Properties

Label 10T13
Degree $10$
Order $120$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_5$

Related objects

Learn more

Group action invariants

Degree $n$:  $10$
Transitive number $t$:  $13$
Group:  $S_5$
CHM label:  $S_{5}(10d)$
Parity:  $-1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,3,5,7,9)(2,4,6,8,10), (1,2)(3,7)(8,9)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: None

Low degree siblings

5T5, 6T14, 10T12, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62

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, 1, 1, 1, 1 $ $10$ $2$ $( 3, 9)( 4, 5)( 7, 8)$
$ 3, 3, 3, 1 $ $20$ $3$ $( 2, 4, 5)( 3, 6, 9)( 7, 8,10)$
$ 2, 2, 2, 2, 1, 1 $ $15$ $2$ $( 2, 7)( 4,10)( 5, 8)( 6, 9)$
$ 6, 3, 1 $ $20$ $6$ $( 2, 7, 5,10, 4, 8)( 3, 6, 9)$
$ 5, 5 $ $24$ $5$ $( 1, 2, 3, 6, 8)( 4, 9, 7, 5,10)$
$ 4, 4, 2 $ $30$ $4$ $( 1, 2, 3, 7)( 4,10)( 5, 9, 6, 8)$

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  [120, 34]
Character table:   
     2  3  2  1  3  1  .  2
     3  1  1  1  .  1  .  .
     5  1  .  .  .  .  1  .

       1a 2a 3a 2b 6a 5a 4a
    2P 1a 1a 3a 1a 3a 5a 2b
    3P 1a 2a 1a 2b 2a 5a 4a
    5P 1a 2a 3a 2b 6a 1a 4a

X.1     1  1  1  1  1  1  1
X.2     1 -1  1  1 -1  1 -1
X.3     4 -2  1  .  1 -1  .
X.4     4  2  1  . -1 -1  .
X.5     5  1 -1  1  1  . -1
X.6     5 -1 -1  1 -1  .  1
X.7     6  .  . -2  .  1  .