↑

Properties

Label 9T27
Degree $9$
Order $504$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PSL(2,8)$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(9, 27);
 

Group action invariants

Degree $n$:  $9$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $27$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSL(2,8)$
CHM label:  $L(9)=PSL(2,8)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,5)(3,6)(4,7)(8,9), (1,9)(2,3)(4,5)(6,7), (1,2,4,3,6,7,5)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

28T70, 36T712

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 7, 1, 1 $ $72$ $7$ $(3,4,6,5,7,9,8)$
$ 7, 1, 1 $ $72$ $7$ $(3,5,8,6,9,4,7)$
$ 7, 1, 1 $ $72$ $7$ $(3,6,7,8,4,5,9)$
$ 2, 2, 2, 2, 1 $ $63$ $2$ $(2,3)(4,9)(5,6)(7,8)$
$ 3, 3, 3 $ $56$ $3$ $(1,2,3)(4,6,7)(5,9,8)$
$ 9 $ $56$ $9$ $(1,2,3,4,8,9,7,6,5)$
$ 9 $ $56$ $9$ $(1,2,3,6,9,5,4,7,8)$
$ 9 $ $56$ $9$ $(1,2,3,7,5,8,6,4,9)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $504=2^{3} \cdot 3^{2} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Label:  504.156
magma: IdentifyGroup(G);
 
Character table:    
     2  3  .  .  .  3  .  .  .  .
     3  2  .  .  .  .  2  2  2  2
     7  1  1  1  1  .  .  .  .  .

       1a 7a 7b 7c 2a 3a 9a 9b 9c
    2P 1a 7c 7a 7b 1a 3a 9b 9c 9a
    3P 1a 7b 7c 7a 2a 1a 3a 3a 3a
    5P 1a 7c 7a 7b 2a 3a 9c 9a 9b
    7P 1a 1a 1a 1a 2a 3a 9b 9c 9a

X.1     1  1  1  1  1  1  1  1  1
X.2     7  .  .  . -1 -2  1  1  1
X.3     7  .  .  . -1  1  D  F  E
X.4     7  .  .  . -1  1  E  D  F
X.5     7  .  .  . -1  1  F  E  D
X.6     8  1  1  1  . -1 -1 -1 -1
X.7     9  A  B  C  1  .  .  .  .
X.8     9  B  C  A  1  .  .  .  .
X.9     9  C  A  B  1  .  .  .  .

A = E(7)^3+E(7)^4
B = E(7)^2+E(7)^5
C = E(7)+E(7)^6
D = -E(9)^4-E(9)^5
E = -E(9)^2-E(9)^7
F = E(9)^2+E(9)^4+E(9)^5+E(9)^7

magma: CharacterTable(G);