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G = C2×3- 1+2order 54 = 2·33

Direct product of C2 and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Aliases: C2×3- 1+2, C18⋊C3, C92C6, C32.C6, C6.2C32, (C3×C6).C3, C3.2(C3×C6), SmallGroup(54,11)

Series: Derived Chief Lower central Upper central

C1C3 — C2×3- 1+2
C1C3C323- 1+2 — C2×3- 1+2
C1C3 — C2×3- 1+2
C1C6 — C2×3- 1+2

Generators and relations for C2×3- 1+2
 G = < a,b,c | a2=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

3C3
3C6

Character table of C2×3- 1+2

 class 123A3B3C3D6A6B6C6D9A9B9C9D9E9F18A18B18C18D18E18F
 size 1111331133333333333333
ρ11111111111111111111111    trivial
ρ21-11111-1-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ31111ζ32ζ311ζ32ζ3ζ3ζ321ζ3ζ321ζ321ζ3ζ3ζ321    linear of order 3
ρ41-111ζ3ζ32-1-1ζ65ζ611ζ32ζ3ζ32ζ3-1ζ6ζ65-1ζ6ζ65    linear of order 6
ρ51111111111ζ32ζ3ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ61111ζ3ζ3211ζ3ζ32ζ3ζ32ζ311ζ32ζ32ζ31ζ31ζ32    linear of order 3
ρ71-111ζ3ζ32-1-1ζ65ζ6ζ3ζ32ζ311ζ32ζ6ζ65-1ζ65-1ζ6    linear of order 6
ρ81-111ζ3ζ32-1-1ζ65ζ6ζ32ζ31ζ32ζ31ζ65-1ζ6ζ6ζ65-1    linear of order 6
ρ91-111ζ32ζ3-1-1ζ6ζ65ζ32ζ3ζ3211ζ3ζ65ζ6-1ζ6-1ζ65    linear of order 6
ρ101111ζ3ζ3211ζ3ζ32ζ32ζ31ζ32ζ31ζ31ζ32ζ32ζ31    linear of order 3
ρ111-11111-1-1-1-1ζ3ζ32ζ32ζ32ζ3ζ3ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ121-11111-1-1-1-1ζ32ζ3ζ3ζ3ζ32ζ32ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ131111111111ζ3ζ32ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ141111ζ32ζ311ζ32ζ3ζ32ζ3ζ3211ζ3ζ3ζ321ζ321ζ3    linear of order 3
ρ151-111ζ32ζ3-1-1ζ6ζ6511ζ3ζ32ζ3ζ32-1ζ65ζ6-1ζ65ζ6    linear of order 6
ρ161111ζ3ζ3211ζ3ζ3211ζ32ζ3ζ32ζ31ζ32ζ31ζ32ζ3    linear of order 3
ρ171-111ζ32ζ3-1-1ζ6ζ65ζ3ζ321ζ3ζ321ζ6-1ζ65ζ65ζ6-1    linear of order 6
ρ181111ζ32ζ311ζ32ζ311ζ3ζ32ζ3ζ321ζ3ζ321ζ3ζ32    linear of order 3
ρ193-3-3+3-3/2-3-3-3/2003+3-3/23-3-3/200000000000000    complex faithful
ρ203-3-3-3-3/2-3+3-3/2003-3-3/23+3-3/200000000000000    complex faithful
ρ2133-3+3-3/2-3-3-3/200-3-3-3/2-3+3-3/200000000000000    complex lifted from 3- 1+2
ρ2233-3-3-3/2-3+3-3/200-3+3-3/2-3-3-3/200000000000000    complex lifted from 3- 1+2

Permutation representations of C2×3- 1+2
On 18 points - transitive group 18T14
Generators in S18
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)

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

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

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

G:=TransitiveGroup(18,14);

Matrix representation of C2×3- 1+2 in GL3(𝔽7) generated by

600
060
006
,
001
500
050
,
200
040
001
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[0,5,0,0,0,5,1,0,0],[2,0,0,0,4,0,0,0,1] >;

C2×3- 1+2 in GAP, Magma, Sage, TeX

C_2\times 3_-^{1+2}
% in TeX

G:=Group("C2xES-(3,1)");
// GroupNames label

G:=SmallGroup(54,11);
// by ID

G=gap.SmallGroup(54,11);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,77,150]);
// Polycyclic

G:=Group<a,b,c|a^2=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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