Properties

Label 5T4
Degree $5$
Order $60$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_5$

Related objects

Learn more about

This is the smallest instance of a non-solvable Galois group, implying that the general quintic cannot be solvable in radicals and motivating the creation of Galois theory.

Group action invariants

Degree $n$:  $5$
Transitive number $t$:  $4$
Group:  $A_5$
CHM label:  $A5$
Parity:  $1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,2,3), (3,4,5)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

6T12, 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 $ $1$ $1$ $()$
$ 3, 1, 1 $ $20$ $3$ $(3,4,5)$
$ 2, 2, 1 $ $15$ $2$ $(2,3)(4,5)$
$ 5 $ $12$ $5$ $(1,2,3,4,5)$
$ 5 $ $12$ $5$ $(1,2,3,5,4)$

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