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G = C32⋊D6order 108 = 22·33

The semidirect product of C32 and D6 acting faithfully

non-abelian, supersoluble, monomial, rational

Aliases: C32⋊D6, He3⋊C22, C3⋊S3⋊S3, C3.2S32, C32⋊C6⋊C2, He3⋊C2⋊C2, SmallGroup(108,17)

Series: Derived Chief Lower central Upper central

C1C3He3 — C32⋊D6
C1C3C32He3C32⋊C6 — C32⋊D6
He3 — C32⋊D6
C1

Generators and relations for C32⋊D6
 G = < a,b,c,d | a3=b3=c6=d2=1, ab=ba, cac-1=dad=a-1b-1, cbc-1=b-1, bd=db, dcd=c-1 >

9C2
9C2
9C2
3C3
3C3
6C3
27C22
3S3
3S3
3S3
3S3
6S3
9S3
9S3
9C6
9C6
9C6
2C32
9D6
9D6
9D6
3C3×S3
3C3×S3
3C3×S3
3C3×S3
6C3×S3
3S32
3S32

Character table of C32⋊D6

 class 12A2B2C3A3B3C3D6A6B6C
 size 199926612181818
ρ111111111111    trivial
ρ211-1-11111-11-1    linear of order 2
ρ31-11-111111-1-1    linear of order 2
ρ41-1-111111-1-11    linear of order 2
ρ52-20022-1-1010    orthogonal lifted from D6
ρ620-202-12-1100    orthogonal lifted from D6
ρ7220022-1-10-10    orthogonal lifted from S3
ρ820202-12-1-100    orthogonal lifted from S3
ρ940004-2-21000    orthogonal lifted from S32
ρ10600-2-3000001    orthogonal faithful
ρ116002-300000-1    orthogonal faithful

Permutation representations of C32⋊D6
On 9 points - transitive group 9T18
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)
(4 9)(5 8)(6 7)

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), (4,9)(5,8)(6,7)>;

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), (4,9)(5,8)(6,7) );

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)], [(4,9),(5,8),(6,7)])

G:=TransitiveGroup(9,18);

On 18 points - transitive group 18T51
Generators in S18
(1 13 9)(2 12 16)(3 10 18)(4 15 7)(5 17 11)(6 8 14)
(1 4 5)(2 6 3)(7 11 9)(8 10 12)(13 15 17)(14 18 16)
(1 2)(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 3)(2 4)(5 6)(7 8)(9 12)(10 11)(13 14)(15 18)(16 17)

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

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

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

G:=TransitiveGroup(18,51);

On 18 points - transitive group 18T55
Generators in S18
(1 11 8)(2 16 13)(3 9 10)(4 7 12)(5 18 17)(6 14 15)
(1 4 3)(2 5 6)(7 9 11)(8 12 10)(13 17 15)(14 16 18)
(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 2)(3 6)(4 5)(7 13)(8 18)(9 17)(10 16)(11 15)(12 14)

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

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

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

G:=TransitiveGroup(18,55);

On 18 points - transitive group 18T56
Generators in S18
(1 18 16)(2 10 6)(3 7 5)(4 13 15)(8 17 12)(9 14 11)
(1 7 14)(2 15 8)(3 9 16)(4 17 10)(5 11 18)(6 13 12)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 3)(4 6)(7 9)(10 12)(13 17)(14 16)

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

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

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

G:=TransitiveGroup(18,56);

On 18 points - transitive group 18T57
Generators in S18
(2 11 16)(3 12 17)(5 13 8)(6 14 9)
(1 15 10)(2 11 16)(3 17 12)(4 7 18)(5 13 8)(6 9 14)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 3)(4 6)(7 9)(10 12)(14 18)(15 17)

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

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

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

G:=TransitiveGroup(18,57);

On 27 points - transitive group 27T29
Generators in S27
(1 26 23)(2 19 16)(3 12 15)(4 10 18)(5 14 27)(6 17 25)(7 21 13)(8 24 11)(9 22 20)
(1 7 4)(2 5 8)(3 9 6)(10 26 21)(11 16 27)(12 22 17)(13 18 23)(14 24 19)(15 20 25)
(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 2)(4 8)(5 7)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)

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

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

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

G:=TransitiveGroup(27,29);

C32⋊D6 is a maximal subgroup of
He3⋊D4  He3⋊D6  He3.D6  He3.2D6  He35D6  He36D6  He3.6D6  C625D6  AGL2(𝔽3)
C32⋊D6 is a maximal quotient of
He32Q8  C6.S32  He32D4  He3⋊(C2×C4)  He33D4  C32⋊D18  He3⋊D6  He3.D6  He3.2D6  He35D6  He36D6  C625D6

Polynomial with Galois group C32⋊D6 over ℚ
actionf(x)Disc(f)
9T18x9-3x8-39x7+167x6-24x5-480x4+136x3+384x2+144x+16231·312·76·373

Matrix representation of C32⋊D6 in GL6(ℤ)

100000
010000
000100
00-1-100
0000-1-1
000010
,
-1-10000
100000
00-1-100
001000
0000-1-1
000010
,
001000
00-1-100
000010
0000-1-1
100000
-1-10000
,
00-1000
000-100
-100000
0-10000
0000-10
00000-1

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],[0,0,0,0,1,-1,0,0,0,0,0,-1,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0],[0,0,-1,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1] >;

C32⋊D6 in GAP, Magma, Sage, TeX

C_3^2\rtimes D_6
% in TeX

G:=Group("C3^2:D6");
// GroupNames label

G:=SmallGroup(108,17);
// by ID

G=gap.SmallGroup(108,17);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,67,483,253,1804,909]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of C32⋊D6 in TeX
Character table of C32⋊D6 in TeX

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