↑

Properties

Label 24T48
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times S_4$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(24, 48);
 

Group action invariants

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

Low degree resolvents

|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: $S_4$

Degree 6: $D_{6}$, $S_4$, $S_4\times C_2$

Degree 8: $S_4\times C_2$

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, 24T46, 24T47, 24T48

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

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
Label:  48.48
magma: IdentifyGroup(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 2B 2B 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);