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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 invariants

Abstract group:  $S_4$
magma: IdentifyGroup(G);
 
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
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $10$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(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

$\card{(G/N)}$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

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{24}$ $1$ $1$ $0$ $()$
2A $2^{12}$ $3$ $2$ $12$ $( 1,15)( 2,16)( 3,19)( 4,20)( 5,10)( 6, 9)( 7,13)( 8,14)(11,17)(12,18)(21,23)(22,24)$
2B $2^{12}$ $6$ $2$ $12$ $( 1,19)( 2,20)( 3,15)( 4,16)( 5, 6)( 7,17)( 8,18)( 9,10)(11,13)(12,14)(21,24)(22,23)$
3A $3^{8}$ $8$ $3$ $16$ $( 1,20, 9)( 2,19,10)( 3,24,11)( 4,23,12)( 5,17, 8)( 6,18, 7)(13,21,15)(14,22,16)$
4A $4^{6}$ $6$ $4$ $18$ $( 1,24,12, 5)( 2,23,11, 6)( 3,20,14, 7)( 4,19,13, 8)( 9,17,21,16)(10,18,22,15)$

Malle's constant $a(G)$:     $1/12$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 3A 4A
Size 1 3 6 8 6
2 P 1A 1A 1A 3A 2A
3 P 1A 2A 2B 1A 4A
Type
24.12.1a R 1 1 1 1 1
24.12.1b R 1 1 −1 1 −1
24.12.2a R 2 2 0 −1 0
24.12.3a R 3 −1 1 0 −1
24.12.3b R 3 −1 −1 0 1

magma: CharacterTable(G);
 

Regular extensions

Data not computed