Show commands:
Magma
magma: G := TransitiveGroup(45, 11);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $11$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_5\times \He_3$ | ||
| Parity: | $1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $15$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,34,23,10,45,33,21,9,41,30,17,5,39,25,14)(2,36,22,12,44,32,20,7,40,29,18,4,37,27,13)(3,35,24,11,43,31,19,8,42,28,16,6,38,26,15), (1,45,40,39,34,32,30,25,22,21,17,13,10,9,4)(2,44,42,37,36,31,29,27,24,20,18,15,12,7,6)(3,43,41,38,35,33,28,26,23,19,16,14,11,8,5) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $5$: $C_5$ $9$: $C_3^2$ $15$: $C_{15}$ x 4 $27$: $C_3^2:C_3$ $45$: 45T2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: $C_5$
Degree 9: $C_3^2:C_3$
Degree 15: $C_{15}$
Low degree siblings
45T11 x 3Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 55 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $135=3^{3} \cdot 5$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | $2$ | ||
| Label: | 135.3 | magma: IdentifyGroup(G);
|
| Character table: not available. |
magma: CharacterTable(G);