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G = C2×C32⋊C6  order 108 = 22·33

Direct product of C2 and C32⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: C2×C32⋊C6, C32⋊2D6, He3⋊2C22, 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

Derived series
C1 — C32 — C2×C32⋊C6
Chief series
C1 — C3 — C32 — He3 — C32⋊C6 — C2×C32⋊C6
Lower central
C32 — C2×C32⋊C6
Upper central
C1 — C2

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 >

C1
C2
9C2
9C2
C3
3C3
3C3
6C3
9C22
C6
3S3
3S3
3C6
3C6
6C6
9S3
9S3
9C6
9C6
C32
C32
2C32
3D6
9C2×C6
9D6
C3⋊S3
C3⋊S3
C3×C6
C3×C6
2C3×C6
3C3×S3
3C3×S3
He3
C2×C3⋊S3
3S3×C6
C2×He3
C32⋊C6
C32⋊C6
C2×C32⋊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 2)(3 5)(4 6)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)

(1 16 9)(2 12 13)(3 10 15)(4 18 7)(5 14 11)(6 8 17)

(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)| (1,2)(3,5)(4,6)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16), (1,16,9)(2,12,13)(3,10,15)(4,18,7)(5,14,11)(6,8,17), (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( (1,2)(3,5)(4,6)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16), (1,16,9)(2,12,13)(3,10,15)(4,18,7)(5,14,11)(6,8,17), (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([[(1,2),(3,5),(4,6),(7,17),(8,18),(9,13),(10,14),(11,15),(12,16)], [(1,16,9),(2,12,13),(3,10,15),(4,18,7),(5,14,11),(6,8,17)], [(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(18,41);
 

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

(1 8 11)(2 9 12)(4 16 13)(6 15 18)

(1 8 11)(2 12 9)(3 10 7)(4 13 16)(5 17 14)(6 15 18)

(1 2 3)(4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
 
G:=sub<Sym(18)| (1,6)(2,4)(3,5)(7,14)(8,15)(9,16)(10,17)(11,18)(12,13), (1,8,11)(2,9,12)(4,16,13)(6,15,18), (1,8,11)(2,12,9)(3,10,7)(4,13,16)(5,17,14)(6,15,18), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)>;
 
G:=Group( (1,6)(2,4)(3,5)(7,14)(8,15)(9,16)(10,17)(11,18)(12,13), (1,8,11)(2,9,12)(4,16,13)(6,15,18), (1,8,11)(2,12,9)(3,10,7)(4,13,16)(5,17,14)(6,15,18), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );
 
G=PermutationGroup([[(1,6),(2,4),(3,5),(7,14),(8,15),(9,16),(10,17),(11,18),(12,13)], [(1,8,11),(2,9,12),(4,16,13),(6,15,18)], [(1,8,11),(2,12,9),(3,10,7),(4,13,16),(5,17,14),(6,15,18)], [(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  He3⋊2D4  He3⋊3D4  He3⋊4D4  He3⋊6D4  C32⋊2GL2(𝔽3)  Q8⋊He3⋊C2
C2×C32⋊C6 is a maximal quotient of   He3⋊3Q8  He3⋊4D4  He3⋊6D4

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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