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Magma
magma: G := TransitiveGroup(20, 4);
Group action invariants
Degree $n$: | $20$ | 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: | $D_{10}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $20$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,15)(2,16)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,19)(18,20), (1,4,6,8,10,12,14,16,18,19)(2,3,5,7,9,11,13,15,17,20) | 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: $C_2$ x 3
Degree 4: $C_2^2$
Degree 5: $D_{5}$
Low degree siblings
10T3 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,19)( 4,20)( 5,18)( 6,17)( 7,16)( 8,15)( 9,14)(10,13)(11,12)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 3)( 2, 4)( 5,19)( 6,20)( 7,18)( 8,17)( 9,16)(10,15)(11,14)(12,13)$ |
$ 10, 10 $ | $2$ | $10$ | $( 1, 4, 6, 8,10,12,14,16,18,19)( 2, 3, 5, 7, 9,11,13,15,17,20)$ |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 6,10,14,18)( 2, 5, 9,13,17)( 3, 7,11,15,20)( 4, 8,12,16,19)$ |
$ 10, 10 $ | $2$ | $10$ | $( 1, 8,14,19, 6,12,18, 4,10,16)( 2, 7,13,20, 5,11,17, 3, 9,15)$ |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,10,18, 6,14)( 2, 9,17, 5,13)( 3,11,20, 7,15)( 4,12,19, 8,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,12)( 2,11)( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,20)(10,19)$ |
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: |
2 2 2 2 1 1 1 1 2 5 1 . . 1 1 1 1 1 1a 2a 2b 10a 5a 10b 5b 2c 2P 1a 1a 1a 5a 5b 5b 5a 1a 3P 1a 2a 2b 10b 5b 10a 5a 2c 5P 1a 2a 2b 2c 1a 2c 1a 2c 7P 1a 2a 2b 10b 5b 10a 5a 2c X.1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 1 X.3 1 -1 1 -1 1 -1 1 -1 X.4 1 1 -1 -1 1 -1 1 -1 X.5 2 . . A *A *A A 2 X.6 2 . . *A A A *A 2 X.7 2 . . -A *A -*A A -2 X.8 2 . . -*A A -A *A -2 A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 |
magma: CharacterTable(G);