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Properties

Label 10T26
Order \(360\)
n \(10\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSL(2,9)$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $26$
Group :  $\PSL(2,9)$
CHM label :  $L(10)=PSL(2,9)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(4,7)(5,8)(9,10), (1,2,10)(3,4,5)(6,7,8), (1,3,2,6)(4,5,8,7)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: None

Low degree siblings

6T15 x 2, 15T20 x 2, 20T89, 30T88 x 2, 36T555, 40T304, 45T49

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1 $ $45$ $2$ $( 3, 6)( 4, 5)( 7, 8)( 9,10)$
$ 4, 4, 1, 1 $ $90$ $4$ $( 3, 9, 6,10)( 4, 8, 5, 7)$
$ 3, 3, 3, 1 $ $40$ $3$ $( 2, 3, 6)( 4, 9, 7)( 5, 8,10)$
$ 3, 3, 3, 1 $ $40$ $3$ $( 2, 4, 5)( 3, 9, 8)( 6, 7,10)$
$ 5, 5 $ $72$ $5$ $( 1, 2, 3, 4, 9)( 5, 7,10, 6, 8)$
$ 5, 5 $ $72$ $5$ $( 1, 2, 3, 8,10)( 4, 7, 5, 9, 6)$

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [360, 118]
Character table:    
     2  3  3  2  .  .  .  .
     3  2  .  .  2  2  .  .
     5  1  .  .  .  .  1  1

       1a 2a 4a 3a 3b 5a 5b
    2P 1a 1a 2a 3a 3b 5b 5a
    3P 1a 2a 4a 1a 1a 5b 5a
    5P 1a 2a 4a 3a 3b 1a 1a

X.1     1  1  1  1  1  1  1
X.2     5  1 -1  2 -1  .  .
X.3     5  1 -1 -1  2  .  .
X.4     8  .  . -1 -1  A *A
X.5     8  .  . -1 -1 *A  A
X.6     9  1  1  .  . -1 -1
X.7    10 -2  .  1  1  .  .

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5