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Properties

Label 24T12155
Degree $24$
Order $15552$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\wr A_4$

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Show commands: Magma

magma: G := TransitiveGroup(24, 12155);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12155$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\wr A_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,24,20,18,7,3,9,16,12,2,23,19,17,8,4,10,15,11)(5,22)(6,21)(13,14), (1,22,12,18,13,3,9,6,20,2,21,11,17,14,4,10,5,19)(7,16)(8,15)(23,24)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$
$96$:  $C_2^4:C_6$
$192$:  $C_2\wr A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $A_4$

Degree 6: None

Degree 8: $A_4\times C_2$

Degree 12: 12T280

Low degree siblings

12T280, 24T12153, 24T12154, 24T12162, 24T12194, 36T10167, 36T10207

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $32$ $3$ $( 1, 9,17)( 2,10,18)( 3,19,11)( 4,20,12)( 7,23,15)( 8,24,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $24$ $3$ $( 3,19,11)( 4,20,12)( 5,13,21)( 6,14,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,13,21)( 6,14,22)( 7,23,15) ( 8,24,16)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 7,15,23)( 8,16,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $81$ $2$ $( 9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $108$ $2$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,15)( 6,16)( 7,21)( 8,22)( 9,12)(10,11)(13,23) (14,24)$
$ 6, 6, 6, 6 $ $432$ $6$ $( 1,12,17,20, 9, 4)( 2,11,18,19,10, 3)( 5, 7,21,23,13,15)( 6, 8,22,24,14,16)$
$ 6, 6, 2, 2, 2, 2, 2, 2 $ $432$ $6$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,23,13, 7,21,15)( 6,24,14, 8,22,16)( 9,12) (10,11)$
$ 4, 4, 4, 4, 2, 2, 2, 2 $ $972$ $4$ $( 1, 5,17,13)( 2, 6,18,14)( 3,16,19,24)( 4,15,20,23)( 7,12)( 8,11)( 9,21) (10,22)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $54$ $2$ $(11,19)(12,20)(13,21)(14,22)$
$ 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ $216$ $6$ $( 1, 9,17)( 2,10,18)( 3,19)( 4,20)( 7,23,15)( 8,24,16)(13,21)(14,22)$
$ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $216$ $6$ $( 7,15,23)( 8,16,24)(11,19)(12,20)(13,21)(14,22)$
$ 9, 9, 1, 1, 1, 1, 1, 1 $ $288$ $9$ $( 3, 6,16,19,22, 8,11,14,24)( 4, 5,15,20,21, 7,12,13,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $288$ $3$ $( 1, 9,17)( 2,10,18)( 3, 6, 8)( 4, 5, 7)(11,14,16)(12,13,15)(19,22,24) (20,21,23)$
$ 9, 9, 3, 3 $ $576$ $9$ $( 1,17, 9)( 2,18,10)( 3, 6,24,11,14, 8,19,22,16)( 4, 5,23,12,13, 7,20,21,15)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $144$ $3$ $( 3, 6,24)( 4, 5,23)( 7,12,13)( 8,11,14)(15,20,21)(16,19,22)$
$ 6, 6, 3, 3, 2, 2, 1, 1 $ $1296$ $6$ $( 3, 6,24)( 4, 5,23)( 7,20,13,15,12,21)( 8,19,14,16,11,22)( 9,17)(10,18)$
$ 9, 9, 1, 1, 1, 1, 1, 1 $ $288$ $9$ $( 3,16,22,11,24, 6,19, 8,14)( 4,15,21,12,23, 5,20, 7,13)$
$ 9, 9, 3, 3 $ $576$ $9$ $( 1, 9,17)( 2,10,18)( 3, 8,14,19,24, 6,11,16,22)( 4, 7,13,20,23, 5,12,15,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $288$ $3$ $( 1,17, 9)( 2,18,10)( 3,24, 6)( 4,23, 5)( 7,13,12)( 8,14,11)(15,21,20) (16,22,19)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $144$ $3$ $( 3, 8,14)( 4, 7,13)( 5,20,23)( 6,19,24)(11,16,22)(12,15,21)$
$ 6, 6, 3, 3, 2, 2, 1, 1 $ $1296$ $6$ $( 3,24, 6,11,16,14)( 4,23, 5,12,15,13)( 7,21,20)( 8,22,19)( 9,17)(10,18)$
$ 18, 2, 2, 2 $ $864$ $18$ $( 1,24,20,18, 7, 3, 9,16,12, 2,23,19,17, 8, 4,10,15,11)( 5,22)( 6,21)(13,14)$
$ 6, 6, 6, 2, 2, 2 $ $432$ $6$ $( 1,24,12, 2,23,11)( 3, 9,16, 4,10,15)( 5, 6)( 7,19,17, 8,20,18)(13,22)(14,21)$
$ 6, 6, 6, 6 $ $864$ $6$ $( 1,16,20,10,23,11)( 2,15,19, 9,24,12)( 3,17, 8, 4,18, 7)( 5,14,21, 6,13,22)$
$ 6, 6, 6, 2, 2, 2 $ $432$ $6$ $( 1,16, 4,18, 7,11)( 2,15, 3,17, 8,12)( 5, 6)( 9,24,20,10,23,19)(13,14)(21,22)$
$ 6, 6, 4, 4, 2, 2 $ $1296$ $12$ $( 1, 3, 9,19,17,11)( 2, 4,10,20,18,12)( 5, 8)( 6, 7)(13,24,21,16)(14,23,22,15)$
$ 4, 4, 2, 2, 2, 2, 2, 2, 2, 2 $ $648$ $4$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,24,21, 8)( 6,23,22, 7)( 9,11)(10,12)(13,16) (14,15)$
$ 6, 6, 6, 2, 2, 2 $ $96$ $6$ $( 1,18, 9, 2,17,10)( 3,12,19, 4,11,20)( 5,22)( 6,21)( 7,16,23, 8,15,24)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 4)( 5,22)( 6,21)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20) (23,24)$
$ 6, 6, 2, 2, 2, 2, 2, 2 $ $144$ $6$ $( 1,18, 9, 2,17,10)( 3, 4)( 5, 6)( 7,16,23, 8,15,24)(11,12)(13,22)(14,21) (19,20)$
$ 6, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $72$ $6$ $( 1, 2)( 3,20,11, 4,19,12)( 5, 6)( 7, 8)( 9,10)(13,22)(14,21)(15,16)(17,18) (23,24)$
$ 6, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $216$ $6$ $( 1,10)( 2, 9)( 3,20)( 4,19)( 5,14,21, 6,13,22)( 7,24)( 8,23)(11,12)(15,16) (17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $108$ $2$ $( 1,10)( 2, 9)( 3, 4)( 5, 6)( 7,24)( 8,23)(11,20)(12,19)(13,14)(15,16)(17,18) (21,22)$
$ 6, 6, 6, 6 $ $864$ $6$ $( 1, 6,12, 2, 5,11)( 3, 9,22,20,18,13)( 4,10,21,19,17,14)( 7,16,23, 8,15,24)$
$ 6, 6, 6, 2, 2, 2 $ $432$ $6$ $( 1, 6, 4,18,13,19)( 2, 5, 3,17,14,20)( 7, 8)( 9,22,12,10,21,11)(15,16)(23,24)$
$ 18, 2, 2, 2 $ $864$ $18$ $( 1, 6,20,10,13, 3,17,22,12, 2, 5,19, 9,14, 4,18,21,11)( 7,24)( 8,23)(15,16)$
$ 6, 6, 6, 2, 2, 2 $ $432$ $6$ $( 1,22,12, 2,21,11)( 3,17,14, 4,18,13)( 5,19, 9, 6,20,10)( 7,24)( 8,23)(15,16)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $15552=2^{6} \cdot 3^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  15552.s
magma: IdentifyGroup(G);
 
Character table:    39 x 39 character table

magma: CharacterTable(G);