Show commands:
Magma
magma: G := TransitiveGroup(40, 20);
Group action invariants
| Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_4\times C_{10}$ | ||
| 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: | (1,34,27,20,10,3,36,26,18,11,2,33,28,19,9,4,35,25,17,12)(5,39,32,24,15,7,38,29,21,13,6,40,31,23,16,8,37,30,22,14), (1,21)(2,22)(3,24)(4,23)(5,28)(6,27)(7,26)(8,25)(9,32)(10,31)(11,29)(12,30)(13,33)(14,34)(15,35)(16,36)(17,38)(18,37)(19,40)(20,39), (1,35,28,18,10)(2,36,27,17,9)(3,33,25,20,11,4,34,26,19,12)(5,37,31,21,15)(6,38,32,22,16)(7,40,30,24,13,8,39,29,23,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $5$: $C_5$ $8$: $D_{4}$ x 2, $C_2^3$ $10$: $C_{10}$ x 7 $16$: $D_4\times C_2$ $20$: 20T3 x 7 $40$: 20T12 x 2, 40T7 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 5: $C_5$
Degree 8: $D_4\times C_2$
Degree 10: $C_{10}$ x 3
Low degree siblings
40T20 x 3Siblings 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: | 80.46 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);