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G = C33⋊C6order 162 = 2·34

1st semidirect product of C33 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: He31S3, C331C6, C3≀C33C2, C33⋊C21C3, C32.6(C3×S3), C3.2(C32⋊C6), SmallGroup(162,11)

Series: Derived Chief Lower central Upper central

C1C33 — C33⋊C6
C1C3C32C33C3≀C3 — C33⋊C6
C33 — C33⋊C6
C1

Generators and relations for C33⋊C6
 G = < a,b,c,d | a3=b3=c3=d6=1, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

27C2
3C3
3C3
3C3
3C3
9C3
9S3
27S3
27S3
27S3
27S3
27C6
3C32
3C32
3C32
3C32
3C32
6C9
3C3⋊S3
9C3⋊S3
9C3⋊S3
9C3×S3
9C3⋊S3
9C3⋊S3
23- 1+2
3C32⋊C6

Character table of C33⋊C6

 class 123A3B3C3D3E3F3G6A6B9A9B
 size 127266669927271818
ρ11111111111111    trivial
ρ21-11111111-1-111    linear of order 2
ρ31111111ζ32ζ3ζ32ζ3ζ3ζ32    linear of order 3
ρ41111111ζ3ζ32ζ3ζ32ζ32ζ3    linear of order 3
ρ51-111111ζ3ζ32ζ65ζ6ζ32ζ3    linear of order 6
ρ61-111111ζ32ζ3ζ6ζ65ζ3ζ32    linear of order 6
ρ7202-1-1-122200-1-1    orthogonal lifted from S3
ρ8202-1-1-12-1+-3-1--300ζ6ζ65    complex lifted from C3×S3
ρ9202-1-1-12-1--3-1+-300ζ65ζ6    complex lifted from C3×S3
ρ1060-303-30000000    orthogonal faithful
ρ1160-3-3030000000    orthogonal faithful
ρ12606000-3000000    orthogonal lifted from C32⋊C6
ρ1360-33-300000000    orthogonal faithful

Permutation representations of C33⋊C6
On 9 points - transitive group 9T22
Generators in S9
(1 4 7)(2 8 5)
(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)| (1,4,7)(2,8,5), (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( (1,4,7)(2,8,5), (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([(1,4,7),(2,8,5)], [(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(9,22);

On 18 points - transitive group 18T85
Generators in S18
(1 8 14)(2 15 9)(4 17 11)(5 12 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:=sub<Sym(18)| (1,8,14)(2,15,9)(4,17,11)(5,12,18), (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( (1,8,14)(2,15,9)(4,17,11)(5,12,18), (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=PermutationGroup([(1,8,14),(2,15,9),(4,17,11),(5,12,18)], [(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:=TransitiveGroup(18,85);

On 27 points - transitive group 27T53
Generators in S27
(1 24 27)(3 5 8)(6 10 17)(9 20 13)(12 26 16)(15 19 23)
(1 27 24)(2 7 4)(3 12 15)(5 26 19)(6 17 10)(8 16 23)(9 13 20)(11 21 25)(14 22 18)
(1 20 17)(2 18 21)(3 16 19)(4 22 11)(5 12 23)(6 24 13)(7 14 25)(8 26 15)(9 10 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)| (1,24,27)(3,5,8)(6,10,17)(9,20,13)(12,26,16)(15,19,23), (1,27,24)(2,7,4)(3,12,15)(5,26,19)(6,17,10)(8,16,23)(9,13,20)(11,21,25)(14,22,18), (1,20,17)(2,18,21)(3,16,19)(4,22,11)(5,12,23)(6,24,13)(7,14,25)(8,26,15)(9,10,27), (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( (1,24,27)(3,5,8)(6,10,17)(9,20,13)(12,26,16)(15,19,23), (1,27,24)(2,7,4)(3,12,15)(5,26,19)(6,17,10)(8,16,23)(9,13,20)(11,21,25)(14,22,18), (1,20,17)(2,18,21)(3,16,19)(4,22,11)(5,12,23)(6,24,13)(7,14,25)(8,26,15)(9,10,27), (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([(1,24,27),(3,5,8),(6,10,17),(9,20,13),(12,26,16),(15,19,23)], [(1,27,24),(2,7,4),(3,12,15),(5,26,19),(6,17,10),(8,16,23),(9,13,20),(11,21,25),(14,22,18)], [(1,20,17),(2,18,21),(3,16,19),(4,22,11),(5,12,23),(6,24,13),(7,14,25),(8,26,15),(9,10,27)], [(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(27,53);

On 27 points - transitive group 27T62
Generators in S27
(1 10 13)(2 16 22)(3 25 19)(4 20 15)(5 18 11)(6 23 24)(7 12 17)(8 14 21)(9 27 26)
(1 7 4)(2 5 6)(3 9 8)(10 12 20)(11 24 22)(13 17 15)(14 25 27)(16 18 23)(19 26 21)
(1 3 2)(4 8 6)(5 7 9)(10 25 16)(11 17 26)(12 27 18)(13 19 22)(14 23 20)(15 21 24)
(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)| (1,10,13)(2,16,22)(3,25,19)(4,20,15)(5,18,11)(6,23,24)(7,12,17)(8,14,21)(9,27,26), (1,7,4)(2,5,6)(3,9,8)(10,12,20)(11,24,22)(13,17,15)(14,25,27)(16,18,23)(19,26,21), (1,3,2)(4,8,6)(5,7,9)(10,25,16)(11,17,26)(12,27,18)(13,19,22)(14,23,20)(15,21,24), (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( (1,10,13)(2,16,22)(3,25,19)(4,20,15)(5,18,11)(6,23,24)(7,12,17)(8,14,21)(9,27,26), (1,7,4)(2,5,6)(3,9,8)(10,12,20)(11,24,22)(13,17,15)(14,25,27)(16,18,23)(19,26,21), (1,3,2)(4,8,6)(5,7,9)(10,25,16)(11,17,26)(12,27,18)(13,19,22)(14,23,20)(15,21,24), (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([(1,10,13),(2,16,22),(3,25,19),(4,20,15),(5,18,11),(6,23,24),(7,12,17),(8,14,21),(9,27,26)], [(1,7,4),(2,5,6),(3,9,8),(10,12,20),(11,24,22),(13,17,15),(14,25,27),(16,18,23),(19,26,21)], [(1,3,2),(4,8,6),(5,7,9),(10,25,16),(11,17,26),(12,27,18),(13,19,22),(14,23,20),(15,21,24)], [(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(27,62);

On 27 points - transitive group 27T63
Generators in S27
(1 10 13)(2 16 19)(3 20 17)(4 23 21)(5 27 11)(6 12 22)(7 18 26)(8 14 24)(9 25 15)
(1 7 4)(3 6 9)(10 18 23)(12 25 20)(13 26 21)(15 17 22)
(1 4 7)(2 8 5)(3 6 9)(10 23 18)(11 19 24)(12 25 20)(13 21 26)(14 27 16)(15 17 22)
(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)| (1,10,13)(2,16,19)(3,20,17)(4,23,21)(5,27,11)(6,12,22)(7,18,26)(8,14,24)(9,25,15), (1,7,4)(3,6,9)(10,18,23)(12,25,20)(13,26,21)(15,17,22), (1,4,7)(2,8,5)(3,6,9)(10,23,18)(11,19,24)(12,25,20)(13,21,26)(14,27,16)(15,17,22), (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( (1,10,13)(2,16,19)(3,20,17)(4,23,21)(5,27,11)(6,12,22)(7,18,26)(8,14,24)(9,25,15), (1,7,4)(3,6,9)(10,18,23)(12,25,20)(13,26,21)(15,17,22), (1,4,7)(2,8,5)(3,6,9)(10,23,18)(11,19,24)(12,25,20)(13,21,26)(14,27,16)(15,17,22), (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([(1,10,13),(2,16,19),(3,20,17),(4,23,21),(5,27,11),(6,12,22),(7,18,26),(8,14,24),(9,25,15)], [(1,7,4),(3,6,9),(10,18,23),(12,25,20),(13,26,21),(15,17,22)], [(1,4,7),(2,8,5),(3,6,9),(10,23,18),(11,19,24),(12,25,20),(13,21,26),(14,27,16),(15,17,22)], [(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(27,63);

C33⋊C6 is a maximal subgroup of   He3⋊D6  (C3×He3)⋊C6  C345C6  C3≀C3.S3
C33⋊C6 is a maximal quotient of   C33⋊C12  C3.C3≀S3  C32⋊C9⋊S3  (C3×He3).C6  C331C18  He3⋊D9  C345C6

Polynomial with Galois group C33⋊C6 over ℚ
actionf(x)Disc(f)
9T22x9-30x7-20x6+279x5+372x4-659x3-1566x2-1044x-232224·39·77·292

Matrix representation of C33⋊C6 in GL6(ℤ)

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

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

C33⋊C6 in GAP, Magma, Sage, TeX

C_3^3\rtimes C_6
% in TeX

G:=Group("C3^3:C6");
// GroupNames label

G:=SmallGroup(162,11);
// by ID

G=gap.SmallGroup(162,11);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,182,187,723,728,2704]);
// Polycyclic

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

Export

Subgroup lattice of C33⋊C6 in TeX
Character table of C33⋊C6 in TeX

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