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Properties

Label 24T18
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_4:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(24, 18);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $18$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4:S_3$
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:  (1,6,10,2,5,9)(3,7,12,4,8,11)(13,17,22)(14,18,21)(15,19,24)(16,20,23), (1,11,2,12)(3,10,4,9)(5,7,6,8)(13,20,14,19)(15,17,16,18)(21,24,22,23), (1,14,2,13)(3,23,4,24)(5,21,6,22)(7,19,8,20)(9,17,10,18)(11,15,12,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$16$:  $Q_8:C_2$
$24$:  $S_3 \times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2

Degree 8: $Q_8:C_2$

Degree 12: $D_6$

Low degree siblings

24T18, 24T23

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

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:  48.39
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 3A 4A 4B1 4B-1 4C 4D 6A 6B 6C 12A
Size 1 1 2 2 6 2 2 3 3 6 6 2 4 4 4
2 P 1A 1A 1A 1A 1A 3A 2A 2A 2A 2A 2A 3A 3A 3A 6A
3 P 1A 2A 2B 2C 2D 1A 4A 4B-1 4B1 4C 4D 2A 2B 2C 4A
Type
48.39.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.39.1b R 1 1 −1 −1 −1 1 1 −1 −1 1 1 1 −1 −1 1
48.39.1c R 1 1 −1 −1 1 1 1 1 1 −1 −1 1 −1 −1 1
48.39.1d R 1 1 −1 1 −1 1 −1 1 1 −1 1 1 −1 1 −1
48.39.1e R 1 1 −1 1 1 1 −1 −1 −1 1 −1 1 −1 1 −1
48.39.1f R 1 1 1 −1 −1 1 −1 1 1 1 −1 1 1 −1 −1
48.39.1g R 1 1 1 −1 1 1 −1 −1 −1 −1 1 1 1 −1 −1
48.39.1h R 1 1 1 1 −1 1 1 −1 −1 −1 −1 1 1 1 1
48.39.2a R 2 2 2 2 0 −1 2 0 0 0 0 −1 −1 −1 −1
48.39.2b R 2 2 −2 −2 0 −1 2 0 0 0 0 −1 1 1 −1
48.39.2c R 2 2 −2 2 0 −1 −2 0 0 0 0 −1 1 −1 1
48.39.2d R 2 2 2 −2 0 −1 −2 0 0 0 0 −1 −1 1 1
48.39.2e1 C 2 −2 0 0 0 2 0 −2i 2i 0 0 −2 0 0 0
48.39.2e2 C 2 −2 0 0 0 2 0 2i −2i 0 0 −2 0 0 0
48.39.4a S 4 −4 0 0 0 −2 0 0 0 0 0 2 0 0 0

magma: CharacterTable(G);