Show commands:
Magma
magma: G := TransitiveGroup(40, 77601);
Group action invariants
| Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $77601$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^6.C_2^8:C_{10}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,23,30,13,39,7,18,26,11,34,2,24,29,14,40,8,17,25,12,33)(3,22,31,16,37,6,19,28,9,35,4,21,32,15,38,5,20,27,10,36), (1,40,15,30,21,7,34,9,26,20,2,39,16,29,22,8,33,10,25,19)(3,37,13,31,23,5,35,11,27,17,4,38,14,32,24,6,36,12,28,18), (1,23,29,10,36,8,18,25,15,38,2,24,30,9,35,7,17,26,16,37)(3,22,31,11,34,6,20,27,14,40,4,21,32,12,33,5,19,28,13,39) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $5$: $C_5$ $10$: $C_{10}$ x 3 $80$: $C_2^4 : C_5$ x 17 $160$: $C_2 \times (C_2^4 : C_5)$ x 51 Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: None
Degree 4: None
Degree 5: $C_5$
Degree 8: None
Degree 10: $C_2^4 : C_5$, $C_2 \times (C_2^4 : C_5)$ x 2
Degree 20: 20T263
Low degree siblings
There are no siblings with degree $\leq 10$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 352 conjugacy class representatives for $C_2^6.C_2^8:C_{10}$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $163840=2^{15} \cdot 5$ | 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: | 163840.nud | magma: IdentifyGroup(G);
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| Character table: | 352 x 352 character table |
magma: CharacterTable(G);