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Magma
magma: G := TransitiveGroup(20, 24);
Group action invariants
Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5\times D_{10}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $10$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,20,5,3,10,7,13,12,17,15)(2,19,6,4,9,8,14,11,18,16), (1,14,5,18,10,2,13,6,17,9)(3,19,15,11,7,4,20,16,12,8) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $5$: $C_5$ $10$: $D_{5}$, $C_{10}$ x 3 $20$: $D_{10}$, 20T3 $50$: $D_5\times C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 5: None
Degree 10: $D_5\times C_5$
Low degree siblings
20T24Siblings 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 $ | $1$ | $1$ | $()$ |
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $5$ | $( 3, 7,12,15,20)( 4, 8,11,16,19)$ |
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $5$ | $( 3,12,20, 7,15)( 4,11,19, 8,16)$ |
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $5$ | $( 3,15, 7,20,12)( 4,16, 8,19,11)$ |
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $5$ | $( 3,20,15,12, 7)( 4,19,16,11, 8)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
$ 10, 2, 2, 2, 2, 2 $ | $2$ | $10$ | $( 1, 2)( 3, 8,12,16,20, 4, 7,11,15,19)( 5, 6)( 9,10)(13,14)(17,18)$ |
$ 10, 2, 2, 2, 2, 2 $ | $2$ | $10$ | $( 1, 2)( 3,11,20, 8,15, 4,12,19, 7,16)( 5, 6)( 9,10)(13,14)(17,18)$ |
$ 10, 2, 2, 2, 2, 2 $ | $2$ | $10$ | $( 1, 2)( 3,16, 7,19,12, 4,15, 8,20,11)( 5, 6)( 9,10)(13,14)(17,18)$ |
$ 10, 2, 2, 2, 2, 2 $ | $2$ | $10$ | $( 1, 2)( 3,19,15,11, 7, 4,20,16,12, 8)( 5, 6)( 9,10)(13,14)(17,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)$ |
$ 10, 10 $ | $5$ | $10$ | $( 1, 3, 5, 7,10,12,13,15,17,20)( 2, 4, 6, 8, 9,11,14,16,18,19)$ |
$ 10, 10 $ | $5$ | $10$ | $( 1, 3,10,12,17,20, 5, 7,13,15)( 2, 4, 9,11,18,19, 6, 8,14,16)$ |
$ 10, 10 $ | $5$ | $10$ | $( 1, 3,13,15, 5, 7,17,20,10,12)( 2, 4,14,16, 6, 8,18,19, 9,11)$ |
$ 10, 10 $ | $5$ | $10$ | $( 1, 3,17,20,13,15,10,12, 5, 7)( 2, 4,18,19,14,16, 9,11, 6, 8)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)$ |
$ 10, 10 $ | $5$ | $10$ | $( 1, 4, 5, 8,10,11,13,16,17,19)( 2, 3, 6, 7, 9,12,14,15,18,20)$ |
$ 10, 10 $ | $5$ | $10$ | $( 1, 4,10,11,17,19, 5, 8,13,16)( 2, 3, 9,12,18,20, 6, 7,14,15)$ |
$ 10, 10 $ | $5$ | $10$ | $( 1, 4,13,16, 5, 8,17,19,10,11)( 2, 3,14,15, 6, 7,18,20, 9,12)$ |
$ 10, 10 $ | $5$ | $10$ | $( 1, 4,17,19,13,16,10,11, 5, 8)( 2, 3,18,20,14,15, 9,12, 6, 7)$ |
$ 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 5,10,13,17)( 2, 6, 9,14,18)( 3, 7,12,15,20)( 4, 8,11,16,19)$ |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 5,10,13,17)( 2, 6, 9,14,18)( 3,12,20, 7,15)( 4,11,19, 8,16)$ |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 5,10,13,17)( 2, 6, 9,14,18)( 3,15, 7,20,12)( 4,16, 8,19,11)$ |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 5,10,13,17)( 2, 6, 9,14,18)( 3,20,15,12, 7)( 4,19,16,11, 8)$ |
$ 10, 10 $ | $1$ | $10$ | $( 1, 6,10,14,17, 2, 5, 9,13,18)( 3, 8,12,16,20, 4, 7,11,15,19)$ |
$ 10, 10 $ | $2$ | $10$ | $( 1, 6,10,14,17, 2, 5, 9,13,18)( 3,11,20, 8,15, 4,12,19, 7,16)$ |
$ 10, 10 $ | $2$ | $10$ | $( 1, 6,10,14,17, 2, 5, 9,13,18)( 3,16, 7,19,12, 4,15, 8,20,11)$ |
$ 10, 10 $ | $2$ | $10$ | $( 1, 6,10,14,17, 2, 5, 9,13,18)( 3,19,15,11, 7, 4,20,16,12, 8)$ |
$ 10, 10 $ | $1$ | $10$ | $( 1, 9,17, 6,13, 2,10,18, 5,14)( 3,11,20, 8,15, 4,12,19, 7,16)$ |
$ 10, 10 $ | $2$ | $10$ | $( 1, 9,17, 6,13, 2,10,18, 5,14)( 3,16, 7,19,12, 4,15, 8,20,11)$ |
$ 10, 10 $ | $2$ | $10$ | $( 1, 9,17, 6,13, 2,10,18, 5,14)( 3,19,15,11, 7, 4,20,16,12, 8)$ |
$ 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,10,17, 5,13)( 2, 9,18, 6,14)( 3,12,20, 7,15)( 4,11,19, 8,16)$ |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,10,17, 5,13)( 2, 9,18, 6,14)( 3,15, 7,20,12)( 4,16, 8,19,11)$ |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,10,17, 5,13)( 2, 9,18, 6,14)( 3,20,15,12, 7)( 4,19,16,11, 8)$ |
$ 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,13, 5,17,10)( 2,14, 6,18, 9)( 3,15, 7,20,12)( 4,16, 8,19,11)$ |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,13, 5,17,10)( 2,14, 6,18, 9)( 3,20,15,12, 7)( 4,19,16,11, 8)$ |
$ 10, 10 $ | $1$ | $10$ | $( 1,14, 5,18,10, 2,13, 6,17, 9)( 3,16, 7,19,12, 4,15, 8,20,11)$ |
$ 10, 10 $ | $2$ | $10$ | $( 1,14, 5,18,10, 2,13, 6,17, 9)( 3,19,15,11, 7, 4,20,16,12, 8)$ |
$ 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,17,13,10, 5)( 2,18,14, 9, 6)( 3,20,15,12, 7)( 4,19,16,11, 8)$ |
$ 10, 10 $ | $1$ | $10$ | $( 1,18,13, 9, 5, 2,17,14,10, 6)( 3,19,15,11, 7, 4,20,16,12, 8)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $100=2^{2} \cdot 5^{2}$ | 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: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 100.14 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);