Show commands:
Magma
magma: G := TransitiveGroup(40, 15);
Group action invariants
| Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_5\times \OD_{16}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
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| Nilpotency class: | $2$ | magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $20$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,21,3,23,2,22,4,24)(5,26,7,28,6,25,8,27)(9,32,11,30,10,31,12,29)(13,35,16,34,14,36,15,33)(17,37,19,40,18,38,20,39), (1,14,25,37,10,24,33,6,18,30,3,15,27,40,12,21,35,8,20,31,2,13,26,38,9,23,34,5,17,29,4,16,28,39,11,22,36,7,19,32) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $5$: $C_5$ $8$: $C_4\times C_2$ $10$: $C_{10}$ x 3 $16$: $C_8:C_2$ $20$: 20T1 x 2, 20T3 $40$: 40T2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: $C_5$
Degree 8: $C_8:C_2$
Degree 10: $C_{10}$
Degree 20: 20T1
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
There are 50 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $80=2^{4} \cdot 5$ | 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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| Label: | 80.24 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);