Show commands:
Magma
magma: G := TransitiveGroup(20, 656);
Group action invariants
| Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $656$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_5^2:C_2^2$ | ||
| 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: | (1,15,9,4,13,20,5,7,17,12)(2,16,10,3,14,19,6,8,18,11), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20), (1,11)(2,12)(3,13,16,9,8,17,19,5)(4,14,15,10,7,18,20,6) | 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 $8$: $D_{4}$ x 2, $C_2^3$ $16$: $D_4\times C_2$ $28800$: $S_5^2 \wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 5: None
Degree 10: $S_5^2 \wr C_2$
Low degree siblings
20T655 x 2, 20T656, 24T16043 x 2, 24T16044 x 2, 40T18814 x 2, 40T18815 x 2, 40T18817 x 2, 40T18820 x 2, 40T18822 x 2, 40T18830 x 2, 40T18839, 40T18840Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 70 conjugacy class representatives for $S_5^2:C_2^2$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $57600=2^{8} \cdot 3^{2} \cdot 5^{2}$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 57600.l | magma: IdentifyGroup(G);
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| Character table: | 70 x 70 character table |
magma: CharacterTable(G);