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Properties

Label 24T39
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2:C_{12}$

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Show commands: Magma

magma: G := TransitiveGroup(24, 39);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $39$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2:C_{12}$
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:  (1,13)(2,14)(5,18)(6,17)(9,21)(10,22), (1,16,5,19,9,23,14,3,17,8,22,11)(2,15,6,20,10,24,13,4,18,7,21,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $C_6$ x 3
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$12$:  $C_{12}$ x 2, $C_6\times C_2$
$16$:  $C_2^2:C_4$
$24$:  $D_4 \times C_3$ x 2, 24T2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 6: $C_6$

Degree 8: $C_2^2:C_4$

Degree 12: $C_{12}$, $D_4 \times C_3$ x 2

Low degree siblings

24T39

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

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:  48.21
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);