C2xS3xC3:S3 - GroupNames

G = C₂×S₃×C₃⋊S₃ order 216 = 2³·3³

Direct product of C₂, S₃ and C₃⋊S₃

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

Aliases: C₂×S₃×C₃⋊S₃, C₃³⋊₃C₂³, C₆⋊₁S₃², (C₃×C₆)⋊₄D₆, (S₃×C₆)⋊₅S₃, (C₃×S₃)⋊₂D₆, (C₃²×C₆)⋊₂C₂², C₃²⋊₅(C₂²×S₃), (S₃×C₃²)⋊₃C₂², C₃³⋊C₂⋊₂C₂², C₃⋊₂(C₂×S₃²), C₆⋊₁(C₂×C₃⋊S₃), (S₃×C₃×C₆)⋊₈C₂, (C₆×C₃⋊S₃)⋊₈C₂, C₃⋊₁(C₂²×C₃⋊S₃), (C₃×C₃⋊S₃)⋊₃C₂², (C₂×C₃³⋊C₂)⋊₅C₂, SmallGroup(216,171)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₂×S₃×C₃⋊S₃

Chief series

C₁ — C₃ — C₃² — C₃³ — S₃×C₃² — S₃×C₃⋊S₃ — C₂×S₃×C₃⋊S₃

Lower central

C₃³ — C₂×S₃×C₃⋊S₃

Upper central

C₁ — C₂

Generators and relations for C₂×S₃×C₃⋊S₃
G = < a,b,c,d,e,f | a²=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: 1252 in 232 conjugacy classes, 56 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₂², S₃, S₃, C₆, C₆, C₆, C₂³, C₃², C₃², C₃², D₆, D₆, C₂×C₆, C₃×S₃, C₃×S₃, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃×C₆, C₂²×S₃, C₃³, S₃², S₃×C₆, S₃×C₆, C₂×C₃⋊S₃, C₂×C₃⋊S₃, C₆², S₃×C₃², C₃×C₃⋊S₃, C₃³⋊C₂, C₃²×C₆, C₂×S₃², C₂²×C₃⋊S₃, S₃×C₃⋊S₃, S₃×C₃×C₆, C₆×C₃⋊S₃, C₂×C₃³⋊C₂, C₂×S₃×C₃⋊S₃
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₃⋊S₃, C₂²×S₃, S₃², C₂×C₃⋊S₃, C₂×S₃², C₂²×C₃⋊S₃, S₃×C₃⋊S₃, C₂×S₃×C₃⋊S₃

Smallest permutation representation of C₂×S₃×C₃⋊S₃
►On 36 points

Generators in S₃₆

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

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

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

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

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

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

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

C₂×S₃×C₃⋊S₃ is a maximal subgroup of
D₆⋊(C₃²⋊C₄) D₆⋊₄S₃² (S₃×C₆)⋊D₆ C₃⋊S₃⋊₄D₁₂ C₁₂⋊S₃² C₆²⋊₂₃D₆ C₂×S₃³
C₂×S₃×C₃⋊S₃ is a maximal quotient of
(C₃×D₁₂)⋊S₃ D₁₂⋊(C₃⋊S₃) C₁₂.₃₉S₃² C₁₂.₄₀S₃² C₃²⋊₉(S₃×Q₈) C₁₂.₇₃S₃² C₁₂.₅₇S₃² C₁₂.₅₈S₃² C₁₂⋊S₃² C₆².₉₀D₆ C₆².₉₁D₆ C₆².₉₃D₆ C₆²⋊₂₃D₆

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	···	3E	3F	3G	3H	3I	6A	···	6E	6F	6G	6H	6I	6J	···	6Q	6R	6S
order	1	2	2	2	2	2	2	2	3	···	3	3	3	3	3	6	···	6	6	6	6	6	6	···	6	6	6
size	1	1	3	3	9	9	27	27	2	···	2	4	4	4	4	2	···	2	4	4	4	4	6	···	6	18	18

36 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

S₃²

C₂×S₃²

kernel

C₂×S₃×C₃⋊S₃

S₃×C₃⋊S₃

S₃×C₃×C₆

C₆×C₃⋊S₃

C₂×C₃³⋊C₂

S₃×C₆

C₂×C₃⋊S₃

C₃×S₃

C₃⋊S₃

C₃×C₆

C₆

C₃

# reps

Matrix representation of C₂×S₃×C₃⋊S₃ ►in GL₆(ℤ)

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	-1	0
0	0	0	0	0	-1

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	-1

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	1	0
0	0	0	0	-1	-1

0	-1	0	0	0	0
1	-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	1

0	-1	0	0	0	0
1	-1	0	0	0	0
0	0	-1	1	0	0
0	0	-1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
1	0	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,Integers())| [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,-1,0,0,0,0,0,0,-1],[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,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1],[0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂×S₃×C₃⋊S₃ in GAP, Magma, Sage, TeX

C_2\times S_3\times C_3\rtimes S_3

% in TeX

G:=Group("C2xS3xC3:S3");

// GroupNames label

G:=SmallGroup(216,171);

// by ID

G=gap.SmallGroup(216,171);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,201,730,5189]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=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 = C2×S3×C3⋊S3 order 216 = 23·33

Direct product of C2, S3 and C3⋊S3

G = C₂×S₃×C₃⋊S₃ order 216 = 2³·3³

Direct product of C₂, S₃ and C₃⋊S₃