Properties

Label 10T7
Order \(60\)
n \(10\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $A_{5}$

Related objects

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Group action invariants

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

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: None

Low degree siblings

5T4, 6T12, 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, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 1 $ $20$ $3$ $( 2, 4, 5)( 3, 6, 9)( 7, 8,10)$
$ 2, 2, 2, 2, 1, 1 $ $15$ $2$ $( 2, 7)( 4,10)( 5, 8)( 6, 9)$
$ 5, 5 $ $12$ $5$ $( 1, 2, 3, 6, 8)( 4, 9, 7, 5,10)$
$ 5, 5 $ $12$ $5$ $( 1, 2, 9, 6, 7)( 3, 8, 4,10, 5)$

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

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