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33) );

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),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,33,35),(32,34,36)], [(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30)], [(1,4),(2,5),(3,6),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(31,34),(32,35),(33,36)], [(1,11,17),(2,12,18),(3,7,13),(4,8,14),(5,9,15),(6,10,16),(19,31,25),(20,32,26),(21,33,27),(22,34,28),(23,35,29),(24,36,30)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,19),(7,28),(8,29),(9,30),(10,25),(11,26),(12,27),(13,34),(14,35),(15,36),(16,31),(17,32),(18,33)]])

54 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	3E	3F	···	3N	4A	4B	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	···	6O	6P	···	6X	6Y	6Z	6AA	6AB	12A	12B	12C	12D
order	1	2	2	2	2	2	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	6	12	12	12	12
size	1	1	3	3	18	18	1	1	2	2	2	8	···	8	18	18	1	1	2	2	2	3	3	3	3	6	···	6	8	···	8	18	18	18	18	18	18	18	18

54 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	3	3	3	3	6	6	6	6
type	+	+	+				+	+	+	+					+	+			+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	S₃	D₆	D₆	C₃×S₃	C₃×S₃	S₃×C₆	S₃×C₆	S₄	C₂×S₄	C₃×S₄	C₆×S₄	C₃⋊S₄	C₂×C₃⋊S₄	C₃×C₃⋊S₄	C₆×C₃⋊S₄
kernel	C₆×C₃⋊S₄	C₃×C₃⋊S₄	A₄×C₃×C₆	C₂×C₃⋊S₄	C₃⋊S₄	C₆×A₄	C₆×A₄	C₂×C₆²	C₃×A₄	C₆²	C₂×A₄	C₂²×C₆	A₄	C₂×C₆	C₃×C₆	C₃²	C₆	C₃	C₆	C₃	C₂	C₁
# reps	1	2	1	2	4	2	3	1	3	1	6	2	6	2	2	2	4	4	1	1	2	2

Matrix representation of C₆×C₃⋊S₄ ►in GL₇(𝔽₁₃)

4	0	0	0	0	0	0
0	4	0	0	0	0	0
0	0	12	0	0	0	0
0	0	0	12	0	0	0
0	0	0	0	12	0	0
0	0	0	0	0	12	0
0	0	0	0	0	0	12

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	3	0	0	0	0
0	0	3	9	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	12	0	0
0	0	0	0	0	12	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	12	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	12

9	0	0	0	0	0	0
0	3	0	0	0	0	0
0	0	9	0	0	0	0
0	0	10	3	0	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1
0	0	0	0	1	0	0

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

G:=sub<GL(7,GF(13))| [4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12],[9,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9,10,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0] >;

C₆×C₃⋊S₄ in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes S_4

% in TeX

G:=Group("C6xC3:S4");

// GroupNames label

G:=SmallGroup(432,761);

// by ID

G=gap.SmallGroup(432,761);

# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,675,2524,9077,782,5298,1350]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^6=b^3=c^2=d^2=e^3=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,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = C6×C3⋊S4 order 432 = 24·33

Direct product of C6 and C3⋊S4

G = C₆×C₃⋊S₄ order 432 = 2⁴·3³

Direct product of C₆ and C₃⋊S₄