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Properties

Label 24T40
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times D_8$

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Show commands: Magma

magma: G := TransitiveGroup(24, 40);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times D_8$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14)(2,13)(3,4)(5,17)(6,18)(9,22)(10,21)(11,12)(19,20), (1,11,9,19,17,3)(2,12,10,20,18,4)(5,15,14,24,22,7)(6,16,13,23,21,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$8$:  $D_{4}$
$12$:  $C_6\times C_2$
$16$:  $D_{8}$
$24$:  $D_4 \times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $D_{4}$

Degree 6: $C_6$

Degree 8: $D_{8}$

Degree 12: $D_4 \times C_3$

Low degree siblings

24T40

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

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:  $3$
Label:  48.25
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);