Show commands:
Magma
magma: G := TransitiveGroup(20, 412);
Group action invariants
| Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $412$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^9.D_{10}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,7)(2,8)(3,6)(4,5)(9,14,11,15)(10,13,12,16)(17,18)(19,20), (1,6,4,8)(2,5,3,7)(9,14,11,15)(10,13,12,16)(17,18)(19,20), (5,13,7,16)(6,14,8,15)(9,17,11,20)(10,18,12,19) | 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 $8$: $C_2^3$ $10$: $D_{5}$ $20$: $D_{10}$ x 3 $40$: 20T8 $160$: $(C_2^4 : C_5) : C_2$ x 5 $320$: $C_2\times (C_2^4 : D_5)$ x 15 $640$: 20T141 x 5 $2560$: 20T240 $5120$: 20T307 x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 5: $D_{5}$
Degree 10: $C_2\times (C_2^4 : D_5)$ x 3
Low degree siblings
20T412 x 719, 40T5903 x 54, 40T7903 x 1080, 40T8176 x 540, 40T8199 x 1080, 40T9339 x 1080, 40T9420 x 1080, 40T9423 x 2160, 40T9457 x 270, 40T9459 x 540, 40T9473 x 180, 40T9619 x 540, 40T9644 x 1080, 40T9972 x 2160, 40T10151 x 1440, 40T10152 x 1440, 40T10205 x 2160, 40T10361 x 36, 40T10365 x 216, 40T10368 x 540, 40T10373 x 360, 40T10376 x 1080Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 208 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $10240=2^{11} \cdot 5$ | 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: | not nilpotent | ||
| Label: | 10240.i | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);