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Properties

Label 24T1
Degree $24$
Order $24$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_{24}$

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Show commands: Magma

magma: G := TransitiveGroup(24, 1);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$8$:  $C_8$
$12$:  $C_{12}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $C_4$

Degree 6: $C_6$

Degree 8: $C_8$

Degree 12: $C_{12}$

Low degree siblings

There are no siblings 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, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 24 $ $1$ $24$ $( 1, 3, 5, 7,10,11,14,15,18,20,22,24, 2, 4, 6, 8, 9,12,13,16,17,19,21,23)$
$ 24 $ $1$ $24$ $( 1, 4, 5, 8,10,12,14,16,18,19,22,23, 2, 3, 6, 7, 9,11,13,15,17,20,21,24)$
$ 12, 12 $ $1$ $12$ $( 1, 5,10,14,18,22, 2, 6, 9,13,17,21)( 3, 7,11,15,20,24, 4, 8,12,16,19,23)$
$ 12, 12 $ $1$ $12$ $( 1, 6,10,13,18,21, 2, 5, 9,14,17,22)( 3, 8,11,16,20,23, 4, 7,12,15,19,24)$
$ 8, 8, 8 $ $1$ $8$ $( 1, 7,14,20, 2, 8,13,19)( 3,10,15,22, 4, 9,16,21)( 5,11,18,24, 6,12,17,23)$
$ 8, 8, 8 $ $1$ $8$ $( 1, 8,14,19, 2, 7,13,20)( 3, 9,15,21, 4,10,16,22)( 5,12,18,23, 6,11,17,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 9,18)( 2,10,17)( 3,12,20)( 4,11,19)( 5,13,22)( 6,14,21)( 7,16,24) ( 8,15,23)$
$ 6, 6, 6, 6 $ $1$ $6$ $( 1,10,18, 2, 9,17)( 3,11,20, 4,12,19)( 5,14,22, 6,13,21)( 7,15,24, 8,16,23)$
$ 24 $ $1$ $24$ $( 1,11,22, 8,17, 3,14,24, 9,19, 5,15, 2,12,21, 7,18, 4,13,23,10,20, 6,16)$
$ 24 $ $1$ $24$ $( 1,12,22, 7,17, 4,14,23, 9,20, 5,16, 2,11,21, 8,18, 3,13,24,10,19, 6,15)$
$ 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,13, 2,14)( 3,16, 4,15)( 5,17, 6,18)( 7,19, 8,20)( 9,22,10,21)(11,23,12,24)$
$ 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,14, 2,13)( 3,15, 4,16)( 5,18, 6,17)( 7,20, 8,19)( 9,21,10,22)(11,24,12,23)$
$ 24 $ $1$ $24$ $( 1,15, 6,19,10,24,13, 3,18, 8,21,11, 2,16, 5,20, 9,23,14, 4,17, 7,22,12)$
$ 24 $ $1$ $24$ $( 1,16, 6,20,10,23,13, 4,18, 7,21,12, 2,15, 5,19, 9,24,14, 3,17, 8,22,11)$
$ 6, 6, 6, 6 $ $1$ $6$ $( 1,17, 9, 2,18,10)( 3,19,12, 4,20,11)( 5,21,13, 6,22,14)( 7,23,16, 8,24,15)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,18, 9)( 2,17,10)( 3,20,12)( 4,19,11)( 5,22,13)( 6,21,14)( 7,24,16) ( 8,23,15)$
$ 8, 8, 8 $ $1$ $8$ $( 1,19,13, 8, 2,20,14, 7)( 3,21,16, 9, 4,22,15,10)( 5,23,17,12, 6,24,18,11)$
$ 8, 8, 8 $ $1$ $8$ $( 1,20,13, 7, 2,19,14, 8)( 3,22,16,10, 4,21,15, 9)( 5,24,17,11, 6,23,18,12)$
$ 12, 12 $ $1$ $12$ $( 1,21,17,13, 9, 6, 2,22,18,14,10, 5)( 3,23,19,16,12, 8, 4,24,20,15,11, 7)$
$ 12, 12 $ $1$ $12$ $( 1,22,17,14, 9, 5, 2,21,18,13,10, 6)( 3,24,19,15,12, 7, 4,23,20,16,11, 8)$
$ 24 $ $1$ $24$ $( 1,23,21,19,17,16,13,12, 9, 8, 6, 4, 2,24,22,20,18,15,14,11,10, 7, 5, 3)$
$ 24 $ $1$ $24$ $( 1,24,21,20,17,15,13,11, 9, 7, 6, 3, 2,23,22,19,18,16,14,12,10, 8, 5, 4)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $24=2^{3} \cdot 3$
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  24.2
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 8A1 8A-1 8A3 8A-3 12A1 12A-1 12A5 12A-5 24A1 24A-1 24A5 24A-5 24A7 24A-7 24A11 24A-11
Size 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 P 1A 1A 3A-1 3A1 2A 2A 3A1 3A-1 4A1 4A-1 4A-1 4A1 6A1 6A-1 6A-1 6A1 12A-5 12A1 12A5 12A-1 12A5 12A1 12A-5 12A-1
3 P 1A 2A 1A 1A 4A-1 4A1 2A 2A 8A3 8A-3 8A1 8A-1 4A1 4A-1 4A1 4A-1 8A-1 8A-3 8A1 8A-1 8A-3 8A1 8A3 8A3
Type
24.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
24.2.1b R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
24.2.1c1 C 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
24.2.1c2 C 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
24.2.1d1 C 1 1 1 1 −1 −1 1 1 −i i i −i −1 −1 −1 −1 i −i i −i −i i −i i
24.2.1d2 C 1 1 1 1 −1 −1 1 1 i −i −i i −1 −1 −1 −1 −i i −i i i −i i −i
24.2.1e1 C 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 −1 −1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
24.2.1e2 C 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 −1 −1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
24.2.1f1 C 1 −1 1 1 −ζ82 ζ82 −1 −1 ζ83 −ζ8 ζ8 −ζ83 ζ82 −ζ82 ζ82 −ζ82 ζ8 −ζ83 −ζ8 ζ83 −ζ83 ζ8 ζ83 −ζ8
24.2.1f2 C 1 −1 1 1 ζ82 −ζ82 −1 −1 −ζ8 ζ83 −ζ83 ζ8 −ζ82 ζ82 −ζ82 ζ82 −ζ83 ζ8 ζ83 −ζ8 ζ8 −ζ83 −ζ8 ζ83
24.2.1f3 C 1 −1 1 1 −ζ82 ζ82 −1 −1 −ζ83 ζ8 −ζ8 ζ83 ζ82 −ζ82 ζ82 −ζ82 −ζ8 ζ83 ζ8 −ζ83 ζ83 −ζ8 −ζ83 ζ8
24.2.1f4 C 1 −1 1 1 ζ82 −ζ82 −1 −1 ζ8 −ζ83 ζ83 −ζ8 −ζ82 ζ82 −ζ82 ζ82 ζ83 −ζ8 −ζ83 ζ8 −ζ8 ζ83 ζ8 −ζ83
24.2.1g1 C 1 1 −ζ122 ζ124 −1 −1 ζ124 −ζ122 −ζ123 ζ123 ζ123 −ζ123 ζ122 −ζ124 −ζ124 ζ122 −ζ12 ζ125 −ζ125 ζ12 ζ12 −ζ125 ζ125 −ζ12
24.2.1g2 C 1 1 ζ124 −ζ122 −1 −1 −ζ122 ζ124 ζ123 −ζ123 −ζ123 ζ123 −ζ124 ζ122 ζ122 −ζ124 ζ125 −ζ12 ζ12 −ζ125 −ζ125 ζ12 −ζ12 ζ125
24.2.1g3 C 1 1 −ζ122 ζ124 −1 −1 ζ124 −ζ122 ζ123 −ζ123 −ζ123 ζ123 ζ122 −ζ124 −ζ124 ζ122 ζ12 −ζ125 ζ125 −ζ12 −ζ12 ζ125 −ζ125 ζ12
24.2.1g4 C 1 1 ζ124 −ζ122 −1 −1 −ζ122 ζ124 −ζ123 ζ123 ζ123 −ζ123 −ζ124 ζ122 ζ122 −ζ124 −ζ125 ζ12 −ζ12 ζ125 ζ125 −ζ12 ζ12 −ζ125
24.2.1h1 C 1 −1 −ζ244 ζ248 −ζ246 ζ246 −ζ248 ζ244 ζ249 −ζ243 ζ243 −ζ249 −ζ2410 ζ242 −ζ242 ζ2410 ζ2411 −ζ24 ζ247 −ζ245 ζ245 −ζ247 ζ24 −ζ2411
24.2.1h2 C 1 −1 ζ248 −ζ244 ζ246 −ζ246 ζ244 −ζ248 −ζ243 ζ249 −ζ249 ζ243 ζ242 −ζ2410 ζ2410 −ζ242 −ζ24 ζ2411 −ζ245 ζ247 −ζ247 ζ245 −ζ2411 ζ24
24.2.1h3 C 1 −1 −ζ244 ζ248 −ζ246 ζ246 −ζ248 ζ244 −ζ249 ζ243 −ζ243 ζ249 −ζ2410 ζ242 −ζ242 ζ2410 −ζ2411 ζ24 −ζ247 ζ245 −ζ245 ζ247 −ζ24 ζ2411
24.2.1h4 C 1 −1 ζ248 −ζ244 ζ246 −ζ246 ζ244 −ζ248 ζ243 −ζ249 ζ249 −ζ243 ζ242 −ζ2410 ζ2410 −ζ242 ζ24 −ζ2411 ζ245 −ζ247 ζ247 −ζ245 ζ2411 −ζ24
24.2.1h5 C 1 −1 −ζ244 ζ248 ζ246 −ζ246 −ζ248 ζ244 −ζ243 ζ249 −ζ249 ζ243 ζ2410 −ζ242 ζ242 −ζ2410 ζ245 −ζ247 ζ24 −ζ2411 ζ2411 −ζ24 ζ247 −ζ245
24.2.1h6 C 1 −1 ζ248 −ζ244 −ζ246 ζ246 ζ244 −ζ248 ζ249 −ζ243 ζ243 −ζ249 −ζ242 ζ2410 −ζ2410 ζ242 −ζ247 ζ245 −ζ2411 ζ24 −ζ24 ζ2411 −ζ245 ζ247
24.2.1h7 C 1 −1 −ζ244 ζ248 ζ246 −ζ246 −ζ248 ζ244 ζ243 −ζ249 ζ249 −ζ243 ζ2410 −ζ242 ζ242 −ζ2410 −ζ245 ζ247 −ζ24 ζ2411 −ζ2411 ζ24 −ζ247 ζ245
24.2.1h8 C 1 −1 ζ248 −ζ244 −ζ246 ζ246 ζ244 −ζ248 −ζ249 ζ243 −ζ243 ζ249 −ζ242 ζ2410 −ζ2410 ζ242 ζ247 −ζ245 ζ2411 −ζ24 ζ24 −ζ2411 ζ245 −ζ247

magma: CharacterTable(G);