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G = He3order 27 = 33

Heisenberg group

p-group, metabelian, nilpotent (class 2), monomial

Aliases: He3, 3+ 1+2, C32⋊C3, C3.1C32, 3-Sylow(SL(3,3)), SmallGroup(27,3)

Series: Derived Chief Lower central Upper central Jennings

C1C3 — He3
C1C3C32 — He3
C1C3 — He3
C1C3 — He3
C1C3 — He3

Generators and relations for He3
 G = < a,b,c | a3=b3=c3=1, ab=ba, cac-1=ab-1, bc=cb >

3C3
3C3
3C3
3C3

Character table of He3

 class 13A3B3C3D3E3F3G3H3I3J
 size 11133333333
ρ111111111111    trivial
ρ2111ζ3ζ321ζ321ζ3ζ3ζ32    linear of order 3
ρ3111ζ321ζ3ζ3ζ321ζ3ζ32    linear of order 3
ρ4111ζ32ζ32ζ32ζ3ζ3ζ311    linear of order 3
ρ5111ζ31ζ32ζ32ζ31ζ32ζ3    linear of order 3
ρ61111ζ3ζ321ζ3ζ32ζ3ζ32    linear of order 3
ρ7111ζ3ζ3ζ3ζ32ζ32ζ3211    linear of order 3
ρ81111ζ32ζ31ζ32ζ3ζ32ζ3    linear of order 3
ρ9111ζ32ζ31ζ31ζ32ζ32ζ3    linear of order 3
ρ103-3+3-3/2-3-3-3/200000000    complex faithful
ρ113-3-3-3/2-3+3-3/200000000    complex faithful

Permutation representations of He3
On 9 points - transitive group 9T7
Generators in S9
(1 2 3)(4 5 6)(7 8 9)
(1 9 5)(2 7 6)(3 8 4)
(1 3 6)(2 9 8)(4 7 5)

G:=sub<Sym(9)| (1,2,3)(4,5,6)(7,8,9), (1,9,5)(2,7,6)(3,8,4), (1,3,6)(2,9,8)(4,7,5)>;

G:=Group( (1,2,3)(4,5,6)(7,8,9), (1,9,5)(2,7,6)(3,8,4), (1,3,6)(2,9,8)(4,7,5) );

G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9)], [(1,9,5),(2,7,6),(3,8,4)], [(1,3,6),(2,9,8),(4,7,5)])

G:=TransitiveGroup(9,7);

Regular action on 27 points - transitive group 27T3
Generators in S27
(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 13 11)(2 14 12)(3 15 10)(4 26 8)(5 27 9)(6 25 7)(16 19 22)(17 20 23)(18 21 24)
(1 5 24)(2 25 19)(3 8 17)(4 20 15)(6 16 12)(7 22 14)(9 21 11)(10 26 23)(13 27 18)

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

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

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

G:=TransitiveGroup(27,3);

Polynomial with Galois group He3 over ℚ
actionf(x)Disc(f)
9T7x9-3x8-15x7+51x6+39x5-219x4+81x3+204x2-132x-826·316·194·22872

Matrix representation of He3 in GL3(𝔽7) generated by

010
001
100
,
200
020
002
,
040
002
100
G:=sub<GL(3,GF(7))| [0,0,1,1,0,0,0,1,0],[2,0,0,0,2,0,0,0,2],[0,0,1,4,0,0,0,2,0] >;

He3 in GAP, Magma, Sage, TeX

{\rm He}_3
% in TeX

G:=Group("He3");
// GroupNames label

G:=SmallGroup(27,3);
// by ID

G=gap.SmallGroup(27,3);
# by ID

G:=PCGroup([3,-3,3,-3,73]);
// Polycyclic

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

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