Show commands:
Magma
magma: G := TransitiveGroup(36, 330);
Group action invariants
| Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $330$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_2\times C_6\times S_4$ | ||
| Parity: | $1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $12$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,5,35,3,7,33)(2,6,36,4,8,34)(9,15,19,12,14,18)(10,16,20,11,13,17)(21,28,29,23,26,32)(22,27,30,24,25,31), (1,28,15,4,25,13)(2,27,16,3,26,14)(5,30,19,7,31,18)(6,29,20,8,32,17)(9,35,24,12,33,22)(10,36,23,11,34,21), (1,11,26,35,15,21,2,12,25,36,16,22)(3,10,28,33,14,23,4,9,27,34,13,24)(5,19,31)(6,20,32)(7,18,30)(8,17,29) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $3$: $C_3$ $4$: $C_2^2$ x 7 $6$: $S_3$, $C_6$ x 7 $8$: $C_2^3$ $12$: $D_{6}$ x 3, $C_6\times C_2$ x 7 $18$: $S_3\times C_3$ $24$: $S_4$, $S_3 \times C_2^2$, 24T3 $36$: $C_6\times S_3$ x 3 $48$: $S_4\times C_2$ x 3 $72$: 12T45, 24T68 $96$: 12T48 $144$: 18T61 x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 6: $C_6$, $D_{6}$, $S_4\times C_2$ x 2
Degree 9: $S_3\times C_3$
Degree 12: 12T48
Degree 18: $S_3 \times C_6$, 18T61 x 2
Low degree siblings
36T330 x 11Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 60 conjugacy class representatives for $C_2\times C_6\times S_4$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $288=2^{5} \cdot 3^{2}$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 288.1033 | magma: IdentifyGroup(G);
| |
| Character table: | 60 x 60 character table |
magma: CharacterTable(G);