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Properties

Label 5T5
Degree $5$
Order $120$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_5$

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Show commands: Magma

magma: G := TransitiveGroup(5, 5);
 

Group action invariants

Degree $n$:  $5$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_5$
CHM label:  $S5$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2), (1,2,3,4,5)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

6T14, 10T12, 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 $ $1$ $1$ $()$
$ 2, 1, 1, 1 $ $10$ $2$ $(4,5)$
$ 3, 1, 1 $ $20$ $3$ $(3,4,5)$
$ 2, 2, 1 $ $15$ $2$ $(2,3)(4,5)$
$ 4, 1 $ $30$ $4$ $(2,3,4,5)$
$ 3, 2 $ $20$ $6$ $(1,2)(3,4,5)$
$ 5 $ $24$ $5$ $(1,2,3,4,5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Label:  120.34
magma: IdentifyGroup(G);
 
Character table:    
     2  3  2  1  3  2  1  .
     3  1  1  1  .  .  1  .
     5  1  .  .  .  .  .  1

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

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

magma: CharacterTable(G);