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G = C2×C32⋊C6order 108 = 22·33

Direct product of C2 and C32⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: C2×C32⋊C6, C322D6, He32C22, C3⋊S3⋊C6, (C3×C6)⋊C6, (C3×C6)⋊1S3, C32⋊(C2×C6), C3.2(S3×C6), C6.5(C3×S3), (C2×He3)⋊1C2, (C2×C3⋊S3)⋊C3, SmallGroup(108,25)

Series: Derived Chief Lower central Upper central

C1C32 — C2×C32⋊C6
C1C3C32He3C32⋊C6 — C2×C32⋊C6
C32 — C2×C32⋊C6
C1C2

Generators and relations for C2×C32⋊C6
 G = < a,b,c,d | a2=b3=c3=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

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

Character table of C2×C32⋊C6

 class 12A2B2C3A3B3C3D3E3F6A6B6C6D6E6F6G6H6I6J
 size 11992336662336669999
ρ111111111111111111111    trivial
ρ211-1-1111111111111-1-1-1-1    linear of order 2
ρ31-11-1111111-1-1-1-1-1-11-11-1    linear of order 2
ρ41-1-11111111-1-1-1-1-1-1-11-11    linear of order 2
ρ51-1-111ζ3ζ32ζ321ζ3-1ζ65ζ6ζ6ζ65-1ζ6ζ3ζ65ζ32    linear of order 6
ρ61-11-11ζ32ζ3ζ31ζ32-1ζ6ζ65ζ65ζ6-1ζ3ζ6ζ32ζ65    linear of order 6
ρ711111ζ32ζ3ζ31ζ321ζ32ζ3ζ3ζ321ζ3ζ32ζ32ζ3    linear of order 3
ρ811-1-11ζ3ζ32ζ321ζ31ζ3ζ32ζ32ζ31ζ6ζ65ζ65ζ6    linear of order 6
ρ911-1-11ζ32ζ3ζ31ζ321ζ32ζ3ζ3ζ321ζ65ζ6ζ6ζ65    linear of order 6
ρ101-1-111ζ32ζ3ζ31ζ32-1ζ6ζ65ζ65ζ6-1ζ65ζ32ζ6ζ3    linear of order 6
ρ111-11-11ζ3ζ32ζ321ζ3-1ζ65ζ6ζ6ζ65-1ζ32ζ65ζ3ζ6    linear of order 6
ρ1211111ζ3ζ32ζ321ζ31ζ3ζ32ζ32ζ31ζ32ζ3ζ3ζ32    linear of order 3
ρ132-200222-1-1-1-2-2-21110000    orthogonal lifted from D6
ρ142200222-1-1-1222-1-1-10000    orthogonal lifted from S3
ρ152-2002-1+-3-1--3ζ6-1ζ65-21--31+-3ζ32ζ310000    complex lifted from S3×C6
ρ1622002-1--3-1+-3ζ65-1ζ62-1--3-1+-3ζ65ζ6-10000    complex lifted from C3×S3
ρ172-2002-1--3-1+-3ζ65-1ζ6-21+-31--3ζ3ζ3210000    complex lifted from S3×C6
ρ1822002-1+-3-1--3ζ6-1ζ652-1+-3-1--3ζ6ζ65-10000    complex lifted from C3×S3
ρ196-600-3000003000000000    orthogonal faithful
ρ206600-300000-3000000000    orthogonal lifted from C32⋊C6

Permutation representations of C2×C32⋊C6
On 18 points - transitive group 18T41
Generators in S18
(1 3)(2 4)(5 6)(7 14)(8 15)(9 16)(10 17)(11 18)(12 13)
(1 11 17)(2 14 8)(3 18 10)(4 7 15)(5 9 13)(6 16 12)
(1 4 5)(2 6 3)(7 9 11)(8 12 10)(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)| (1,3)(2,4)(5,6)(7,14)(8,15)(9,16)(10,17)(11,18)(12,13), (1,11,17)(2,14,8)(3,18,10)(4,7,15)(5,9,13)(6,16,12), (1,4,5)(2,6,3)(7,9,11)(8,12,10)(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( (1,3)(2,4)(5,6)(7,14)(8,15)(9,16)(10,17)(11,18)(12,13), (1,11,17)(2,14,8)(3,18,10)(4,7,15)(5,9,13)(6,16,12), (1,4,5)(2,6,3)(7,9,11)(8,12,10)(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([(1,3),(2,4),(5,6),(7,14),(8,15),(9,16),(10,17),(11,18),(12,13)], [(1,11,17),(2,14,8),(3,18,10),(4,7,15),(5,9,13),(6,16,12)], [(1,4,5),(2,6,3),(7,9,11),(8,12,10),(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(18,41);

On 18 points - transitive group 18T42
Generators in S18
(1 5)(2 6)(3 4)(7 15)(8 16)(9 17)(10 18)(11 13)(12 14)
(1 8 11)(2 9 12)(5 16 13)(6 17 14)
(1 8 11)(2 12 9)(3 10 7)(4 18 15)(5 16 13)(6 14 17)
(1 2 3)(4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,42);

C2×C32⋊C6 is a maximal subgroup of   C6.S32  He32D4  He33D4  He34D4  He36D4  C322GL2(𝔽3)  Q8⋊He3⋊C2
C2×C32⋊C6 is a maximal quotient of   He33Q8  He34D4  He36D4

Matrix representation of C2×C32⋊C6 in GL6(ℤ)

-100000
0-10000
00-1000
000-100
0000-10
00000-1
,
001000
000100
000010
000001
100000
010000
,
-1-10000
100000
00-1-100
001000
0000-1-1
000010
,
100000
-1-10000
000001
000010
00-1-100
000100

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

C2×C32⋊C6 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes C_6
% in TeX

G:=Group("C2xC3^2:C6");
// GroupNames label

G:=SmallGroup(108,25);
// by ID

G=gap.SmallGroup(108,25);
# by ID

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

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

Export

Subgroup lattice of C2×C32⋊C6 in TeX
Character table of C2×C32⋊C6 in TeX

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