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Properties

Label 24T19
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{12}:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(24, 19);
 

Group action invariants

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

24T24 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, 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 12, 12 $ $2$ $12$ $( 1, 3,10,12, 5, 7, 2, 4, 9,11, 6, 8)(13,19,21,16,18,24,14,20,22,15,17,23)$
$ 12, 12 $ $2$ $12$ $( 1, 3,10,12, 5, 7, 2, 4, 9,11, 6, 8)(13,20,21,15,18,23,14,19,22,16,17,24)$
$ 12, 12 $ $2$ $12$ $( 1, 4,10,11, 5, 8, 2, 3, 9,12, 6, 7)(13,19,21,16,18,24,14,20,22,15,17,23)$
$ 12, 12 $ $2$ $12$ $( 1, 4,10,11, 5, 8, 2, 3, 9,12, 6, 7)(13,20,21,15,18,23,14,19,22,16,17,24)$
$ 6, 6, 3, 3, 3, 3 $ $2$ $6$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)(13,17,22,14,18,21)(15,20,24,16,19,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)(13,18,22)(14,17,21)(15,19,24) (16,20,23)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 6, 9, 2, 5,10)( 3, 8,11, 4, 7,12)(13,17,22,14,18,21)(15,20,24,16,19,23)$
$ 6, 6, 3, 3, 3, 3 $ $2$ $6$ $( 1, 6, 9, 2, 5,10)( 3, 8,11, 4, 7,12)(13,18,22)(14,17,21)(15,19,24)(16,20,23)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,11, 2,12)( 3, 6, 4, 5)( 7,10, 8, 9)(13,15,14,16)(17,20,18,19)(21,23,22,24)$
$ 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,11, 2,12)( 3, 6, 4, 5)( 7,10, 8, 9)(13,16,14,15)(17,19,18,20)(21,24,22,23)$
$ 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,12, 2,11)( 3, 5, 4, 6)( 7, 9, 8,10)(13,15,14,16)(17,20,18,19)(21,23,22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1,13)( 2,14)( 3,23)( 4,24)( 5,22)( 6,21)( 7,20)( 8,19)( 9,18)(10,17)(11,16) (12,15)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1,13, 2,14)( 3,23, 4,24)( 5,22, 6,21)( 7,20, 8,19)( 9,18,10,17)(11,16,12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1,15)( 2,16)( 3,22)( 4,21)( 5,24)( 6,23)( 7,18)( 8,17)( 9,19)(10,20)(11,13) (12,14)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1,15, 2,16)( 3,22, 4,21)( 5,24, 6,23)( 7,18, 8,17)( 9,19,10,20)(11,13,12,14)$

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:  48.37
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 3A 4A1 4A-1 4B 4C 4D 6A 6B1 6B-1 12A1 12A-1 12B1 12B5
Size 1 1 2 6 6 2 1 1 2 6 6 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 3A 2A 2A 2A 2A 2A 3A 3A 3A 6A 6A 6A 6A
3 P 1A 2A 2B 2C 2D 1A 4A-1 4A1 4B 4C 4D 2B 2B 2A 4A1 4B 4B 4A-1
Type
48.37.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.37.1b R 1 1 −1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1
48.37.1c R 1 1 −1 −1 1 1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1
48.37.1d R 1 1 −1 1 −1 1 1 1 −1 1 −1 1 −1 −1 1 1 −1 −1
48.37.1e R 1 1 −1 1 1 1 −1 −1 1 −1 −1 1 −1 −1 −1 −1 1 1
48.37.1f R 1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 1 1 1 1 1
48.37.1g R 1 1 1 −1 1 1 −1 −1 −1 1 −1 1 1 1 −1 −1 −1 −1
48.37.1h R 1 1 1 1 −1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1
48.37.2a R 2 2 2 0 0 −1 2 2 2 0 0 −1 −1 −1 −1 −1 −1 −1
48.37.2b R 2 2 −2 0 0 −1 −2 −2 2 0 0 −1 1 1 1 1 −1 −1
48.37.2c R 2 2 −2 0 0 −1 2 2 −2 0 0 −1 1 1 −1 −1 1 1
48.37.2d R 2 2 2 0 0 −1 −2 −2 −2 0 0 −1 −1 −1 1 1 1 1
48.37.2e1 C 2 −2 0 0 0 2 −2i 2i 0 0 0 −2 0 0 2i −2i 0 0
48.37.2e2 C 2 −2 0 0 0 2 2i −2i 0 0 0 −2 0 0 −2i 2i 0 0
48.37.2f1 C 2 −2 0 0 0 −1 −2ζ123 2ζ123 0 0 0 1 1−2ζ122 −1+2ζ122 −ζ123 ζ123 −ζ12−1−ζ12 ζ12−1+ζ12
48.37.2f2 C 2 −2 0 0 0 −1 2ζ123 −2ζ123 0 0 0 1 −1+2ζ122 1−2ζ122 ζ123 −ζ123 −ζ12−1−ζ12 ζ12−1+ζ12
48.37.2f3 C 2 −2 0 0 0 −1 −2ζ123 2ζ123 0 0 0 1 −1+2ζ122 1−2ζ122 −ζ123 ζ123 ζ12−1+ζ12 −ζ12−1−ζ12
48.37.2f4 C 2 −2 0 0 0 −1 2ζ123 −2ζ123 0 0 0 1 1−2ζ122 −1+2ζ122 ζ123 −ζ123 ζ12−1+ζ12 −ζ12−1−ζ12

magma: CharacterTable(G);