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Properties

Label 9T27
Order \(504\)
n \(9\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSL(2,8)$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $27$
Group :  $\PSL(2,8)$
CHM label :  $L(9)=PSL(2,8)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,5)(3,6)(4,7)(8,9), (1,9)(2,3)(4,5)(6,7), (1,2,4,3,6,7,5)
$|\Aut(F/K)|$:  $1$

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 $ $56$ $9$ $(1,2,3,7,5,8,6,4,9)$

Group invariants

Order:  $504=2^{3} \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [504, 156]
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