Show commands:
Magma
magma: G := TransitiveGroup(16, 1869);
Group action invariants
| Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $1869$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_2^6.S_4\wr C_2$ | ||
| 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: | (1,15,5,11,4,10,8,14)(2,16,6,12,3,9,7,13), (1,13,5,9,3,16,2,14,6,10,4,15)(7,11,8,12) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ $72$: $C_3^2:D_4$ $1152$: $S_4\wr C_2$ $18432$: 16T1792 $36864$: 32T1515322 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 8: $S_4\wr C_2$
Low degree siblings
16T1869 x 3, 32T1831791 x 2, 32T1831792 x 2, 32T1831793 x 2, 32T1831932 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 77 conjugacy class representatives for $C_2^6.S_4\wr C_2$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $73728=2^{13} \cdot 3^{2}$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 73728.i | magma: IdentifyGroup(G);
| |
| Character table: | 77 x 77 character table |
magma: CharacterTable(G);