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Magma
magma: G := TransitiveGroup(25, 40);
Group action invariants
| Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{25}:C_{20}$ | ||
| 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: | (1,2,5,4)(6,18,21,11,7,16,25,13,8,19,24,15,9,17,23,12,10,20,22,14), (1,10,12,17,23,2,6,13,18,24,3,7,14,19,25,4,8,15,20,21,5,9,11,16,22) | 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
There are no siblings 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 $ | $25$ | $20$ | $( 2, 3, 5, 4)( 6,11,23,19,10,14,24,16, 9,12,25,18, 8,15,21,20, 7,13,22,17)$ |
| $ 20, 4, 1 $ | $25$ | $20$ | $( 2, 3, 5, 4)( 6,12,22,16, 7,14,21,19, 8,11,25,17, 9,13,24,20,10,15,23,18)$ |
| $ 20, 4, 1 $ | $25$ | $20$ | $( 2, 3, 5, 4)( 6,13,21,18, 9,14,23,17, 7,15,25,16,10,11,22,20, 8,12,24,19)$ |
| $ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 3, 5, 4)( 6,14,25,20)( 7,11,24,18)( 8,13,23,16)( 9,15,22,19)(10,12,21,17)$ |
| $ 20, 4, 1 $ | $25$ | $20$ | $( 2, 3, 5, 4)( 6,15,24,17, 8,14,22,18,10,13,25,19, 7,12,23,20, 9,11,21,16)$ |
| $ 20, 4, 1 $ | $25$ | $20$ | $( 2, 4, 5, 3)( 6,16,21,11, 9,20,23,12, 7,19,25,13,10,18,22,14, 8,17,24,15)$ |
| $ 20, 4, 1 $ | $25$ | $20$ | $( 2, 4, 5, 3)( 6,17,22,13, 7,20,21,15, 8,18,25,12, 9,16,24,14,10,19,23,11)$ |
| $ 20, 4, 1 $ | $25$ | $20$ | $( 2, 4, 5, 3)( 6,18,23,15,10,20,24,13, 9,17,25,11, 8,19,21,14, 7,16,22,12)$ |
| $ 20, 4, 1 $ | $25$ | $20$ | $( 2, 4, 5, 3)( 6,19,24,12, 8,20,22,11,10,16,25,15, 7,17,23,14, 9,18,21,13)$ |
| $ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 4, 5, 3)( 6,20,25,14)( 7,18,24,11)( 8,16,23,13)( 9,19,22,15)(10,17,21,12)$ |
| $ 10, 10, 2, 2, 1 $ | $25$ | $10$ | $( 2, 5)( 3, 4)( 6,21, 9,23, 7,25,10,22, 8,24)(11,20,12,19,13,18,14,17,15,16)$ |
| $ 10, 10, 2, 2, 1 $ | $25$ | $10$ | $( 2, 5)( 3, 4)( 6,22, 7,21, 8,25, 9,24,10,23)(11,17,13,20,15,18,12,16,14,19)$ |
| $ 10, 10, 2, 2, 1 $ | $25$ | $10$ | $( 2, 5)( 3, 4)( 6,23,10,24, 9,25, 8,21, 7,22)(11,19,14,16,12,18,15,20,13,17)$ |
| $ 10, 10, 2, 2, 1 $ | $25$ | $10$ | $( 2, 5)( 3, 4)( 6,24, 8,22,10,25, 7,23, 9,21)(11,16,15,17,14,18,13,19,12,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,25)( 7,24)( 8,23)( 9,22)(10,21)(11,18)(12,17)(13,16)(14,20) (15,19)$ |
| $ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
| $ 25 $ | $20$ | $25$ | $( 1, 6,11,16,24, 2, 7,12,17,25, 3, 8,13,18,21, 4, 9,14,19,22, 5,10,15,20,23)$ |
| $ 25 $ | $20$ | $25$ | $( 1, 6,12,19,25, 2, 7,13,20,21, 3, 8,14,16,22, 4, 9,15,17,23, 5,10,11,18,24)$ |
| $ 25 $ | $20$ | $25$ | $( 1, 6,13,17,21, 2, 7,14,18,22, 3, 8,15,19,23, 4, 9,11,20,24, 5,10,12,16,25)$ |
| $ 25 $ | $20$ | $25$ | $( 1, 6,14,20,22, 2, 7,15,16,23, 3, 8,11,17,24, 4, 9,12,18,25, 5,10,13,19,21)$ |
| $ 25 $ | $20$ | $25$ | $( 1, 6,15,18,23, 2, 7,11,19,24, 3, 8,12,20,25, 4, 9,13,16,21, 5,10,14,17,22)$ |
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: | 500.18 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);