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G = 3- 1+2order 27 = 33

Extraspecial group

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

Aliases: 3- 1+2, C9⋊C3, C32.C3, C3.2C32, 3-Sylow(AGammaL(1,64)), SmallGroup(27,4)

Series: Derived Chief Lower central Upper central Jennings

C1C3 — 3- 1+2
C1C3C32 — 3- 1+2
C1C3 — 3- 1+2
C1C3 — 3- 1+2
C1C3C3 — 3- 1+2

Generators and relations for 3- 1+2
 G = < a,b | a9=b3=1, bab-1=a4 >

3C3

Character table of 3- 1+2

 class 13A3B3C3D9A9B9C9D9E9F
 size 11133333333
ρ111111111111    trivial
ρ2111ζ3ζ32ζ3ζ32ζ311ζ32    linear of order 3
ρ3111ζ32ζ3ζ311ζ32ζ3ζ32    linear of order 3
ρ411111ζ3ζ3ζ32ζ3ζ32ζ32    linear of order 3
ρ511111ζ32ζ32ζ3ζ32ζ3ζ3    linear of order 3
ρ6111ζ3ζ32ζ3211ζ3ζ32ζ3    linear of order 3
ρ7111ζ32ζ3ζ32ζ3ζ3211ζ3    linear of order 3
ρ8111ζ32ζ31ζ32ζ3ζ3ζ321    linear of order 3
ρ9111ζ3ζ321ζ3ζ32ζ32ζ31    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 3- 1+2
On 9 points - transitive group 9T6
Generators in S9
(1 2 3 4 5 6 7 8 9)
(2 8 5)(3 6 9)

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

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

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

G:=TransitiveGroup(9,6);

Regular action on 27 points - transitive group 27T5
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 16 24)(2 14 19)(3 12 23)(4 10 27)(5 17 22)(6 15 26)(7 13 21)(8 11 25)(9 18 20)

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

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

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

G:=TransitiveGroup(27,5);

Polynomial with Galois group 3- 1+2 over ℚ
actionf(x)Disc(f)
9T6x9-21x7-7x6+126x5+42x4-273x3-63x2+189x+7312·78·2151532

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

001
400
040
,
100
040
002
G:=sub<GL(3,GF(7))| [0,4,0,0,0,4,1,0,0],[1,0,0,0,4,0,0,0,2] >;

3- 1+2 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(27,4);
// by ID

G=gap.SmallGroup(27,4);
# by ID

G:=PCGroup([3,-3,3,-3,27,73]);
// Polycyclic

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

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