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Properties

Label 24T34
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{24}$

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Show commands: Magma

magma: G := TransitiveGroup(24, 34);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $34$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{24}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8)(2,7)(3,6)(4,5)(9,23)(10,24)(11,21)(12,22)(13,19)(14,20)(15,18)(16,17), (1,6)(2,5)(7,23)(8,24)(9,21)(10,22)(11,19)(12,20)(13,17)(14,18)(15,16)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Degree 8: $D_{8}$

Degree 12: $D_{12}$

Low degree siblings

24T34

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

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:  48.7
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A 6A 8A1 8A3 12A1 12A5 24A1 24A5 24A7 24A11
Size 1 1 12 12 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A 2A 3A 4A 4A 6A 6A 12A1 12A5 12A1 12A5
3 P 1A 2A 2B 2C 1A 4A 2A 8A3 8A1 4A 4A 8A1 8A1 8A3 8A3
Type
48.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.7.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1
48.7.1c R 1 1 −1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1
48.7.1d R 1 1 1 −1 1 1 1 −1 −1 1 1 −1 −1 −1 −1
48.7.2a R 2 2 0 0 −1 2 −1 2 2 −1 −1 −1 −1 −1 −1
48.7.2b R 2 2 0 0 2 −2 2 0 0 −2 −2 0 0 0 0
48.7.2c R 2 2 0 0 −1 2 −1 −2 −2 −1 −1 1 1 1 1
48.7.2d1 R 2 −2 0 0 2 0 −2 −ζ8−1−ζ8 ζ8−1+ζ8 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8
48.7.2d2 R 2 −2 0 0 2 0 −2 ζ8−1+ζ8 −ζ8−1−ζ8 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8
48.7.2e1 R 2 2 0 0 −1 −2 −1 0 0 1 1 −ζ12−1−ζ12 ζ12−1+ζ12 ζ12−1+ζ12 −ζ12−1−ζ12
48.7.2e2 R 2 2 0 0 −1 −2 −1 0 0 1 1 ζ12−1+ζ12 −ζ12−1−ζ12 −ζ12−1−ζ12 ζ12−1+ζ12
48.7.2f1 R 2 −2 0 0 −1 0 1 −ζ24−3−ζ243 ζ24−3+ζ243 −ζ24−2−ζ242 ζ24−2+ζ242 ζ24−5+ζ245 ζ24−1+ζ24 −ζ24−1−ζ24 −ζ24−5−ζ245
48.7.2f2 R 2 −2 0 0 −1 0 1 −ζ24−3−ζ243 ζ24−3+ζ243 ζ24−2+ζ242 −ζ24−2−ζ242 −ζ24−1−ζ24 −ζ24−5−ζ245 ζ24−5+ζ245 ζ24−1+ζ24
48.7.2f3 R 2 −2 0 0 −1 0 1 ζ24−3+ζ243 −ζ24−3−ζ243 −ζ24−2−ζ242 ζ24−2+ζ242 −ζ24−5−ζ245 −ζ24−1−ζ24 ζ24−1+ζ24 ζ24−5+ζ245
48.7.2f4 R 2 −2 0 0 −1 0 1 ζ24−3+ζ243 −ζ24−3−ζ243 ζ24−2+ζ242 −ζ24−2−ζ242 ζ24−1+ζ24 ζ24−5+ζ245 −ζ24−5−ζ245 −ζ24−1−ζ24

magma: CharacterTable(G);