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Magma
magma: G := TransitiveGroup(24, 2);
Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $2$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2\times C_{12}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $24$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24), (1,24,22,19,18,16,13,11,9,7,5,4)(2,23,21,20,17,15,14,12,10,8,6,3) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_4$ x 2, $C_2^2$ $6$: $C_6$ x 3 $8$: $C_4\times C_2$ $12$: $C_{12}$ x 2, $C_6\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: $C_3$
Degree 6: $C_6$ x 3
Degree 8: $C_4\times C_2$
Degree 12: $C_{12}$ x 2, $C_6\times C_2$
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
| Cycle Type | Size | Order | Representative |
| $ 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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$ |
| $ 12, 12 $ | $1$ | $12$ | $( 1, 3, 5, 8, 9,12,13,15,18,20,22,23)( 2, 4, 6, 7,10,11,14,16,17,19,21,24)$ |
| $ 12, 12 $ | $1$ | $12$ | $( 1, 4, 5, 7, 9,11,13,16,18,19,22,24)( 2, 3, 6, 8,10,12,14,15,17,20,21,23)$ |
| $ 6, 6, 6, 6 $ | $1$ | $6$ | $( 1, 5, 9,13,18,22)( 2, 6,10,14,17,21)( 3, 8,12,15,20,23)( 4, 7,11,16,19,24)$ |
| $ 6, 6, 6, 6 $ | $1$ | $6$ | $( 1, 6, 9,14,18,21)( 2, 5,10,13,17,22)( 3, 7,12,16,20,24)( 4, 8,11,15,19,23)$ |
| $ 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 7,13,19)( 2, 8,14,20)( 3,10,15,21)( 4, 9,16,22)( 5,11,18,24)( 6,12,17,23)$ |
| $ 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 8,13,20)( 2, 7,14,19)( 3, 9,15,22)( 4,10,16,21)( 5,12,18,23)( 6,11,17,24)$ |
| $ 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 $ | $1$ | $6$ | $( 1,10,18, 2, 9,17)( 3,11,20, 4,12,19)( 5,14,22, 6,13,21)( 7,15,24, 8,16,23)$ |
| $ 12, 12 $ | $1$ | $12$ | $( 1,11,22, 7,18, 4,13,24, 9,19, 5,16)( 2,12,21, 8,17, 3,14,23,10,20, 6,15)$ |
| $ 12, 12 $ | $1$ | $12$ | $( 1,12,22, 8,18, 3,13,23, 9,20, 5,15)( 2,11,21, 7,17, 4,14,24,10,19, 6,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,13)( 2,14)( 3,15)( 4,16)( 5,18)( 6,17)( 7,19)( 8,20)( 9,22)(10,21)(11,24) (12,23)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,14)( 2,13)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,21)(10,22)(11,23) (12,24)$ |
| $ 12, 12 $ | $1$ | $12$ | $( 1,15, 5,20, 9,23,13, 3,18, 8,22,12)( 2,16, 6,19,10,24,14, 4,17, 7,21,11)$ |
| $ 12, 12 $ | $1$ | $12$ | $( 1,16, 5,19, 9,24,13, 4,18, 7,22,11)( 2,15, 6,20,10,23,14, 3,17, 8,21,12)$ |
| $ 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 $ | $1$ | $3$ | $( 1,18, 9)( 2,17,10)( 3,20,12)( 4,19,11)( 5,22,13)( 6,21,14)( 7,24,16) ( 8,23,15)$ |
| $ 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,19,13, 7)( 2,20,14, 8)( 3,21,15,10)( 4,22,16, 9)( 5,24,18,11)( 6,23,17,12)$ |
| $ 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,20,13, 8)( 2,19,14, 7)( 3,22,15, 9)( 4,21,16,10)( 5,23,18,12)( 6,24,17,11)$ |
| $ 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,21,18,14, 9, 6)( 2,22,17,13,10, 5)( 3,24,20,16,12, 7)( 4,23,19,15,11, 8)$ |
| $ 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,22,18,13, 9, 5)( 2,21,17,14,10, 6)( 3,23,20,15,12, 8)( 4,24,19,16,11, 7)$ |
| $ 12, 12 $ | $1$ | $12$ | $( 1,23,22,20,18,15,13,12, 9, 8, 5, 3)( 2,24,21,19,17,16,14,11,10, 7, 6, 4)$ |
| $ 12, 12 $ | $1$ | $12$ | $( 1,24,22,19,18,16,13,11, 9, 7, 5, 4)( 2,23,21,20,17,15,14,12,10, 8, 6, 3)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $24=2^{3} \cdot 3$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | yes | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | $1$ | ||
| Label: | 24.9 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);