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Properties

Label 24T24
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, 24);
 

Group action invariants

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

24T19, 24T24

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 $ $6$ $2$ $( 3,12)( 4,11)( 5,22)( 6,21)( 7, 8)( 9,17)(10,18)(15,23)(16,24)(19,20)$
$ 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 $ $6$ $2$ $( 1, 3)( 2, 4)( 5,23)( 6,24)( 7,21)( 8,22)( 9,19)(10,20)(11,17)(12,18)(13,15) (14,16)$
$ 12, 12 $ $2$ $12$ $( 1, 3,18,20, 9,11, 2, 4,17,19,10,12)( 5, 8,21,24,14,16, 6, 7,22,23,13,15)$
$ 12, 12 $ $2$ $12$ $( 1, 4,18,19, 9,12, 2, 3,17,20,10,11)( 5, 7,21,23,14,15, 6, 8,22,24,13,16)$
$ 12, 12 $ $2$ $12$ $( 1, 5,10,13,17,22, 2, 6, 9,14,18,21)( 3, 8,12,15,19,23, 4, 7,11,16,20,24)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1, 5, 2, 6)( 3,15, 4,16)( 7,12, 8,11)( 9,22,10,21)(13,17,14,18)(19,24,20,23)$
$ 12, 12 $ $2$ $12$ $( 1, 6,10,14,17,21, 2, 5, 9,13,18,22)( 3, 7,12,16,19,24, 4, 8,11,15,20,23)$
$ 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 $ $2$ $2$ $( 1, 7)( 2, 8)( 3,22)( 4,21)( 5,11)( 6,12)( 9,15)(10,16)(13,20)(14,19)(17,24) (18,23)$
$ 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 $ $1$ $4$ $( 1,13, 2,14)( 3,15, 4,16)( 5,17, 6,18)( 7,20, 8,19)( 9,21,10,22)(11,24,12,23)$
$ 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,14, 2,13)( 3,16, 4,15)( 5,18, 6,17)( 7,19, 8,20)( 9,22,10,21)(11,23,12,24)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1,15,17, 7, 9,24)( 2,16,18, 8,10,23)( 3, 5,19,22,11,14)( 4, 6,20,21,12,13)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1,16,17, 8, 9,23)( 2,15,18, 7,10,24)( 3, 6,19,21,11,13)( 4, 5,20,22,12,14)$
$ 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:  48.37
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);