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G = C32⋊C6order 54 = 2·33

The semidirect product of C32 and C6 acting faithfully

metabelian, supersoluble, monomial

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

Series: Derived Chief Lower central Upper central

C1C32 — C32⋊C6
C1C3C32He3 — C32⋊C6
C32 — C32⋊C6
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 >

9C2
3C3
3C3
6C3
3S3
9C6
9S3
2C32
3C3×S3

Character table of C32⋊C6

 class 123A3B3C3D3E3F6A6B
 size 1923366699
ρ11111111111    trivial
ρ21-1111111-1-1    linear of order 2
ρ3111ζ3ζ321ζ3ζ32ζ32ζ3    linear of order 3
ρ41-11ζ3ζ321ζ3ζ32ζ6ζ65    linear of order 6
ρ5111ζ32ζ31ζ32ζ3ζ3ζ32    linear of order 3
ρ61-11ζ32ζ31ζ32ζ3ζ65ζ6    linear of order 6
ρ720222-1-1-100    orthogonal lifted from S3
ρ8202-1--3-1+-3-1ζ6ζ6500    complex lifted from C3×S3
ρ9202-1+-3-1--3-1ζ65ζ600    complex lifted from C3×S3
ρ1060-30000000    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 9 15)(3 10 16)(5 18 12)(6 13 7)
(1 14 8)(2 9 15)(3 16 10)(4 11 17)(5 18 12)(6 7 13)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(27,11);

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)

001000
000100
000010
000001
100000
010000
,
010000
-1-10000
000100
00-1-100
000001
0000-1-1
,
100000
-1-10000
0000-1-1
000001
000100
001000

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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