C3^3:S3 - GroupNames

G = C₃³⋊S₃ order 162 = 2·3⁴

2^nd semidirect product of C₃³ and S₃ acting faithfully

non-abelian, supersoluble, monomial

Aliases: He₃⋊₂S₃, C₃³⋊₂S₃, 3_-¹⁺²⋊₁S₃, C₃≀C₃⋊₁C₂, C₃².₁(C₃⋊S₃), C₃.₂(He₃⋊C₂), SmallGroup(162,19)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃≀C₃ — C₃³⋊S₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃≀C₃ — C₃³⋊S₃

Lower central

C₃≀C₃ — C₃³⋊S₃

Upper central

C₁

Generators and relations for C₃³⋊S₃
G = < a,b,c,d,e | a³=b³=c³=d³=e²=1, dad^-1=ab=ba, ac=ca, eae=a^-1, dbd^-1=bc=cb, ebe=bc^-1, cd=dc, ece=c^-1, ede=d^-1 >

C₁

₂₇C₂

C₃

₃C₃

₉C₃

₉S₃

₂₇C₆

₂₇S₃

C₃²

₃C₉

₃C₃²

₃D₉

₃C₃⋊S₃

₉C₃×S₃

3_-¹⁺²

C₃³

He₃

3_-¹⁺²

₃C₉⋊C₆

₃C₃×C₃⋊S₃

₃C₃²⋊C₆

₃C₉⋊C₆

C₃≀C₃

C₃³⋊S₃

Character table of C₃³⋊S₃

class	1	2	3A	3B	3C	3D	3E	3F	3G	6A	6B	9A	9B
size	1	27	2	3	3	6	6	6	18	27	27	18	18
ρ₁	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	linear of order 2
ρ₃	2	0	2	2	2	-1	-1	-1	-1	0	0	2	-1	orthogonal lifted from S₃
ρ₄	2	0	2	2	2	2	2	2	-1	0	0	-1	-1	orthogonal lifted from S₃
ρ₅	2	0	2	2	2	-1	-1	-1	2	0	0	-1	-1	orthogonal lifted from S₃
ρ₆	2	0	2	2	2	-1	-1	-1	-1	0	0	-1	2	orthogonal lifted from S₃
ρ₇	3	1	3	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	0	ζ₃²	ζ₃	0	0	complex lifted from He₃⋊C₂
ρ₈	3	1	3	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	0	ζ₃	ζ₃²	0	0	complex lifted from He₃⋊C₂
ρ₉	3	-1	3	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	0	ζ₆⁵	ζ₆	0	0	complex lifted from He₃⋊C₂
ρ₁₀	3	-1	3	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	0	ζ₆	ζ₆⁵	0	0	complex lifted from He₃⋊C₂
ρ₁₁	6	0	-3	0	0	-3	0	3	0	0	0	0	0	orthogonal faithful
ρ₁₂	6	0	-3	0	0	0	3	-3	0	0	0	0	0	orthogonal faithful
ρ₁₃	6	0	-3	0	0	3	-3	0	0	0	0	0	0	orthogonal faithful

Permutation representations of C₃³⋊S₃
►On 9 points - transitive group 9T21

Generators in S₉

(4 5 6)(7 8 9)

(1 2 3)(7 9 8)

(1 3 2)(4 6 5)(7 9 8)

(1 4 7)(2 5 8)(3 6 9)

(2 3)(4 7)(5 9)(6 8)

G:=sub<Sym(9)| (4,5,6)(7,8,9), (1,2,3)(7,9,8), (1,3,2)(4,6,5)(7,9,8), (1,4,7)(2,5,8)(3,6,9), (2,3)(4,7)(5,9)(6,8)>;

G:=Group( (4,5,6)(7,8,9), (1,2,3)(7,9,8), (1,3,2)(4,6,5)(7,9,8), (1,4,7)(2,5,8)(3,6,9), (2,3)(4,7)(5,9)(6,8) );

G=PermutationGroup([[(4,5,6),(7,8,9)], [(1,2,3),(7,9,8)], [(1,3,2),(4,6,5),(7,9,8)], [(1,4,7),(2,5,8),(3,6,9)], [(2,3),(4,7),(5,9),(6,8)]])

G:=TransitiveGroup(9,21);

►On 18 points - transitive group 18T88

Generators in S₁₈

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

(1 2 3)(4 6 5)(7 9 8)(10 12 11)

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

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

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

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

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

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

G:=TransitiveGroup(18,88);

►On 27 points - transitive group 27T51

Generators in S₂₇

(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)

(1 8 5)(2 9 6)(3 7 4)(10 11 12)(13 14 15)(16 17 18)(19 25 22)(20 26 23)(21 27 24)

(1 2 3)(4 5 6)(7 8 9)(10 15 17)(11 13 18)(12 14 16)(19 26 24)(20 27 22)(21 25 23)

(1 17 25)(2 10 23)(3 15 21)(4 16 26)(5 12 24)(6 14 19)(7 11 22)(8 13 20)(9 18 27)

(2 3)(5 6)(7 8)(10 21)(11 20)(12 19)(13 22)(14 24)(15 23)(16 26)(17 25)(18 27)

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

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), (1,8,5)(2,9,6)(3,7,4)(10,11,12)(13,14,15)(16,17,18)(19,25,22)(20,26,23)(21,27,24), (1,2,3)(4,5,6)(7,8,9)(10,15,17)(11,13,18)(12,14,16)(19,26,24)(20,27,22)(21,25,23), (1,17,25)(2,10,23)(3,15,21)(4,16,26)(5,12,24)(6,14,19)(7,11,22)(8,13,20)(9,18,27), (2,3)(5,6)(7,8)(10,21)(11,20)(12,19)(13,22)(14,24)(15,23)(16,26)(17,25)(18,27) );

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)], [(1,8,5),(2,9,6),(3,7,4),(10,11,12),(13,14,15),(16,17,18),(19,25,22),(20,26,23),(21,27,24)], [(1,2,3),(4,5,6),(7,8,9),(10,15,17),(11,13,18),(12,14,16),(19,26,24),(20,27,22),(21,25,23)], [(1,17,25),(2,10,23),(3,15,21),(4,16,26),(5,12,24),(6,14,19),(7,11,22),(8,13,20),(9,18,27)], [(2,3),(5,6),(7,8),(10,21),(11,20),(12,19),(13,22),(14,24),(15,23),(16,26),(17,25),(18,27)]])

G:=TransitiveGroup(27,51);

►On 27 points - transitive group 27T52

Generators in S₂₇

(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)

(1 12 15)(2 10 13)(3 11 14)(4 26 8)(5 27 9)(6 25 7)

(1 15 12)(2 13 10)(3 14 11)(4 26 8)(5 27 9)(6 25 7)(16 22 19)(17 23 20)(18 24 21)

(1 21 5)(2 16 25)(3 23 8)(4 14 20)(6 10 19)(7 13 22)(9 12 24)(11 17 26)(15 18 27)

(2 3)(4 19)(5 21)(6 20)(7 17)(8 16)(9 18)(10 14)(11 13)(12 15)(22 26)(23 25)(24 27)

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

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), (1,12,15)(2,10,13)(3,11,14)(4,26,8)(5,27,9)(6,25,7), (1,15,12)(2,13,10)(3,14,11)(4,26,8)(5,27,9)(6,25,7)(16,22,19)(17,23,20)(18,24,21), (1,21,5)(2,16,25)(3,23,8)(4,14,20)(6,10,19)(7,13,22)(9,12,24)(11,17,26)(15,18,27), (2,3)(4,19)(5,21)(6,20)(7,17)(8,16)(9,18)(10,14)(11,13)(12,15)(22,26)(23,25)(24,27) );

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)], [(1,12,15),(2,10,13),(3,11,14),(4,26,8),(5,27,9),(6,25,7)], [(1,15,12),(2,13,10),(3,14,11),(4,26,8),(5,27,9),(6,25,7),(16,22,19),(17,23,20),(18,24,21)], [(1,21,5),(2,16,25),(3,23,8),(4,14,20),(6,10,19),(7,13,22),(9,12,24),(11,17,26),(15,18,27)], [(2,3),(4,19),(5,21),(6,20),(7,17),(8,16),(9,18),(10,14),(11,13),(12,15),(22,26),(23,25),(24,27)]])

G:=TransitiveGroup(27,52);

►On 27 points - transitive group 27T67

Generators in S₂₇

(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)

(1 6 14)(2 4 15)(3 5 13)(7 24 16)(8 22 17)(9 23 18)(10 27 19)(11 25 20)(12 26 21)

(1 26 8)(2 27 9)(3 25 7)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)

(2 10 4)(3 24 13)(5 11 25)(6 22 21)(7 20 16)(9 15 23)(12 17 14)(18 19 27)

(2 3)(4 24)(5 23)(6 22)(7 27)(8 26)(9 25)(10 13)(11 15)(12 14)(16 18)(19 20)

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

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), (1,6,14)(2,4,15)(3,5,13)(7,24,16)(8,22,17)(9,23,18)(10,27,19)(11,25,20)(12,26,21), (1,26,8)(2,27,9)(3,25,7)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (2,10,4)(3,24,13)(5,11,25)(6,22,21)(7,20,16)(9,15,23)(12,17,14)(18,19,27), (2,3)(4,24)(5,23)(6,22)(7,27)(8,26)(9,25)(10,13)(11,15)(12,14)(16,18)(19,20) );

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)], [(1,6,14),(2,4,15),(3,5,13),(7,24,16),(8,22,17),(9,23,18),(10,27,19),(11,25,20),(12,26,21)], [(1,26,8),(2,27,9),(3,25,7),(4,19,23),(5,20,24),(6,21,22),(10,18,15),(11,16,13),(12,17,14)], [(2,10,4),(3,24,13),(5,11,25),(6,22,21),(7,20,16),(9,15,23),(12,17,14),(18,19,27)], [(2,3),(4,24),(5,23),(6,22),(7,27),(8,26),(9,25),(10,13),(11,15),(12,14),(16,18),(19,20)]])

G:=TransitiveGroup(27,67);

C₃³⋊S₃ is a maximal subgroup of
He₃⋊D₆ C₃³⋊(C₃×S₃) C₃⁴⋊₇S₃ C₃≀C₃⋊S₃
C₃³⋊S₃ is a maximal quotient of
C₃³⋊Dic₃ (C₃×He₃)⋊S₃ (C₃×He₃).S₃ C₃³.(C₃⋊S₃) C₃²⋊C₉⋊₆S₃ C₃.(C₃³⋊S₃) C₃³⋊₂D₉ He₃⋊₂D₉ 3_-¹⁺²⋊D₉ C₃⁴⋊₇S₃

Polynomial with Galois group C₃³⋊S₃ over ℚ

action	f(x)	Disc(f)
9T21	x⁹-x⁸-36x⁷+118x⁶+56x⁵-766x⁴+1316x³-952x²+291x-25	2⁸·5⁴·17²·37⁴·109²

Matrix representation of C₃³⋊S₃ ►in GL₆(ℤ)

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

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

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

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

C₃³⋊S₃ in GAP, Magma, Sage, TeX

C_3^3\rtimes S_3

% in TeX

G:=Group("C3^3:S3");

// GroupNames label

G:=SmallGroup(162,19);

// by ID

G=gap.SmallGroup(162,19);

# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,41,182,187,2523,728,2704]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₃³⋊S₃ in TeX
Character table of C₃³⋊S₃ in TeX

G = C33⋊S3 order 162 = 2·34

2nd semidirect product of C33 and S3 acting faithfully

G = C₃³⋊S₃ order 162 = 2·3⁴

2^nd semidirect product of C₃³ and S₃ acting faithfully