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Properties

Label 20T30
Degree $20$
Order $120$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 30);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $30$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_5$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3)(2,4)(5,13)(6,14)(15,17)(16,18)(19,20), (1,6,10,13,17)(2,5,9,14,18)(3,8,12,15,20)(4,7,11,16,19)
magma: Generators(G);
 

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: $S_5$

Degree 10: $S_5$

Low degree siblings

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

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

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

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

magma: CharacterTable(G);