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Properties

Label 24T26
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times Q_8$

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Show commands: Magma

magma: G := TransitiveGroup(24, 26);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $26$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times Q_8$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (3,11)(4,12)(5,22)(6,21)(9,17)(10,18)(15,24)(16,23), (1,14,2,13)(3,24,4,23)(5,10,6,9)(7,20,8,19)(11,15,12,16)(17,22,18,21), (1,24,10,8,17,15,2,23,9,7,18,16)(3,13,12,22,19,6,4,14,11,21,20,5)
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
$6$:  $S_3$
$8$:  $C_2^3$, $Q_8$ x 2
$12$:  $D_{6}$ x 3
$16$:  $Q_8\times C_2$
$24$:  $S_3 \times C_2^2$

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: $Q_8$

Degree 12: $S_3 \times C_2^2$

Low degree siblings

24T26

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, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3,11)( 4,12)( 5,22)( 6,21)( 9,17)(10,18)(15,24)(16,23)$
$ 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 $ $3$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,21)( 6,22)( 7, 8)( 9,18)(10,17)(13,14)(15,23)(16,24) (19,20)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1, 3, 2, 4)( 5,23, 6,24)( 7,22, 8,21)( 9,19,10,20)(11,18,12,17)(13,15,14,16)$
$ 12, 12 $ $4$ $12$ $( 1, 3,18,20, 9,11, 2, 4,17,19,10,12)( 5, 8,21,24,14,16, 6, 7,22,23,13,15)$
$ 12, 12 $ $4$ $12$ $( 1, 5,10,13,17,22, 2, 6, 9,14,18,21)( 3, 7,12,16,19,24, 4, 8,11,15,20,23)$
$ 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 $ $6$ $4$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,24,10,23)(11,21,12,22)(13,20,14,19)(15,18,16,17)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 7, 2, 8)( 3,21, 4,22)( 5,11, 6,12)( 9,15,10,16)(13,20,14,19)(17,24,18,23)$
$ 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 $ $2$ $6$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,13, 2,14)( 3,16, 4,15)( 5,17, 6,18)( 7,19, 8,20)( 9,21,10,22)(11,23,12,24)$
$ 12, 12 $ $4$ $12$ $( 1,15,18, 8, 9,24, 2,16,17, 7,10,23)( 3, 6,20,22,11,13, 4, 5,19,21,12,14)$
$ 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:  48.40
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A 4B 4C 4D 4E 4F 6A 12A 12B 12C
Size 1 1 3 3 2 2 2 2 6 6 6 2 4 4 4
2 P 1A 1A 1A 1A 3A 2A 2A 2A 2A 2A 2A 3A 6A 6A 6A
3 P 1A 2A 2B 2C 1A 4A 4B 4C 4D 4E 4F 2A 4C 4A 4B
Type
48.40.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.40.1b R 1 1 −1 −1 1 −1 −1 1 1 −1 1 1 −1 −1 1
48.40.1c R 1 1 −1 −1 1 −1 1 −1 −1 1 1 1 −1 1 −1
48.40.1d R 1 1 −1 −1 1 1 −1 −1 1 1 −1 1 1 −1 −1
48.40.1e R 1 1 −1 −1 1 1 1 1 −1 −1 −1 1 1 1 1
48.40.1f R 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1
48.40.1g R 1 1 1 1 1 −1 1 −1 1 −1 −1 1 −1 1 −1
48.40.1h R 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1
48.40.2a R 2 2 0 0 −1 2 2 2 0 0 0 −1 −1 −1 −1
48.40.2b R 2 2 0 0 −1 −2 −2 2 0 0 0 −1 1 1 −1
48.40.2c R 2 2 0 0 −1 −2 2 −2 0 0 0 −1 1 −1 1
48.40.2d R 2 2 0 0 −1 2 −2 −2 0 0 0 −1 −1 1 1
48.40.2e S 2 −2 −2 2 2 0 0 0 0 0 0 −2 0 0 0
48.40.2f S 2 −2 2 −2 2 0 0 0 0 0 0 −2 0 0 0
48.40.4a S 4 −4 0 0 −2 0 0 0 0 0 0 2 0 0 0

magma: CharacterTable(G);