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Properties

Label 24T23
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, 23);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $23$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4:S_3$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,21,2,22)(3,8,4,7)(5,17,6,18)(9,13,10,14)(11,23,12,24)(15,19,16,20), (1,20,2,19)(3,9,4,10)(5,24,6,23)(7,13,8,14)(11,17,12,18)(15,21,16,22), (3,11)(4,12)(5,21)(6,22)(7,8)(9,17)(10,18)(13,14)(15,23)(16,24)
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: $D_{6}$ x 3

Degree 8: $Q_8:C_2$

Degree 12: $S_3 \times C_2^2$

Low degree siblings

24T18 x 2

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

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);