Show commands:
Magma
magma: G := TransitiveGroup(40, 16);
Group action invariants
| Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^2:C_{20}$ | ||
| 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: | (1,29,19,6,35,24,12,38,28,13,4,32,18,7,33,21,9,40,26,16)(2,30,20,5,36,23,11,37,27,14,3,31,17,8,34,22,10,39,25,15), (1,20,35,11,28,3,18,34,9,25)(2,19,36,12,27,4,17,33,10,26)(5,23,37,14,31,8,22,39,15,30)(6,24,38,13,32,7,21,40,16,29) | 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, $C_2^2$ $5$: $C_5$ $8$: $D_{4}$ x 2, $C_4\times C_2$ $10$: $C_{10}$ x 3 $16$: $C_2^2:C_4$ $20$: 20T1 x 2, 20T3 $40$: 20T12 x 2, 40T2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 5: $C_5$
Degree 8: $C_2^2:C_4$
Degree 10: $C_{10}$
Low degree siblings
40T16Siblings are shown 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: | 80.21 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);