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Properties

Label 30T25
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, 25);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $25$
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);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14,17,12)(2,13,18,11)(3,24,8,10)(4,23,7,9)(5,25)(6,26)(15,30,19,21)(16,29,20,22)(27,28), (1,9,13,3,5)(2,10,14,4,6)(7,22,25,15,18)(8,21,26,16,17)(11,28,23,20,30)(12,27,24,19,29)
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: $C_2$

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, 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, 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 $ $15$ $2$ $( 3,13)( 4,14)( 5, 9)( 6,10)( 7,22)( 8,21)(11,30)(12,29)(17,26)(18,25)(19,27) (20,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3,12)( 4,11)( 5, 7)( 6, 8)( 9,10)(13,25)(14,26)(15,23)(16,24)(17,19) (18,20)(21,22)(27,30)(28,29)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ $30$ $4$ $( 1, 2)( 3,25,20,29)( 4,26,19,30)( 5,10, 8,22)( 6, 9, 7,21)(11,27,17,14) (12,28,18,13)(15,23)(16,24)$
$ 5, 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 3, 9, 5,13)( 2, 4,10, 6,14)( 7,15,22,18,25)( 8,16,21,17,26) (11,20,28,30,23)(12,19,27,29,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 3,28)( 2, 4,27)( 5,16,21)( 6,15,22)( 7,14,29)( 8,13,30)( 9,17,20) (10,18,19)(11,26,23)(12,25,24)$
$ 6, 6, 6, 6, 6 $ $20$ $6$ $( 1, 4,23,22,16,12)( 2, 3,24,21,15,11)( 5,18,30,27,20, 7)( 6,17,29,28,19, 8) ( 9,14,26,10,13,25)$

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

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