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Properties

Label 24T46
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times S_4$

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Show commands: Magma

magma: G := TransitiveGroup(24, 46);
 

Group invariants

Abstract group:  $C_2\times S_4$
magma: IdentifyGroup(G);
 
Order:  $48=2^{4} \cdot 3$
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 $t$:  $46$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(5,17)(6,18)(7,8)(9,10)(11,23)(12,24)(19,20)(21,22)$, $(1,3)(2,4)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,15)(14,16)$, $(1,9,18)(2,10,17)(3,11,20)(4,12,19)(5,14,22)(6,13,21)(7,16,24)(8,15,23)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$, $S_4$, $S_4$, $S_4\times C_2$ x 2

Degree 8: None

Degree 12: $S_4$, $C_2\times S_4$, $C_2 \times S_4$

Low degree siblings

6T11 x 2, 8T24 x 2, 12T21, 12T22, 12T23 x 2, 12T24 x 2, 16T61, 24T47, 24T48 x 2

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

Malle's constant $a(G)$:     $1/8$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 3A 4A 4B 6A
Size 1 1 3 3 6 6 8 6 6 8
2 P 1A 1A 1A 1A 1A 1A 3A 2C 2C 3A
3 P 1A 2A 2B 2C 2D 2E 1A 4A 4B 2A
Type
48.48.1a R 1 1 1 1 1 1 1 1 1 1
48.48.1b R 1 −1 −1 1 −1 1 1 1 −1 −1
48.48.1c R 1 −1 −1 1 1 −1 1 −1 1 −1
48.48.1d R 1 1 1 1 −1 −1 1 −1 −1 1
48.48.2a R 2 2 2 2 0 0 −1 0 0 −1
48.48.2b R 2 −2 −2 2 0 0 −1 0 0 1
48.48.3a R 3 3 −1 −1 1 1 0 −1 −1 0
48.48.3b R 3 3 −1 −1 −1 −1 0 1 1 0
48.48.3c R 3 −3 1 −1 −1 1 0 −1 1 0
48.48.3d R 3 −3 1 −1 1 −1 0 1 −1 0

magma: CharacterTable(G);
 

Regular extensions

Data not computed