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G = C2×He3order 54 = 2·33

Direct product of C2 and He3

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

Aliases: C2×He3, C322C6, C6.1C32, (C3×C6)⋊C3, C3.1(C3×C6), SmallGroup(54,10)

Series: Derived Chief Lower central Upper central

C1C3 — C2×He3
C1C3C32He3 — C2×He3
C1C3 — C2×He3
C1C6 — C2×He3

Generators and relations for C2×He3
 G = < a,b,c,d | a2=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

3C3
3C3
3C3
3C3
3C6
3C6
3C6
3C6

Character table of C2×He3

 class 123A3B3C3D3E3F3G3H3I3J6A6B6C6D6E6F6G6H6I6J
 size 1111333333331133333333
ρ11111111111111111111111    trivial
ρ21-11111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ31-111ζ3ζ31ζ32ζ32ζ31ζ32-1-1ζ6ζ65ζ65-1ζ6ζ6ζ65-1    linear of order 6
ρ41111ζ3ζ31ζ32ζ32ζ31ζ3211ζ32ζ3ζ31ζ32ζ32ζ31    linear of order 3
ρ51-1111ζ32ζ32ζ32ζ3ζ3ζ31-1-1-1-1ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ611111ζ3ζ3ζ3ζ32ζ32ζ3211111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ71111ζ32ζ3ζ321ζ321ζ3ζ311ζ3ζ32ζ3ζ321ζ321ζ3    linear of order 3
ρ81111ζ32ζ321ζ3ζ3ζ321ζ311ζ3ζ32ζ321ζ3ζ3ζ321    linear of order 3
ρ91-111ζ31ζ32ζ31ζ32ζ3ζ32-1-1ζ6ζ65-1ζ6ζ65-1ζ6ζ65    linear of order 6
ρ1011111ζ32ζ32ζ32ζ3ζ3ζ311111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ111-111ζ3ζ32ζ31ζ31ζ32ζ32-1-1ζ6ζ65ζ6ζ65-1ζ65-1ζ6    linear of order 6
ρ121111ζ321ζ3ζ321ζ3ζ32ζ311ζ3ζ321ζ3ζ321ζ3ζ32    linear of order 3
ρ131111ζ31ζ32ζ31ζ32ζ3ζ3211ζ32ζ31ζ32ζ31ζ32ζ3    linear of order 3
ρ141-111ζ321ζ3ζ321ζ3ζ32ζ3-1-1ζ65ζ6-1ζ65ζ6-1ζ65ζ6    linear of order 6
ρ151-111ζ32ζ3ζ321ζ321ζ3ζ3-1-1ζ65ζ6ζ65ζ6-1ζ6-1ζ65    linear of order 6
ρ161-111ζ32ζ321ζ3ζ3ζ321ζ3-1-1ζ65ζ6ζ6-1ζ65ζ65ζ6-1    linear of order 6
ρ171111ζ3ζ32ζ31ζ31ζ32ζ3211ζ32ζ3ζ32ζ31ζ31ζ32    linear of order 3
ρ181-1111ζ3ζ3ζ3ζ32ζ32ζ321-1-1-1-1ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ1933-3-3-3/2-3+3-3/200000000-3+3-3/2-3-3-3/200000000    complex lifted from He3
ρ203-3-3-3-3/2-3+3-3/2000000003-3-3/23+3-3/200000000    complex faithful
ρ2133-3+3-3/2-3-3-3/200000000-3-3-3/2-3+3-3/200000000    complex lifted from He3
ρ223-3-3+3-3/2-3-3-3/2000000003+3-3/23-3-3/200000000    complex faithful

Permutation representations of C2×He3
On 18 points - transitive group 18T15
Generators in S18
(1 6)(2 4)(3 5)(7 16)(8 17)(9 18)(10 13)(11 14)(12 15)
(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 4)(2 6 3)(7 9 8)(10 11 12)(13 14 15)(16 18 17)
(1 16 14)(2 8 10)(3 9 12)(4 17 13)(5 18 15)(6 7 11)

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

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

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

G:=TransitiveGroup(18,15);

C2×He3 is a maximal subgroup of   C32⋊C12  He33C4  Q8⋊He3  D7⋊He3
C2×He3 is a maximal quotient of   D7⋊He3

Matrix representation of C2×He3 in GL3(𝔽7) generated by

600
060
006
,
400
020
001
,
400
040
004
,
060
006
100
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[4,0,0,0,2,0,0,0,1],[4,0,0,0,4,0,0,0,4],[0,0,1,6,0,0,0,6,0] >;

C2×He3 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3
% in TeX

G:=Group("C2xHe3");
// GroupNames label

G:=SmallGroup(54,10);
// by ID

G=gap.SmallGroup(54,10);
# by ID

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

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

Export

Subgroup lattice of C2×He3 in TeX
Character table of C2×He3 in TeX

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