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Properties

Label 24T17
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_4:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(24, 17);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $17$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4:C_6$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $12$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7,5,12,9,3,2,8,6,11,10,4)(13,19,17,24,22,16,14,20,18,23,21,15), (1,24,5,16,9,20,2,23,6,15,10,19)(3,17,8,22,11,14,4,18,7,21,12,13), (1,3,10,12,6,7,2,4,9,11,5,8)(13,15,21,23,18,20,14,16,22,24,17,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$3$:  $C_3$
$4$:  $C_2^2$ x 7
$6$:  $C_6$ x 7
$8$:  $C_2^3$
$12$:  $C_6\times C_2$ x 7
$16$:  $Q_8:C_2$
$24$:  24T3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$

Degree 4: $C_2^2$

Degree 6: $C_6$ x 3

Degree 8: $Q_8:C_2$

Degree 12: $C_6\times C_2$

Low degree siblings

24T17 x 2

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

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:  $2$
Label:  48.47
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);