Show commands:
Magma
magma: G := TransitiveGroup(45, 12);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_9:C_{15}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $15$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,8,13,20,26,31,38,45,6,10,18,23,30,35,40,2,9,15,19,25,33,39,44,5,12,16,22,29,34,42,3,7,14,21,27,32,37,43,4,11,17,24,28,36,41), (1,20,38,10,30,2,19,39,12,29,3,21,37,11,28)(4,23,42,13,33)(5,24,40,14,31)(6,22,41,15,32)(7,25,45,17,34,9,26,43,16,35,8,27,44,18,36) | magma: Generators(G);
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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 $27$: $C_9:C_3$ $45$: 45T2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: $C_5$
Degree 9: $C_9:C_3$
Degree 15: $C_{15}$
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
The 55 conjugacy class representatives for $C_9:C_{15}$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $135=3^{3} \cdot 5$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | $2$ | ||
| Label: | 135.4 | magma: IdentifyGroup(G);
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| Character table: | 55 x 55 character table |
magma: CharacterTable(G);