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Magma
magma: G := TransitiveGroup(20, 47);
Group action invariants
Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $47$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5^2:Q_8$ | ||
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: | (1,9,14,6)(2,10,13,5)(3,20,12,15)(4,19,11,16)(7,8)(17,18), (1,16,10,3)(2,15,9,4)(5,19)(6,20)(7,17,12,14)(8,18,11,13) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $Q_8$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 5: None
Degree 10: $C_5^2 : Q_8$
Low degree siblings
10T20 x 3, 20T47 x 2, 25T17, 40T166 x 3Siblings 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, 1, 1, 1, 1 $ | $25$ | $2$ | $( 5,17)( 6,18)( 7,19)( 8,20)( 9,13)(10,14)(11,15)(12,16)$ |
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $5$ | $( 3, 7,12,16,19)( 4, 8,11,15,20)$ |
$ 4, 4, 4, 4, 2, 2 $ | $50$ | $4$ | $( 1, 2)( 3, 4)( 5, 9,17,13)( 6,10,18,14)( 7,15,19,11)( 8,16,20,12)$ |
$ 4, 4, 4, 4, 2, 2 $ | $50$ | $4$ | $( 1, 3)( 2, 4)( 5, 7,17,19)( 6, 8,18,20)( 9,11,13,15)(10,12,14,16)$ |
$ 4, 4, 4, 4, 2, 2 $ | $50$ | $4$ | $( 1, 4)( 2, 3)( 5,11,17,15)( 6,12,18,16)( 7, 9,19,13)( 8,10,20,14)$ |
$ 5, 5, 5, 5 $ | $8$ | $5$ | $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3, 7,12,16,19)( 4, 8,11,15,20)$ |
$ 5, 5, 5, 5 $ | $8$ | $5$ | $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3,12,19, 7,16)( 4,11,20, 8,15)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $200=2^{3} \cdot 5^{2}$ | 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: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 200.44 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 . 2 2 2 . . 5 2 . 2 . . . 2 2 1a 2a 5a 4a 4b 4c 5b 5c 2P 1a 1a 5a 2a 2a 2a 5b 5c 3P 1a 2a 5a 4a 4b 4c 5b 5c 5P 1a 2a 1a 4a 4b 4c 1a 1a 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 -2 2 . . . 2 2 X.6 8 . 3 . . . -2 -2 X.7 8 . -2 . . . -2 3 X.8 8 . -2 . . . 3 -2 |
magma: CharacterTable(G);