Show commands:
Magma
magma: G := TransitiveGroup(24, 10);
Group invariants
Abstract group: | $S_4$ | magma: IdentifyGroup(G);
| |
Order: | $24=2^{3} \cdot 3$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | magma: NilpotencyClass(G);
|
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Parity: | $1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| |
$\card{\Aut(F/K)}$: | $24$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | $(1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,24)$, $(1,5,12,24)(2,6,11,23)(3,7,14,20)(4,8,13,19)(9,16,21,17)(10,15,22,18)$ | magma: Generators(G);
|
Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: $S_4$
Degree 8: $S_4$
Low degree siblings
4T5, 6T7, 6T8, 8T14, 12T8, 12T9Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Index | Representative |
1A | $1^{24}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{12}$ | $3$ | $2$ | $12$ | $( 1,15)( 2,16)( 3,19)( 4,20)( 5,10)( 6, 9)( 7,13)( 8,14)(11,17)(12,18)(21,23)(22,24)$ |
2B | $2^{12}$ | $6$ | $2$ | $12$ | $( 1,19)( 2,20)( 3,15)( 4,16)( 5, 6)( 7,17)( 8,18)( 9,10)(11,13)(12,14)(21,24)(22,23)$ |
3A | $3^{8}$ | $8$ | $3$ | $16$ | $( 1,20, 9)( 2,19,10)( 3,24,11)( 4,23,12)( 5,17, 8)( 6,18, 7)(13,21,15)(14,22,16)$ |
4A | $4^{6}$ | $6$ | $4$ | $18$ | $( 1,24,12, 5)( 2,23,11, 6)( 3,20,14, 7)( 4,19,13, 8)( 9,17,21,16)(10,18,22,15)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 3A | 4A | ||
Size | 1 | 3 | 6 | 8 | 6 | |
2 P | 1A | 1A | 1A | 3A | 2A | |
3 P | 1A | 2A | 2B | 1A | 4A | |
Type | ||||||
24.12.1a | R | |||||
24.12.1b | R | |||||
24.12.2a | R | |||||
24.12.3a | R | |||||
24.12.3b | R |
magma: CharacterTable(G);
Regular extensions
Data not computed