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Magma
magma: G := TransitiveGroup(10, 3);
Group action invariants
Degree $n$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $3$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_{10}$ | ||
CHM label: | $D_{10}(10)=[D(5)]2$ | ||
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: | (1,2,3,4,5,6,7,8,9,10), (1,8)(2,7)(3,6)(4,5)(9,10) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $10$: $D_{5}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 5: $D_{5}$
Low degree siblings
10T3, 20T4Siblings 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 $ | $5$ | $2$ | $( 2,10)( 3, 9)( 4, 8)( 5, 7)$ | |
$ 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,10)( 4, 9)( 5, 8)( 6, 7)$ | |
$ 10 $ | $2$ | $10$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$ | |
$ 5, 5 $ | $2$ | $5$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ | |
$ 10 $ | $2$ | $10$ | $( 1, 4, 7,10, 3, 6, 9, 2, 5, 8)$ | |
$ 5, 5 $ | $2$ | $5$ | $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$ | |
$ 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ |
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: | 20.4 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 5A1 | 5A2 | 10A1 | 10A3 | ||
Size | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 5A2 | 5A1 | 5A1 | 5A2 | |
5 P | 1A | 2A | 2B | 2C | 1A | 1A | 2A | 2A | |
Type | |||||||||
20.4.1a | R | ||||||||
20.4.1b | R | ||||||||
20.4.1c | R | ||||||||
20.4.1d | R | ||||||||
20.4.2a1 | R | ||||||||
20.4.2a2 | R | ||||||||
20.4.2b1 | R | ||||||||
20.4.2b2 | R |
magma: CharacterTable(G);