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Magma
magma: G := TransitiveGroup(25, 34);
Group action invariants
Degree $n$: | $25$ | 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: | $C_5^2:C_{20}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,15,18,6,5,12,19,9,4,14,20,7,3,11,16,10,2,13,17,8)(21,24,23,25), (1,17,9,23,15)(2,18,10,24,11)(3,19,6,25,12)(4,20,7,21,13)(5,16,8,22,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $5$: $C_5$ $10$: $C_{10}$ $20$: $F_5$, 20T1 $100$: 20T29 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
Low degree siblings
25T37Siblings 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, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 6, 7, 8, 9,10)(11,13,15,12,14)(16,19,17,20,18)(21,25,24,23,22)$ |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 6, 8,10, 7, 9)(11,15,14,13,12)(16,17,18,19,20)(21,24,22,25,23)$ |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 6, 9, 7,10, 8)(11,12,13,14,15)(16,20,19,18,17)(21,23,25,22,24)$ |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 6,10, 9, 8, 7)(11,14,12,15,13)(16,18,20,17,19)(21,22,23,24,25)$ |
$ 20, 4, 1 $ | $25$ | $20$ | $( 2, 3, 5, 4)( 6,11,24,17,10,14,25,19, 9,12,21,16, 8,15,22,18, 7,13,23,20)$ |
$ 20, 4, 1 $ | $25$ | $20$ | $( 2, 3, 5, 4)( 6,12,23,19, 7,14,22,17, 8,11,21,20, 9,13,25,18,10,15,24,16)$ |
$ 20, 4, 1 $ | $25$ | $20$ | $( 2, 3, 5, 4)( 6,13,22,16, 9,14,24,20, 7,15,21,19,10,11,23,18, 8,12,25,17)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 3, 5, 4)( 6,14,21,18)( 7,11,25,16)( 8,13,24,19)( 9,15,23,17)(10,12,22,20)$ |
$ 20, 4, 1 $ | $25$ | $20$ | $( 2, 3, 5, 4)( 6,15,25,20, 8,14,23,16,10,13,21,17, 7,12,24,18, 9,11,22,19)$ |
$ 20, 4, 1 $ | $25$ | $20$ | $( 2, 4, 5, 3)( 6,16,24,15,10,18,25,13, 9,20,21,11, 8,17,22,14, 7,19,23,12)$ |
$ 20, 4, 1 $ | $25$ | $20$ | $( 2, 4, 5, 3)( 6,17,25,12, 8,18,23,11,10,19,21,15, 7,20,24,14, 9,16,22,13)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 4, 5, 3)( 6,18,21,14)( 7,16,25,11)( 8,19,24,13)( 9,17,23,15)(10,20,22,12)$ |
$ 20, 4, 1 $ | $25$ | $20$ | $( 2, 4, 5, 3)( 6,19,22,11, 9,18,24,12, 7,17,21,13,10,16,23,14, 8,20,25,15)$ |
$ 20, 4, 1 $ | $25$ | $20$ | $( 2, 4, 5, 3)( 6,20,23,13, 7,18,22,15, 8,16,21,12, 9,19,25,14,10,17,24,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ |
$ 10, 10, 2, 2, 1 $ | $25$ | $10$ | $( 2, 5)( 3, 4)( 6,22, 9,24, 7,21,10,23, 8,25)(11,18,12,17,13,16,14,20,15,19)$ |
$ 10, 10, 2, 2, 1 $ | $25$ | $10$ | $( 2, 5)( 3, 4)( 6,23, 7,22, 8,21, 9,25,10,24)(11,20,13,18,15,16,12,19,14,17)$ |
$ 10, 10, 2, 2, 1 $ | $25$ | $10$ | $( 2, 5)( 3, 4)( 6,24,10,25, 9,21, 8,22, 7,23)(11,17,14,19,12,16,15,18,13,20)$ |
$ 10, 10, 2, 2, 1 $ | $25$ | $10$ | $( 2, 5)( 3, 4)( 6,25, 8,23,10,21, 7,24, 9,22)(11,19,15,20,14,16,13,17,12,18)$ |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,11,19,23)( 2, 7,12,20,24)( 3, 8,13,16,25)( 4, 9,14,17,21) ( 5,10,15,18,22)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,12,17,24)( 2, 7,13,18,25)( 3, 8,14,19,21)( 4, 9,15,20,22) ( 5,10,11,16,23)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,13,20,25)( 2, 7,14,16,21)( 3, 8,15,17,22)( 4, 9,11,18,23) ( 5,10,12,19,24)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,14,18,21)( 2, 7,15,19,22)( 3, 8,11,20,23)( 4, 9,12,16,24) ( 5,10,13,17,25)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,15,16,22)( 2, 7,11,17,23)( 3, 8,12,18,24)( 4, 9,13,19,25) ( 5,10,14,20,21)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $500=2^{2} \cdot 5^{3}$ | 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: | 500.17 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);