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Magma
magma: G := TransitiveGroup(25, 13);
Group action invariants
| Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $13$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{25}:C_5$ | ||
| 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: | (1,19,10,24,13,4,17,8,22,11,2,20,6,25,14,5,18,9,23,12,3,16,7,21,15), (1,10,11,16,25,2,6,12,17,21,3,7,13,18,22,4,8,14,19,23,5,9,15,20,24) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $5$: $C_5$ x 6 $25$: 25T2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $C_5$
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 $ | $5$ | $5$ | $( 6,10, 9, 8, 7)(11,14,12,15,13)(16,18,20,17,19)(21,22,23,24,25)$ |
| $ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
| $ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ |
| $ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$ |
| $ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$ |
| $ 25 $ | $5$ | $25$ | $( 1, 6,11,18,22, 2, 7,12,19,23, 3, 8,13,20,24, 4, 9,14,16,25, 5,10,15,17,21)$ |
| $ 25 $ | $5$ | $25$ | $( 1, 6,12,16,23, 2, 7,13,17,24, 3, 8,14,18,25, 4, 9,15,19,21, 5,10,11,20,22)$ |
| $ 25 $ | $5$ | $25$ | $( 1, 6,13,19,24, 2, 7,14,20,25, 3, 8,15,16,21, 4, 9,11,17,22, 5,10,12,18,23)$ |
| $ 25 $ | $5$ | $25$ | $( 1, 6,14,17,25, 2, 7,15,18,21, 3, 8,11,19,22, 4, 9,12,20,23, 5,10,13,16,24)$ |
| $ 25 $ | $5$ | $25$ | $( 1, 6,15,20,21, 2, 7,11,16,22, 3, 8,12,17,23, 4, 9,13,18,24, 5,10,14,19,25)$ |
| $ 25 $ | $5$ | $25$ | $( 1,11,23,10,20, 3,13,25, 7,17, 5,15,22, 9,19, 2,12,24, 6,16, 4,14,21, 8,18)$ |
| $ 25 $ | $5$ | $25$ | $( 1,11,25, 6,17, 3,13,22, 8,19, 5,15,24,10,16, 2,12,21, 7,18, 4,14,23, 9,20)$ |
| $ 25 $ | $5$ | $25$ | $( 1,11,22, 7,19, 3,13,24, 9,16, 5,15,21, 6,18, 2,12,23, 8,20, 4,14,25,10,17)$ |
| $ 25 $ | $5$ | $25$ | $( 1,11,24, 8,16, 3,13,21,10,18, 5,15,23, 7,20, 2,12,25, 9,17, 4,14,22, 6,19)$ |
| $ 25 $ | $5$ | $25$ | $( 1,11,21, 9,18, 3,13,23, 6,20, 5,15,25, 8,17, 2,12,22,10,19, 4,14,24, 7,16)$ |
| $ 25 $ | $5$ | $25$ | $( 1,16, 6,21,13, 4,19, 9,24,11, 2,17, 7,22,14, 5,20,10,25,12, 3,18, 8,23,15)$ |
| $ 25 $ | $5$ | $25$ | $( 1,16, 9,25,11, 4,19, 7,23,14, 2,17,10,21,12, 5,20, 8,24,15, 3,18, 6,22,13)$ |
| $ 25 $ | $5$ | $25$ | $( 1,16, 7,24,14, 4,19,10,22,12, 2,17, 8,25,15, 5,20, 6,23,13, 3,18, 9,21,11)$ |
| $ 25 $ | $5$ | $25$ | $( 1,16,10,23,12, 4,19, 8,21,15, 2,17, 6,24,13, 5,20, 9,22,11, 3,18, 7,25,14)$ |
| $ 25 $ | $5$ | $25$ | $( 1,16, 8,22,15, 4,19, 6,25,13, 2,17, 9,23,11, 5,20, 7,21,14, 3,18,10,24,12)$ |
| $ 25 $ | $5$ | $25$ | $( 1,21,17,15,10, 5,25,16,14, 9, 4,24,20,13, 8, 3,23,19,12, 7, 2,22,18,11, 6)$ |
| $ 25 $ | $5$ | $25$ | $( 1,21,16,12, 9, 5,25,20,11, 8, 4,24,19,15, 7, 3,23,18,14, 6, 2,22,17,13,10)$ |
| $ 25 $ | $5$ | $25$ | $( 1,21,20,14, 8, 5,25,19,13, 7, 4,24,18,12, 6, 3,23,17,11,10, 2,22,16,15, 9)$ |
| $ 25 $ | $5$ | $25$ | $( 1,21,19,11, 7, 5,25,18,15, 6, 4,24,17,14,10, 3,23,16,13, 9, 2,22,20,12, 8)$ |
| $ 25 $ | $5$ | $25$ | $( 1,21,18,13, 6, 5,25,17,12,10, 4,24,16,11, 9, 3,23,20,15, 8, 2,22,19,14, 7)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $125=5^{3}$ | 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: | $2$ | ||
| Label: | 125.4 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);