C3^2xS4 - GroupNames

G = C₃²×S₄ order 216 = 2³·3³

Direct product of C₃² and S₄

direct product, non-abelian, soluble, monomial

Aliases: C₃²×S₄, C₆²⋊₁S₃, A₄⋊(C₃×C₆), (C₃×A₄)⋊₃C₆, C₂²⋊(S₃×C₃²), (C₃²×A₄)⋊₁C₂, (C₂×C₆)⋊₁(C₃×S₃), SmallGroup(216,163)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — A₄ — C₃²×S₄

Chief series

C₁ — C₂² — A₄ — C₃×A₄ — C₃²×A₄ — C₃²×S₄

Lower central

A₄ — C₃²×S₄

Upper central

C₁ — C₃²

Generators and relations for C₃²×S₄
G = < a,b,c,d,e,f | a³=b³=c²=d²=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece^-1=fcf=cd=dc, ede^-1=c, df=fd, fef=e^-1 >

Subgroups: 260 in 82 conjugacy classes, 24 normal (8 characteristic)
C₁, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, D₄, C₃², C₃², C₁₂, A₄, A₄, C₂×C₆, C₂×C₆, C₃×S₃, C₃×C₆, C₃×D₄, S₄, C₃³, C₃×C₁₂, C₃×A₄, C₃×A₄, C₆², C₆², S₃×C₃², D₄×C₃², C₃×S₄, C₃²×A₄, C₃²×S₄
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃×C₆, S₄, S₃×C₃², C₃×S₄, C₃²×S₄

Smallest permutation representation of C₃²×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 23 28)(2 24 29)(3 22 30)(4 31 26)(5 32 27)(6 33 25)(7 15 35)(8 13 36)(9 14 34)(10 21 18)(11 19 16)(12 20 17)

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

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

(4 9 21)(5 7 19)(6 8 20)(10 26 34)(11 27 35)(12 25 36)(13 17 33)(14 18 31)(15 16 32)

(7 19)(8 20)(9 21)(10 34)(11 35)(12 36)(13 17)(14 18)(15 16)

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

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

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

45 conjugacy classes

class	1	2A	2B	3A	···	3H	3I	···	3Q	4	6A	···	6H	6I	···	6P	12A	···	12H
order	1	2	2	3	···	3	3	···	3	4	6	···	6	6	···	6	12	···	12
size	1	3	6	1	···	1	8	···	8	6	3	···	3	6	···	6	6	···	6

45 irreducible representations

dim	1	1	1	1	2	2	3	3
type	+	+			+		+
image	C₁	C₂	C₃	C₆	S₃	C₃×S₃	S₄	C₃×S₄
kernel	C₃²×S₄	C₃²×A₄	C₃×S₄	C₃×A₄	C₆²	C₂×C₆	C₃²	C₃
# reps	1	1	8	8	1	8	2	16

Matrix representation of C₃²×S₄ ►in GL₇(𝔽₁₃)

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

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

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

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

C₃²×S₄ in GAP, Magma, Sage, TeX

C_3^2\times S_4

% in TeX

G:=Group("C3^2xS4");

// GroupNames label

G:=SmallGroup(216,163);

// by ID

G=gap.SmallGroup(216,163);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,867,3244,202,1949,347]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=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,b*f=f*b,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 = C32×S4 order 216 = 23·33

Direct product of C32 and S4

G = C₃²×S₄ order 216 = 2³·3³

Direct product of C₃² and S₄