Properties

Label 20T35
Order \(120\)
n \(20\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_5$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $35$
Group :  $S_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3)(2,4)(5,13)(6,14)(7,8)(9,10)(11,12)(15,17)(16,18), (1,6,10,14,18)(2,5,9,13,17)(3,8,11,16,19)(4,7,12,15,20)
$|\Aut(F/K)|$:  $2$

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 4: None

Degree 5: None

Degree 10: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 1, 1 $ $20$ $3$ $( 3, 7,10)( 4, 8, 9)( 5,11,17)( 6,12,18)(13,16,20)(14,15,19)$
$ 6, 6, 6, 1, 1 $ $20$ $6$ $( 3,14,10,19, 7,15)( 4,13, 9,20, 8,16)( 5,12,17, 6,11,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $10$ $2$ $( 3,19)( 4,20)( 5, 6)( 7,14)( 8,13)( 9,16)(10,15)(11,12)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,20)( 8,19)( 9,15)(10,16)(11,18)(12,17)$
$ 5, 5, 5, 5 $ $24$ $5$ $( 1, 3, 6,12,15)( 2, 4, 5,11,16)( 7,17,14, 9,20)( 8,18,13,10,19)$
$ 4, 4, 4, 4, 4 $ $30$ $4$ $( 1, 3, 8,10)( 2, 4, 7, 9)( 5,12,16,20)( 6,11,15,19)(13,18,14,17)$

Group invariants

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

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

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