C3xS3xC3:S3 - GroupNames

G = C₃×S₃×C₃⋊S₃ order 324 = 2²·3⁴

Direct product of C₃, S₃ and C₃⋊S₃

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

Aliases: C₃×S₃×C₃⋊S₃, C₃³⋊₁₃D₆, C₃⁴⋊₂C₂², C₃²⋊₇S₃², C₃³⋊₇(C₂×C₆), C₃²⋊₆(S₃×C₆), (S₃×C₃²)⋊₃S₃, (S₃×C₃²)⋊₄C₆, (S₃×C₃³)⋊₂C₂, C₃³⋊C₂⋊₄C₆, C₃⋊₁(C₃×S₃²), (C₃×S₃)⋊(C₃×S₃), C₃⋊₁(C₆×C₃⋊S₃), (C₃×C₃⋊S₃)⋊₄C₆, C₃²⋊₆(C₂×C₃⋊S₃), (C₃²×C₃⋊S₃)⋊₂C₂, (C₃×C₃³⋊C₂)⋊₁C₂, SmallGroup(324,166)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₃×S₃×C₃⋊S₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃⁴ — S₃×C₃³ — C₃×S₃×C₃⋊S₃

Lower central

C₃³ — C₃×S₃×C₃⋊S₃

Upper central

C₁ — C₃

Generators and relations for C₃×S₃×C₃⋊S₃
G = < a,b,c,d,e,f | a³=b³=c²=d³=e³=f²=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, de=ed, fdf=d^-1, fef=e^-1 >

Subgroups: 1000 in 224 conjugacy classes, 44 normal (20 characteristic)
C₁, C₂, C₃, C₃, C₃, C₂², S₃, S₃, C₆, C₃², C₃², C₃², D₆, C₂×C₆, C₃×S₃, C₃×S₃, C₃×S₃, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₃³, C₃³, C₃³, S₃², S₃×C₆, C₂×C₃⋊S₃, S₃×C₃², S₃×C₃², S₃×C₃², C₃×C₃⋊S₃, C₃×C₃⋊S₃, C₃³⋊C₂, C₃²×C₆, C₃⁴, C₃×S₃², S₃×C₃⋊S₃, C₆×C₃⋊S₃, S₃×C₃³, C₃²×C₃⋊S₃, C₃×C₃³⋊C₂, C₃×S₃×C₃⋊S₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₆, C₂×C₆, C₃×S₃, C₃⋊S₃, S₃², S₃×C₆, C₂×C₃⋊S₃, C₃×C₃⋊S₃, C₃×S₃², S₃×C₃⋊S₃, C₆×C₃⋊S₃, C₃×S₃×C₃⋊S₃

Smallest permutation representation of C₃×S₃×C₃⋊S₃
►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 12 14)(2 10 15)(3 11 13)(4 7 36)(5 8 34)(6 9 35)(16 22 19)(17 23 20)(18 24 21)(25 28 31)(26 29 32)(27 30 33)

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

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

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3Q	3R	···	3AC	6A	6B	6C	···	6N	6O	6P	6Q	6R	6S	6T	6U
order	1	2	2	2	3	3	3	···	3	3	···	3	6	6	6	···	6	6	6	6	6	6	6	6
size	1	3	9	27	1	1	2	···	2	4	···	4	3	3	6	···	6	9	9	18	18	18	27	27

54 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+					+	+	+				+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	S₃	D₆	C₃×S₃	C₃×S₃	S₃×C₆	S₃²	C₃×S₃²
kernel	C₃×S₃×C₃⋊S₃	S₃×C₃³	C₃²×C₃⋊S₃	C₃×C₃³⋊C₂	S₃×C₃⋊S₃	S₃×C₃²	C₃×C₃⋊S₃	C₃³⋊C₂	S₃×C₃²	C₃×C₃⋊S₃	C₃³	C₃×S₃	C₃⋊S₃	C₃²	C₃²	C₃
# reps	1	1	1	1	2	2	2	2	4	1	5	8	2	10	4	8

Matrix representation of C₃×S₃×C₃⋊S₃ ►in GL₆(𝔽₇)

2	0	0	0	0	0
0	2	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

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

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

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

C₃×S₃×C₃⋊S₃ in GAP, Magma, Sage, TeX

C_3\times S_3\times C_3\rtimes S_3

% in TeX

G:=Group("C3xS3xC3:S3");

// GroupNames label

G:=SmallGroup(324,166);

// by ID

G=gap.SmallGroup(324,166);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,297,1090,7781]);

// Polycyclic

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

// generators/relations

G = C3×S3×C3⋊S3 order 324 = 22·34

Direct product of C3, S3 and C3⋊S3

G = C₃×S₃×C₃⋊S₃ order 324 = 2²·3⁴

Direct product of C₃, S₃ and C₃⋊S₃