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Properties

Label 10T12
Order \(120\)
n \(10\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_5$

Related objects

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $S_5$

Low degree siblings

5T5, 6T14, 10T13, 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 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1, 1 $ $20$ $3$ $( 3, 5, 9)( 4, 8,10)$
$ 2, 2, 2, 2, 1, 1 $ $15$ $2$ $( 2, 4)( 3, 5)( 7, 9)( 8,10)$
$ 6, 2, 2 $ $20$ $6$ $( 1, 2)( 3, 4, 5, 8, 9,10)( 6, 7)$
$ 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3, 8)( 4, 9)( 5,10)( 6, 7)$
$ 4, 4, 2 $ $30$ $4$ $( 1, 2, 3, 4)( 5,10)( 6, 7, 8, 9)$
$ 5, 5 $ $24$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$

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  3  1  2  2  .
     3  1  1  .  1  1  .  .
     5  1  .  .  .  .  .  1

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

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