S3^2xA4 - GroupNames

G = S₃²×A₄ order 432 = 2⁴·3³

Direct product of S₃, S₃ and A₄

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

Aliases: S₃²×A₄, (C₃×A₄)⋊₈D₆, C₆²⋊₂(C₂×C₆), C₃²⋊₂(C₂²×A₄), (C₃²×A₄)⋊₅C₂², (S₃×C₂×C₆)⋊C₆, C₃⋊₁(C₂×S₃×A₄), (C₃×S₃)⋊(C₂×A₄), (C₃×S₃×A₄)⋊₃C₂, (C₂²×S₃²)⋊C₃, C₃⋊S₃⋊₂(C₂×A₄), (A₄×C₃⋊S₃)⋊₃C₂, (C₂×C₆)⋊₂(S₃×C₆), C₂²⋊₂(C₃×S₃²), (C₂²×C₃⋊S₃)⋊₄C₆, (C₂²×S₃)⋊₂(C₃×S₃), SmallGroup(432,749)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆² — S₃²×A₄

Chief series

C₁ — C₃ — C₃² — C₆² — C₃²×A₄ — C₃×S₃×A₄ — S₃²×A₄

Lower central

C₆² — S₃²×A₄

Upper central

C₁

Generators and relations for S₃²×A₄
G = < a,b,c,d,e,f,g | a³=b²=c³=d²=e²=f²=g³=1, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, dcd=c^-1, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=ef=fe, gfg^-1=e >

Subgroups: 1372 in 188 conjugacy classes, 30 normal (12 characteristic)
C₁, C₂, C₃, C₃, C₂², C₂², S₃, S₃, C₆, C₂³, C₃², C₃², A₄, A₄, D₆, C₂×C₆, C₂×C₆, C₂⁴, C₃×S₃, C₃×S₃, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₂×A₄, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₃³, S₃², S₃², C₃×A₄, C₃×A₄, S₃×C₆, C₂×C₃⋊S₃, C₆², C₂²×A₄, S₃×C₂³, S₃×C₃², C₃×C₃⋊S₃, S₃×A₄, S₃×A₄, C₂×S₃², C₆×A₄, S₃×C₂×C₆, C₂²×C₃⋊S₃, C₃×S₃², C₃²×A₄, C₂×S₃×A₄, C₂²×S₃², C₃×S₃×A₄, A₄×C₃⋊S₃, S₃²×A₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, A₄, D₆, C₂×C₆, C₃×S₃, C₂×A₄, S₃², S₃×C₆, C₂²×A₄, S₃×A₄, C₃×S₃², C₂×S₃×A₄, S₃²×A₄

Permutation representations of S₃²×A₄
►On 24 points - transitive group 24T1338

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(24,1338);

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	3F	3G	3H	3I	3J	3K	6A	6B	6C	6D	6E	···	6I	6J	6K	6L	6M	6N	6O	6P	6Q
order	1	2	2	2	2	2	2	2	3	3	3	3	3	3	3	3	3	3	3	6	6	6	6	6	···	6	6	6	6	6	6	6	6	6
size	1	3	3	3	9	9	9	27	2	2	4	4	4	8	8	8	8	16	16	6	6	6	6	12	···	12	18	18	24	24	24	24	36	36

36 irreducible representations

dim	1	1	1	1	1	1	12	2	2	2	2	3	3	3	4	4	6	6
type	+	+	+				+	+	+			+	+	+	+		+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃²×A₄	S₃	D₆	C₃×S₃	S₃×C₆	A₄	C₂×A₄	C₂×A₄	S₃²	C₃×S₃²	S₃×A₄	C₂×S₃×A₄
kernel	S₃²×A₄	C₃×S₃×A₄	A₄×C₃⋊S₃	C₂²×S₃²	S₃×C₂×C₆	C₂²×C₃⋊S₃	C₁	S₃×A₄	C₃×A₄	C₂²×S₃	C₂×C₆	S₃²	C₃×S₃	C₃⋊S₃	A₄	C₂²	S₃	C₃
# reps	1	2	1	2	4	2	1	2	2	4	4	1	2	1	1	2	2	2

Matrix representation of S₃²×A₄ ►in GL₇(𝔽₇)

0	1	0	0	0	0	0
6	6	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

6	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	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	0	1	0	0	0
0	0	6	6	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

6	0	0	0	0	0	0
0	6	0	0	0	0	0
0	0	1	0	0	0	0
0	0	6	6	0	0	0
0	0	0	0	6	0	0
0	0	0	0	0	6	0
0	0	0	0	0	0	6

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	1	6
0	0	0	0	1	0	6
0	0	0	0	0	0	6

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	6	0	0
0	0	0	0	6	0	1
0	0	0	0	6	1	0

2	0	0	0	0	0	0
0	2	0	0	0	0	0
0	0	2	0	0	0	0
0	0	0	2	0	0	0
0	0	0	0	0	0	1
0	0	0	0	1	0	0
0	0	0	0	0	1	0

G:=sub<GL(7,GF(7))| [0,6,0,0,0,0,0,1,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,1,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,6,6,6],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,6,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0] >;

S₃²×A₄ in GAP, Magma, Sage, TeX

S_3^2\times A_4

% in TeX

G:=Group("S3^2xA4");

// GroupNames label

G:=SmallGroup(432,749);

// by ID

G=gap.SmallGroup(432,749);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-3,-3,269,123,2028,14118]);

// Polycyclic

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

// generators/relations

G = S32×A4 order 432 = 24·33

Direct product of S3, S3 and A4

G = S₃²×A₄ order 432 = 2⁴·3³

Direct product of S₃, S₃ and A₄