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## G = C32⋊C6order 54 = 2·33

### The semidirect product of C32 and C6 acting faithfully

Aliases: C32⋊C6, He31C2, C321S3, C3⋊S3⋊C3, C3.2(C3×S3), SmallGroup(54,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊C6
 Chief series C1 — C3 — C32 — He3 — C32⋊C6
 Lower central C32 — C32⋊C6
 Upper central C1

Generators and relations for C32⋊C6
G = < a,b,c | a3=b3=c6=1, ab=ba, cac-1=a-1b-1, cbc-1=b-1 >

Character table of C32⋊C6

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B size 1 9 2 3 3 6 6 6 9 9 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 ζ3 ζ32 1 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ4 1 -1 1 ζ3 ζ32 1 ζ3 ζ32 ζ6 ζ65 linear of order 6 ρ5 1 1 1 ζ32 ζ3 1 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ6 1 -1 1 ζ32 ζ3 1 ζ32 ζ3 ζ65 ζ6 linear of order 6 ρ7 2 0 2 2 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ8 2 0 2 -1-√-3 -1+√-3 -1 ζ6 ζ65 0 0 complex lifted from C3×S3 ρ9 2 0 2 -1+√-3 -1-√-3 -1 ζ65 ζ6 0 0 complex lifted from C3×S3 ρ10 6 0 -3 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C32⋊C6
On 9 points - transitive group 9T11
Generators in S9
(1 7 4)(2 5 6)(3 9 8)
(1 3 2)(4 8 6)(5 7 9)
(2 3)(4 5 6 7 8 9)

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

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

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

G:=TransitiveGroup(9,11);

On 9 points - transitive group 9T13
Generators in S9
(2 5 8)(3 6 9)
(1 7 4)(2 5 8)(3 9 6)
(1 2 3)(4 5 6 7 8 9)

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

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

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

G:=TransitiveGroup(9,13);

On 18 points - transitive group 18T20
Generators in S18
(2 12 15)(3 7 16)(5 18 9)(6 13 10)
(1 14 11)(2 12 15)(3 16 7)(4 8 17)(5 18 9)(6 10 13)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,20);

On 18 points - transitive group 18T21
Generators in S18
(1 14 11)(2 8 17)(3 12 13)(4 16 9)(5 18 7)(6 10 15)
(1 4 5)(2 6 3)(7 11 9)(8 10 12)(13 17 15)(14 16 18)
(1 2)(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,21);

On 18 points - transitive group 18T22
Generators in S18
(1 11 3)(2 13 15)(4 6 8)(5 18 16)(7 14 9)(10 12 17)
(1 7 16)(2 17 8)(3 9 18)(4 13 10)(5 11 14)(6 15 12)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,22);

On 27 points - transitive group 27T11
Generators in S27
(1 8 5)(2 21 18)(3 12 15)(4 16 24)(6 11 23)(7 27 19)(9 26 14)(10 20 25)(13 22 17)
(1 22 25)(2 26 23)(3 24 27)(4 19 12)(5 13 20)(6 21 14)(7 15 16)(8 17 10)(9 11 18)
(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)

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

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

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

G:=TransitiveGroup(27,11);

C32⋊C6 is a maximal subgroup of
C32⋊D6  C33⋊C6  He3.S3  He3.2S3  C33⋊S3  He3.3S3  He3⋊S3  He34S3  He3.4S3  C62⋊S3  C62⋊C6  ASL2(𝔽3)  He3⋊D5  C7⋊He3⋊C2  C32⋊F7  He3⋊D7
C32⋊C6 is a maximal quotient of
C32⋊C12  C32⋊C18  C32⋊D9  C3≀S3  C33⋊C6  He3.C6  He3.S3  He3.2C6  He3.2S3  He34S3  C62⋊S3  C62⋊C6  He3⋊D5  C7⋊He3⋊C2  C32⋊F7  He3⋊D7

Polynomial with Galois group C32⋊C6 over ℚ
actionf(x)Disc(f)
9T11x9-27x7+30x6+189x5-378x4-21x3+378x2-126x-42214·318·76·30232
9T13x9-4x8-30x7+142x6+79x5-680x4-247x3+998x2+716x+104218·33·76·414·126132

Matrix representation of C32⋊C6 in GL6(ℤ)

 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 -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
,
 1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0

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

C32⋊C6 in GAP, Magma, Sage, TeX

C_3^2\rtimes C_6
% in TeX

G:=Group("C3^2:C6");
// GroupNames label

G:=SmallGroup(54,5);
// by ID

G=gap.SmallGroup(54,5);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,146,150,579]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^6=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=b^-1>;
// generators/relations

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