C4xS3xA4 - GroupNames

G = C₄xS₃xA₄ order 288 = 2⁵·3²

Direct product of C₄, S₃ and A₄

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

Aliases: C₄xS₃xA₄, C₁₂:₂(C₂xA₄), (C₁₂xA₄):₆C₂, D₆.₃(C₂xA₄), (S₃xC₂³).C₆, C₂²:₂(S₃xC₁₂), (C₂²xC₁₂):₂C₆, (Dic₃xA₄):₅C₂, Dic₃:₂(C₂xA₄), (C₂xA₄).₁₄D₆, C₆.₂(C₂²xA₄), (C₂²xS₃):₂C₁₂, C₂³.₁₇(S₃xC₆), (C₆xA₄).₁₉C₂², (C₂²xDic₃):₂C₆, C₃:₁(C₂xC₄xA₄), (S₃xC₂²xC₄):C₃, C₂.₁(C₂xS₃xA₄), (C₂xS₃xA₄).₂C₂, (C₃xA₄):₆(C₂xC₄), (C₂xC₆):₁(C₂xC₁₂), (C₂²xC₄):₄(C₃xS₃), (C₂²xC₆).₂(C₂xC₆), SmallGroup(288,919)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₄xS₃xA₄

Chief series

C₁ — C₃ — C₂xC₆ — C₂²xC₆ — C₆xA₄ — C₂xS₃xA₄ — C₄xS₃xA₄

Lower central

C₂xC₆ — C₄xS₃xA₄

Upper central

C₁ — C₄

Generators and relations for C₄xS₃xA₄
G = < a,b,c,d,e,f | a⁴=b³=c²=d²=e²=f³=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, fdf^-1=de=ed, fef^-1=d >

Subgroups: 578 in 138 conjugacy classes, 33 normal (27 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₂xC₄, C₂³, C₂³, C₃², Dic₃, Dic₃, C₁₂, C₁₂, A₄, A₄, D₆, D₆, C₂xC₆, C₂xC₆, C₂²xC₄, C₂²xC₄, C₂⁴, C₃xS₃, C₃xC₆, C₄xS₃, C₄xS₃, C₂xDic₃, C₂xC₁₂, C₂xA₄, C₂xA₄, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂³xC₄, C₃xDic₃, C₃xC₁₂, C₃xA₄, S₃xC₆, C₄xA₄, C₄xA₄, S₃xC₂xC₄, C₂²xDic₃, C₂²xC₁₂, C₂²xA₄, S₃xC₂³, S₃xC₁₂, S₃xA₄, C₆xA₄, C₂xC₄xA₄, S₃xC₂²xC₄, Dic₃xA₄, C₁₂xA₄, C₂xS₃xA₄, C₄xS₃xA₄
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, C₁₂, A₄, D₆, C₂xC₆, C₃xS₃, C₄xS₃, C₂xC₁₂, C₂xA₄, S₃xC₆, C₄xA₄, C₂²xA₄, S₃xC₁₂, S₃xA₄, C₂xC₄xA₄, C₂xS₃xA₄, C₄xS₃xA₄

Smallest permutation representation of C₄xS₃xA₄
►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 29 14)(2 30 15)(3 31 16)(4 32 13)(5 10 33)(6 11 34)(7 12 35)(8 9 36)(17 22 27)(18 23 28)(19 24 25)(20 21 26)

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

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(29 31)(30 32)(33 35)(34 36)

(1 3)(2 4)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)

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

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

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

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

48 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	12E	12F	12G	12H	12I	12J	12K	12L	12M	12N	12O	12P
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	12	12	12	12	12	12	12	12	12	12	12	12
size	1	1	3	3	3	3	9	9	2	4	4	8	8	1	1	3	3	3	3	9	9	2	4	4	6	6	8	8	12	12	12	12	2	2	4	4	4	4	6	6	8	8	8	8	12	12	12	12

48 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	3	3	3	3	3	6	6	6
type	+	+	+	+							+	+					+	+	+	+		+	+
image	C₁	C₂	C₂	C₂	C₃	C₄	C₆	C₆	C₆	C₁₂	S₃	D₆	C₃xS₃	C₄xS₃	S₃xC₆	S₃xC₁₂	A₄	C₂xA₄	C₂xA₄	C₂xA₄	C₄xA₄	S₃xA₄	C₂xS₃xA₄	C₄xS₃xA₄
kernel	C₄xS₃xA₄	Dic₃xA₄	C₁₂xA₄	C₂xS₃xA₄	S₃xC₂²xC₄	S₃xA₄	C₂²xDic₃	C₂²xC₁₂	S₃xC₂³	C₂²xS₃	C₄xA₄	C₂xA₄	C₂²xC₄	A₄	C₂³	C₂²	C₄xS₃	Dic₃	C₁₂	D₆	S₃	C₄	C₂	C₁
# reps	1	1	1	1	2	4	2	2	2	8	1	1	2	2	2	4	1	1	1	1	4	1	1	2

Matrix representation of C₄xS₃xA₄ ►in GL₅(F₁₃)

5	0	0	0	0
0	5	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

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

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

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	10	12	0
0	0	9	0	12

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

3	0	0	0	0
0	3	0	0	0
0	0	3	0	8
0	0	0	0	1
0	0	0	4	10

G:=sub<GL(5,GF(13))| [5,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,1,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,10,9,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,3,0,0,0,0,1,0,0,0,0,0,12],[3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,8,1,10] >;

C₄xS₃xA₄ in GAP, Magma, Sage, TeX

C_4\times S_3\times A_4

% in TeX

G:=Group("C4xS3xA4");

// GroupNames label

G:=SmallGroup(288,919);

// by ID

G=gap.SmallGroup(288,919);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,92,648,271,9414]);

// Polycyclic

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

// generators/relations

G = C4xS3xA4 order 288 = 25·32

Direct product of C4, S3 and A4

G = C₄xS₃xA₄ order 288 = 2⁵·3²

Direct product of C₄, S₃ and A₄