Properties

Label 6T12
Order \(60\)
n \(6\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSL(2,5)$

Related objects

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Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $12$
Group :  $\PSL(2,5)$
CHM label :  $L(6) = PSL(2,5) = A_{5}(6)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(5,6), (1,2,3,4,6)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Low degree siblings

5T4, 10T7, 12T33, 15T5, 20T15, 30T9

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 $ $1$ $1$ $()$
$ 2, 2, 1, 1 $ $15$ $2$ $(3,6)(4,5)$
$ 5, 1 $ $12$ $5$ $(2,3,5,4,6)$
$ 5, 1 $ $12$ $5$ $(2,4,3,6,5)$
$ 3, 3 $ $20$ $3$ $(1,2,3)(4,5,6)$

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [60, 5]
Character table:   
     2  2  2  .  .  .
     3  1  .  .  .  1
     5  1  .  1  1  .

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

X.1     1  1  1  1  1
X.2     3 -1  A *A  .
X.3     3 -1 *A  A  .
X.4     4  . -1 -1  1
X.5     5  1  .  . -1

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