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Magma
magma: G := TransitiveGroup(25, 35);
Group action invariants
Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5^2:F_5$ | ||
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: | (1,18,14,21)(2,19,15,22)(3,20,11,23)(4,16,12,24)(5,17,13,25), (1,24,18,15,9)(2,25,19,11,10)(3,21,20,12,6)(4,22,16,13,7)(5,23,17,14,8) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $20$: $F_5$ x 2 $100$: $C_5^2 : C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
Low degree siblings
25T35Siblings 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 $ | $20$ | $5$ | $( 6, 7, 8, 9,10)(11,13,15,12,14)(16,19,17,20,18)(21,25,24,23,22)$ |
$ 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ | $25$ | $4$ | $( 6,11,21,19)( 7,12,22,20)( 8,13,23,16)( 9,14,24,17)(10,15,25,18)$ |
$ 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ | $25$ | $4$ | $( 6,16,24,12)( 7,17,25,13)( 8,18,21,14)( 9,19,22,15)(10,20,23,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $25$ | $2$ | $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,19)(12,20)(13,16)(14,17)(15,18)$ |
$ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1, 2, 3, 4, 5)( 6,11,25,20,10,15,24,19, 9,14,23,18, 8,13,22,17, 7,12,21,16)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1, 2, 3, 4, 5)( 6,16,22,12,10,20,21,11, 9,19,25,15, 8,18,24,14, 7,17,23,13)$ |
$ 10, 10, 5 $ | $25$ | $10$ | $( 1, 2, 3, 4, 5)( 6,21, 8,23,10,25, 7,22, 9,24)(11,18,13,20,15,17,12,19,14,16)$ |
$ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1, 3, 5, 2, 4)( 6,11,24,16, 9,14,22,19, 7,12,25,17,10,15,23,20, 8,13,21,18)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1, 3, 5, 2, 4)( 6,16,25,12, 9,19,23,15, 7,17,21,13,10,20,24,11, 8,18,22,14)$ |
$ 10, 10, 5 $ | $25$ | $10$ | $( 1, 3, 5, 2, 4)( 6,21,10,25, 9,24, 8,23, 7,22)(11,17,15,16,14,20,13,19,12,18)$ |
$ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1, 4, 2, 5, 3)( 6,11,23,17, 8,13,25,19,10,15,22,16, 7,12,24,18, 9,14,21,20)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1, 4, 2, 5, 3)( 6,16,23,12, 8,18,25,14,10,20,22,11, 7,17,24,13, 9,19,21,15)$ |
$ 10, 10, 5 $ | $25$ | $10$ | $( 1, 4, 2, 5, 3)( 6,21, 7,22, 8,23, 9,24,10,25)(11,16,12,17,13,18,14,19,15,20)$ |
$ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1, 5, 4, 3, 2)( 6,11,22,18, 7,12,23,19, 8,13,24,20, 9,14,25,16,10,15,21,17)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1, 5, 4, 3, 2)( 6,16,21,12, 7,17,22,13, 8,18,23,14, 9,19,24,15,10,20,25,11)$ |
$ 10, 10, 5 $ | $25$ | $10$ | $( 1, 5, 4, 3, 2)( 6,21, 9,24, 7,22,10,25, 8,23)(11,20,14,18,12,16,15,19,13,17)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,11,20,24)( 2, 7,12,16,25)( 3, 8,13,17,21)( 4, 9,14,18,22) ( 5,10,15,19,23)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,12,18,25)( 2, 7,13,19,21)( 3, 8,14,20,22)( 4, 9,15,16,23) ( 5,10,11,17,24)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,13,16,21)( 2, 7,14,17,22)( 3, 8,15,18,23)( 4, 9,11,19,24) ( 5,10,12,20,25)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,14,19,22)( 2, 7,15,20,23)( 3, 8,11,16,24)( 4, 9,12,17,25) ( 5,10,13,18,21)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,15,17,23)( 2, 7,11,18,24)( 3, 8,12,19,25)( 4, 9,13,20,21) ( 5,10,14,16,22)$ |
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: | 500.25 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);