C6^2:S3 - GroupNames

G = C₆²:S₃ order 216 = 2³·3³

4^th semidirect product of C₆² and S₃ acting faithfully

Aliases: C₆²:₄S₃, C₃²:₁S₄, C₃:S₄:C₃, (C₃xA₄):C₆, C₃.₃(C₃xS₄), C₃²:A₄:₁C₂, C₂²:(C₃²:C₆), (C₂xC₆).₄(C₃xS₃), SmallGroup(216,92)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃xA₄ — C₆²:S₃

Chief series

C₁ — C₂² — C₂xC₆ — C₃xA₄ — C₃²:A₄ — C₆²:S₃

Lower central

C₃xA₄ — C₆²:S₃

Upper central

C₁

Generators and relations for C₆²:S₃
G = < a,b,c,d,e,f | a³=b³=c²=d²=e³=f²=1, ab=ba, ac=ca, ad=da, eae^-1=ab^-1, af=fa, bc=cb, bd=db, be=eb, fbf=b^-1, ece^-1=fcf=cd=dc, ede^-1=c, df=fd, fef=e^-1 >

Subgroups: 270 in 45 conjugacy classes, 10 normal (all characteristic)
Quotients: C₁, C₂, C₃, S₃, C₆, C₃xS₃, S₄, C₃²:C₆, C₃xS₄, C₆²:S₃

C₁

₃C₂

₁₈C₂

C₃

₃C₃

₁₂C₃

₂₄C₃

C₂²

₉C₄

₉C₂²

₃C₆

₆C₆

₆S₃

₁₈C₆

₃₆S₃

C₃²

₄C₃²

₈C₃²

₉D₄

C₂xC₆

₃D₆

₃C₂xC₆

₃Dic₃

₃A₄

₆A₄

₉C₂xC₆

₉C₁₂

₃C₃xC₆

₄C₃:S₃

₆C₃xS₃

₄He₃

₃C₃:D₄

₉C₃xD₄

₉S₄

C₆²

C₃xA₄

₂C₃xA₄

₃C₃xDic₃

₃S₃xC₆

₄C₃²:C₆

C₃:S₄

₃C₃xC₃:D₄

C₃²:A₄

C₆²:S₃

Character table of C₆²:S₃

class	1	2A	2B	3A	3B	3C	3D	3E	3F	4	6A	6B	6C	6D	6E	6F	6G	12A	12B
size	1	3	18	2	3	3	24	24	24	18	3	3	6	6	6	18	18	18	18
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	1	1	1	1	-1	1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₃	1	1	-1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	-1	ζ₃	ζ₃²	ζ₃²	1	ζ₃	ζ₆	ζ₆⁵	ζ₆	ζ₆⁵	linear of order 6
ρ₄	1	1	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	1	ζ₃²	ζ₃	ζ₃	1	ζ₃²	ζ₃	ζ₃²	ζ₃	ζ₃²	linear of order 3
ρ₅	1	1	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	1	ζ₃	ζ₃²	ζ₃²	1	ζ₃	ζ₃²	ζ₃	ζ₃²	ζ₃	linear of order 3
ρ₆	1	1	-1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	-1	ζ₃²	ζ₃	ζ₃	1	ζ₃²	ζ₆⁵	ζ₆	ζ₆⁵	ζ₆	linear of order 6
ρ₇	2	2	0	2	2	2	-1	-1	-1	0	2	2	2	2	2	0	0	0	0	orthogonal lifted from S₃
ρ₈	2	2	0	2	-1+√-3	-1-√-3	ζ₆⁵	ζ₆	-1	0	-1+√-3	-1-√-3	-1-√-3	2	-1+√-3	0	0	0	0	complex lifted from C₃xS₃
ρ₉	2	2	0	2	-1-√-3	-1+√-3	ζ₆	ζ₆⁵	-1	0	-1-√-3	-1+√-3	-1+√-3	2	-1-√-3	0	0	0	0	complex lifted from C₃xS₃
ρ₁₀	3	-1	-1	3	3	3	0	0	0	1	-1	-1	-1	-1	-1	-1	-1	1	1	orthogonal lifted from S₄
ρ₁₁	3	-1	1	3	3	3	0	0	0	-1	-1	-1	-1	-1	-1	1	1	-1	-1	orthogonal lifted from S₄
ρ₁₂	3	-1	1	3	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	-1	ζ₆⁵	ζ₆	ζ₆	-1	ζ₆⁵	ζ₃²	ζ₃	ζ₆	ζ₆⁵	complex lifted from C₃xS₄
ρ₁₃	3	-1	1	3	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	-1	ζ₆	ζ₆⁵	ζ₆⁵	-1	ζ₆	ζ₃	ζ₃²	ζ₆⁵	ζ₆	complex lifted from C₃xS₄
ρ₁₄	3	-1	-1	3	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	1	ζ₆⁵	ζ₆	ζ₆	-1	ζ₆⁵	ζ₆	ζ₆⁵	ζ₃²	ζ₃	complex lifted from C₃xS₄
ρ₁₅	3	-1	-1	3	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	1	ζ₆	ζ₆⁵	ζ₆⁵	-1	ζ₆	ζ₆⁵	ζ₆	ζ₃	ζ₃²	complex lifted from C₃xS₄
ρ₁₆	6	-2	0	-3	0	0	0	0	0	0	4	4	-2	1	-2	0	0	0	0	orthogonal faithful
ρ₁₇	6	6	0	-3	0	0	0	0	0	0	0	0	0	-3	0	0	0	0	0	orthogonal lifted from C₃²:C₆
ρ₁₈	6	-2	0	-3	0	0	0	0	0	0	-2+2√-3	-2-2√-3	1+√-3	1	1-√-3	0	0	0	0	complex faithful
ρ₁₉	6	-2	0	-3	0	0	0	0	0	0	-2-2√-3	-2+2√-3	1-√-3	1	1+√-3	0	0	0	0	complex faithful

Permutation representations of C₆²:S₃
►On 18 points - transitive group 18T97

Generators in S₁₈

(7 8 9)(10 11 12)(13 14 15)(16 17 18)

(1 5 3)(2 6 4)(7 8 9)(10 12 11)(13 15 14)(16 17 18)

(1 2)(3 4)(5 6)(10 15)(11 13)(12 14)

(7 16)(8 17)(9 18)(10 15)(11 13)(12 14)

(1 14 8)(2 12 17)(3 15 7)(4 10 16)(5 13 9)(6 11 18)

(1 2)(3 6)(4 5)(7 11)(8 12)(9 10)(13 16)(14 17)(15 18)

G:=sub<Sym(18)| (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,3)(2,6,4)(7,8,9)(10,12,11)(13,15,14)(16,17,18), (1,2)(3,4)(5,6)(10,15)(11,13)(12,14), (7,16)(8,17)(9,18)(10,15)(11,13)(12,14), (1,14,8)(2,12,17)(3,15,7)(4,10,16)(5,13,9)(6,11,18), (1,2)(3,6)(4,5)(7,11)(8,12)(9,10)(13,16)(14,17)(15,18)>;

G:=Group( (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,3)(2,6,4)(7,8,9)(10,12,11)(13,15,14)(16,17,18), (1,2)(3,4)(5,6)(10,15)(11,13)(12,14), (7,16)(8,17)(9,18)(10,15)(11,13)(12,14), (1,14,8)(2,12,17)(3,15,7)(4,10,16)(5,13,9)(6,11,18), (1,2)(3,6)(4,5)(7,11)(8,12)(9,10)(13,16)(14,17)(15,18) );

G=PermutationGroup([[(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,5,3),(2,6,4),(7,8,9),(10,12,11),(13,15,14),(16,17,18)], [(1,2),(3,4),(5,6),(10,15),(11,13),(12,14)], [(7,16),(8,17),(9,18),(10,15),(11,13),(12,14)], [(1,14,8),(2,12,17),(3,15,7),(4,10,16),(5,13,9),(6,11,18)], [(1,2),(3,6),(4,5),(7,11),(8,12),(9,10),(13,16),(14,17),(15,18)]])

G:=TransitiveGroup(18,97);

►On 18 points - transitive group 18T99

Generators in S₁₈

(7 8 9)(10 11 12)(13 14 15)(16 17 18)

(1 5 3)(2 6 4)(7 9 8)(10 11 12)(13 15 14)(16 17 18)

(1 2)(3 4)(5 6)(7 14)(8 15)(9 13)

(7 14)(8 15)(9 13)(10 16)(11 17)(12 18)

(1 8 16)(2 15 10)(3 9 18)(4 13 12)(5 7 17)(6 14 11)

(3 5)(4 6)(7 18)(8 16)(9 17)(10 15)(11 13)(12 14)

G:=sub<Sym(18)| (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,3)(2,6,4)(7,9,8)(10,11,12)(13,15,14)(16,17,18), (1,2)(3,4)(5,6)(7,14)(8,15)(9,13), (7,14)(8,15)(9,13)(10,16)(11,17)(12,18), (1,8,16)(2,15,10)(3,9,18)(4,13,12)(5,7,17)(6,14,11), (3,5)(4,6)(7,18)(8,16)(9,17)(10,15)(11,13)(12,14)>;

G:=Group( (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,3)(2,6,4)(7,9,8)(10,11,12)(13,15,14)(16,17,18), (1,2)(3,4)(5,6)(7,14)(8,15)(9,13), (7,14)(8,15)(9,13)(10,16)(11,17)(12,18), (1,8,16)(2,15,10)(3,9,18)(4,13,12)(5,7,17)(6,14,11), (3,5)(4,6)(7,18)(8,16)(9,17)(10,15)(11,13)(12,14) );

G=PermutationGroup([[(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,5,3),(2,6,4),(7,9,8),(10,11,12),(13,15,14),(16,17,18)], [(1,2),(3,4),(5,6),(7,14),(8,15),(9,13)], [(7,14),(8,15),(9,13),(10,16),(11,17),(12,18)], [(1,8,16),(2,15,10),(3,9,18),(4,13,12),(5,7,17),(6,14,11)], [(3,5),(4,6),(7,18),(8,16),(9,17),(10,15),(11,13),(12,14)]])

G:=TransitiveGroup(18,99);

C₆²:S₃ is a maximal subgroup of C₆²:₅D₆
C₆²:S₃ is a maximal quotient of C₃²:CSU₂(F₃) C₃²:₂GL₂(F₃) C₆²:₅Dic₃

Matrix representation of C₆²:S₃ ►in GL₆(Z)

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

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

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

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,-1],[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,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0] >;

C₆²:S₃ in GAP, Magma, Sage, TeX

C_6^2\rtimes S_3

% in TeX

G:=Group("C6^2:S3");

// GroupNames label

G:=SmallGroup(216,92);

// by ID

G=gap.SmallGroup(216,92);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,218,224,867,3244,556,1949,989]);

// 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,e*a*e^-1=a*b^-1,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,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

Export

Subgroup lattice of C₆²:S₃ in TeX
Character table of C₆²:S₃ in TeX

G = C62:S3 order 216 = 23·33

4th semidirect product of C62 and S3 acting faithfully

G = C₆²:S₃ order 216 = 2³·3³

4^th semidirect product of C₆² and S₃ acting faithfully