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Magma
magma: G := TransitiveGroup(20, 34);
Group action invariants
Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5\times S_4$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $5$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20), (1,9,17,5,13)(2,10,18,6,14)(3,12,19,8,15,4,11,20,7,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $6$: $S_3$ $10$: $C_{10}$ $24$: $S_4$ $30$: $S_3 \times C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $S_4$
Degree 5: $C_5$
Degree 10: None
Low degree siblings
30T33, 30T34, 40T64Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 3, 4)( 7, 8)(11,12)(15,16)(19,20)$ |
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
$ 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10,11,12)(13,14,15,16)(17,18,19,20)$ |
$ 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3, 7,11,15,19)( 4, 8,12,16,20)$ |
$ 10, 5, 5 $ | $6$ | $10$ | $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3, 8,11,16,19, 4, 7,12,15,20)$ |
$ 15, 5 $ | $8$ | $15$ | $( 1, 5, 9,13,17)( 2, 7,12,14,19, 4, 6,11,16,18, 3, 8,10,15,20)$ |
$ 10, 10 $ | $3$ | $10$ | $( 1, 6, 9,14,17, 2, 5,10,13,18)( 3, 8,11,16,19, 4, 7,12,15,20)$ |
$ 20 $ | $6$ | $20$ | $( 1, 6,11,16,17, 2, 7,12,13,18, 3, 8, 9,14,19, 4, 5,10,15,20)$ |
$ 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 9,17, 5,13)( 2,10,18, 6,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$ |
$ 10, 5, 5 $ | $6$ | $10$ | $( 1, 9,17, 5,13)( 2,10,18, 6,14)( 3,12,19, 8,15, 4,11,20, 7,16)$ |
$ 15, 5 $ | $8$ | $15$ | $( 1, 9,17, 5,13)( 2,11,20, 6,15, 4,10,19, 8,14, 3,12,18, 7,16)$ |
$ 10, 10 $ | $3$ | $10$ | $( 1,10,17, 6,13, 2, 9,18, 5,14)( 3,12,19, 8,15, 4,11,20, 7,16)$ |
$ 20 $ | $6$ | $20$ | $( 1,10,19, 8,13, 2,11,20, 5,14, 3,12,17, 6,15, 4, 9,18, 7,16)$ |
$ 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,13, 5,17, 9)( 2,14, 6,18,10)( 3,15, 7,19,11)( 4,16, 8,20,12)$ |
$ 10, 5, 5 $ | $6$ | $10$ | $( 1,13, 5,17, 9)( 2,14, 6,18,10)( 3,16, 7,20,11, 4,15, 8,19,12)$ |
$ 15, 5 $ | $8$ | $15$ | $( 1,13, 5,17, 9)( 2,15, 8,18,11, 4,14, 7,20,10, 3,16, 6,19,12)$ |
$ 10, 10 $ | $3$ | $10$ | $( 1,14, 5,18, 9, 2,13, 6,17,10)( 3,16, 7,20,11, 4,15, 8,19,12)$ |
$ 20 $ | $6$ | $20$ | $( 1,14, 7,20, 9, 2,15, 8,17,10, 3,16, 5,18,11, 4,13, 6,19,12)$ |
$ 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,17,13, 9, 5)( 2,18,14,10, 6)( 3,19,15,11, 7)( 4,20,16,12, 8)$ |
$ 10, 5, 5 $ | $6$ | $10$ | $( 1,17,13, 9, 5)( 2,18,14,10, 6)( 3,20,15,12, 7, 4,19,16,11, 8)$ |
$ 15, 5 $ | $8$ | $15$ | $( 1,17,13, 9, 5)( 2,19,16,10, 7, 4,18,15,12, 6, 3,20,14,11, 8)$ |
$ 10, 10 $ | $3$ | $10$ | $( 1,18,13,10, 5, 2,17,14, 9, 6)( 3,20,15,12, 7, 4,19,16,11, 8)$ |
$ 20 $ | $6$ | $20$ | $( 1,18,15,12, 5, 2,19,16, 9, 6, 3,20,13,10, 7, 4,17,14,11, 8)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $120=2^{3} \cdot 3 \cdot 5$ | 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: | 120.37 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);