C3xS3xA4 - GroupNames

G = C₃×S₃×A₄ order 216 = 2³·3³

Direct product of C₃, S₃ and A₄

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

Aliases: C₃×S₃×A₄, C₆²⋊₅C₆, C₃⋊(C₆×A₄), (C₃×A₄)⋊₅C₆, C₃²⋊₃(C₂×A₄), (C₃²×A₄)⋊₄C₂, (C₂²×S₃)⋊C₃², C₂²⋊₂(S₃×C₃²), (S₃×C₂×C₆)⋊C₃, (C₂×C₆)⋊(C₃×C₆), (C₂×C₆)⋊₄(C₃×S₃), SmallGroup(216,166)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₃×S₃×A₄

Chief series

C₁ — C₃ — C₂×C₆ — C₆² — C₃²×A₄ — C₃×S₃×A₄

Lower central

C₂×C₆ — C₃×S₃×A₄

Upper central

C₁ — C₃

Generators and relations for C₃×S₃×A₄
G = < a,b,c,d,e,f | a³=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: 308 in 81 conjugacy classes, 24 normal (15 characteristic)
C₁, C₂, C₃, C₃, C₂², C₂², S₃, S₃, C₆, C₂³, C₃², C₃², A₄, A₄, D₆, C₂×C₆, C₂×C₆, C₃×S₃, C₃×S₃, C₃×C₆, C₂×A₄, C₂²×S₃, C₂²×C₆, C₃³, C₃×A₄, C₃×A₄, C₃×A₄, S₃×C₆, C₆², S₃×C₃², S₃×A₄, C₆×A₄, S₃×C₂×C₆, C₃²×A₄, C₃×S₃×A₄
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², A₄, C₃×S₃, C₃×C₆, C₂×A₄, C₃×A₄, S₃×C₃², S₃×A₄, C₆×A₄, C₃×S₃×A₄

Permutation representations of C₃×S₃×A₄
►On 24 points - transitive group 24T563

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

(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)

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

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

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

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

G:=TransitiveGroup(24,563);

36 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	···	3K	3L	···	3Q	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	···	6O
order	1	2	2	2	3	3	3	3	3	3	···	3	3	···	3	6	6	6	6	6	6	6	6	6	6	···	6
size	1	3	3	9	1	1	2	2	2	4	···	4	8	···	8	3	3	3	3	6	6	6	9	9	12	···	12

36 irreducible representations

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

Matrix representation of C₃×S₃×A₄ ►in GL₅(𝔽₇)

4	0	0	0	0
0	4	0	0	0
0	0	4	0	0
0	0	0	4	0
0	0	0	0	4

2	0	0	0	0
0	4	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

0	6	0	0	0
6	0	0	0	0
0	0	6	0	0
0	0	0	6	0
0	0	0	0	6

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

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

4	0	0	0	0
0	4	0	0	0
0	0	2	0	0
0	0	5	5	5
0	0	0	2	0

G:=sub<GL(5,GF(7))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[2,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,6,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,6,0,1,0,0,6,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,0,1,0,6,0,0,0,0,6],[4,0,0,0,0,0,4,0,0,0,0,0,2,5,0,0,0,0,5,2,0,0,0,5,0] >;

C₃×S₃×A₄ in GAP, Magma, Sage, TeX

C_3\times S_3\times A_4

% in TeX

G:=Group("C3xS3xA4");

// GroupNames label

G:=SmallGroup(216,166);

// by ID

G=gap.SmallGroup(216,166);

# by ID

G:=PCGroup([6,-2,-3,-3,-2,2,-3,657,280,5189]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=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 = C3×S3×A4 order 216 = 23·33

Direct product of C3, S3 and A4

G = C₃×S₃×A₄ order 216 = 2³·3³

Direct product of C₃, S₃ and A₄