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Properties

Label 24T10
Degree $24$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_4$

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Show commands: Magma

magma: G := TransitiveGroup(24, 10);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $10$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $24$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,24), (1,5,12,24)(2,6,11,23)(3,7,14,20)(4,8,13,19)(9,16,21,17)(10,15,22,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $S_4$

Degree 6: $S_3$, $S_4$, $S_4$

Degree 8: $S_4$

Degree 12: $S_4$, $S_4$

Low degree siblings

4T5, 6T7, 6T8, 8T14, 12T8, 12T9

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $24=2^{3} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  24.12
magma: IdentifyGroup(G);
 
Character table:    
     2  3  2  .  2  3
     3  1  .  1  .  .

       1a 2a 3a 4a 2b
    2P 1a 1a 3a 2b 1a
    3P 1a 2a 1a 4a 2b

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

magma: CharacterTable(G);