Show commands:
Magma
magma: G := TransitiveGroup(20, 781);
Group action invariants
| Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $781$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_5^2:D_4$ | ||
| 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: | (1,13,2,14)(3,11,4,12)(5,19,6,20)(7,17,10,16)(8,18,9,15), (1,18,3,12,7,20,5,14)(2,17,4,11,8,19,6,13)(9,16)(10,15), (1,16,7,12)(2,15,8,11)(3,20)(4,19)(5,18,9,14)(6,17,10,13) | 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 6, $C_2^3$ $16$: $D_4\times C_2$ x 3 $32$: $C_2^2 \wr C_2$ $28800$: $S_5^2 \wr C_2$ $57600$: 20T655 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 5: None
Degree 10: $S_5^2 \wr C_2$
Low degree siblings
20T781 x 7, 24T17918 x 8, 40T45487 x 4, 40T45488 x 8, 40T45489 x 8, 40T45497 x 4, 40T45508 x 4, 40T45542 x 8, 40T45559 x 4, 40T45560 x 4Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 119 conjugacy class representatives for $S_5^2:D_4$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $115200=2^{9} \cdot 3^{2} \cdot 5^{2}$ | 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: | 115200.d | magma: IdentifyGroup(G);
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| Character table: | 119 x 119 character table |
magma: CharacterTable(G);