↑

Properties

Label 15T10
Order \(120\)
n \(15\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_5$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $10$
Group :  $S_5$
CHM label :  $S_{5}(15)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8), (1,4,10)(2,5,8)(3,7,11)(6,9,15)(12,14,13), (1,4)(2,6)(3,7)(5,15)(8,9)(12,13)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 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, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $15$ $2$ $( 2, 5)( 3,13)( 4,10)( 7,14)( 9,15)(11,12)$
$ 4, 4, 4, 2, 1 $ $30$ $4$ $( 2, 7, 5,14)( 3,10,13, 4)( 6, 8)( 9,11,15,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $10$ $2$ $( 2, 9)( 3,13)( 5,15)( 6, 8)( 7,12)(11,14)$
$ 5, 5, 5 $ $24$ $5$ $( 1, 2, 7,14, 5)( 3,13,15, 8, 9)( 4,11,12,10, 6)$
$ 6, 6, 3 $ $20$ $6$ $( 1, 2, 8, 4, 6, 9)( 3,13,15, 7,12, 5)(10,11,14)$
$ 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 2,12)( 3,11, 7)( 4,13, 6)( 5,10,15)( 8, 9,14)$

Group invariants

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

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

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  .  .