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Properties

Label 30T27
Degree $30$
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(30, 27);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $27$
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:  (1,17,7,4)(2,18,8,3)(5,6)(9,30,12,15)(10,29,11,16)(13,22,24,19)(14,21,23,20)(25,27,26,28), (1,11,13,28)(2,12,14,27)(3,29,5,23)(4,30,6,24)(7,25,15,21)(8,26,16,22)(9,19,10,20)(17,18)
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 3: None

Degree 5: $S_5$

Degree 6: None

Degree 10: $S_5$

Degree 15: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3,11)( 4,12)( 5, 7)( 6, 8)(13,25)(14,26)(15,23)(16,24)(17,19)(18,20)(27,30) (28,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $15$ $2$ $( 3,20)( 4,19)( 5, 7)( 6, 8)( 9,22)(10,21)(11,18)(12,17)(13,27)(14,28)(15,16) (23,24)(25,30)(26,29)$
$ 4, 4, 4, 4, 4, 4, 4, 2 $ $30$ $4$ $( 1, 2)( 3,25,20,30)( 4,26,19,29)( 5, 9, 7,22)( 6,10, 8,21)(11,27,18,13) (12,28,17,14)(15,23,16,24)$
$ 5, 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 3, 9, 6,14)( 2, 4,10, 5,13)( 7,15,22,17,26)( 8,16,21,18,25) (11,19,27,29,24)(12,20,28,30,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 3,28)( 2, 4,27)( 5,15,21)( 6,16,22)( 7,13,29)( 8,14,30)( 9,17,20) (10,18,19)(11,25,23)(12,26,24)$
$ 6, 6, 6, 6, 3, 3 $ $20$ $6$ $( 1, 4,24,21,16,12)( 2, 3,23,22,15,11)( 5,17,30,28,20, 8)( 6,18,29,27,19, 7) ( 9,13,25)(10,14,26)$

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

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

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);