Show commands:
Magma
magma: G := TransitiveGroup(10, 4);
Group action invariants
| Degree $n$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $4$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $F_5$ | ||
| CHM label: | $1/2[F(5)]2$ | ||
| 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: | (1,3,5,7,9)(2,4,6,8,10), (1,2,9,8)(3,6,7,4)(5,10) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 5: $F_5$
Low degree siblings
5T3, 20T5Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 2, 2, 1, 1 $ | $5$ | $2$ | $( 2,10)( 3, 9)( 4, 8)( 5, 7)$ | |
| $ 4, 4, 2 $ | $5$ | $4$ | $( 1, 2, 5, 4)( 3, 8)( 6, 7,10, 9)$ | |
| $ 4, 4, 2 $ | $5$ | $4$ | $( 1, 2, 9, 8)( 3, 6, 7, 4)( 5,10)$ | |
| $ 5, 5 $ | $4$ | $5$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $20=2^{2} \cdot 5$ | 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: | 20.3 | magma: IdentifyGroup(G);
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| Character table: |
| 1A | 2A | 4A1 | 4A-1 | 5A | ||
| Size | 1 | 5 | 5 | 5 | 4 | |
| 2 P | 1A | 1A | 2A | 2A | 5A | |
| 5 P | 1A | 2A | 4A1 | 4A-1 | 1A | |
| Type | ||||||
| 20.3.1a | R | |||||
| 20.3.1b | R | |||||
| 20.3.1c1 | C | |||||
| 20.3.1c2 | C | |||||
| 20.3.4a | R |
magma: CharacterTable(G);