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Magma
magma: G := TransitiveGroup(25, 37);
Group action invariants
Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $37$ | 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: | (1,22,15,17,9,4,21,11,19,6,3,23,14,20,7,5,24,13,18,10)(2,25,12,16,8), (1,8,16,11,22,5,10,18,12,21,3,9,17,14,24,4,7,20,13,25)(2,6,19,15,23) | 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: $C_5$
Low degree siblings
25T34Siblings 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 $ | $20$ | $5$ | $( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)$ |
$ 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ | $25$ | $4$ | $( 2, 3, 5, 4)( 7, 8,10, 9)(12,13,15,14)(17,18,20,19)(22,23,25,24)$ |
$ 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ | $25$ | $4$ | $( 2, 4, 5, 3)( 7, 9,10, 8)(12,14,15,13)(17,19,20,18)(22,24,25,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$ |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 2, 3, 4, 5)( 6,10, 9, 8, 7)(11,14,12,15,13)(16,18,20,17,19) (21,22,23,24,25)$ |
$ 10, 10, 5 $ | $25$ | $10$ | $( 1, 6,16,11,21)( 2, 7,18,13,22, 5,10,19,14,25)( 3, 8,20,15,23, 4, 9,17,12,24)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1, 6,16,11,21)( 2, 8,19,12,22, 4, 7,20,14,24, 5, 9,18,15,25, 3,10,17,13,23)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1, 6,16,11,21)( 2, 9,19,15,22, 3, 7,17,14,23, 5, 8,18,12,25, 4,10,20,13,24)$ |
$ 5, 5, 5, 5, 5 $ | $5$ | $5$ | $( 1, 6,16,11,21)( 2,10,18,14,22)( 3, 9,20,12,23)( 4, 8,17,15,24) ( 5, 7,19,13,25)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,17,13,24)( 2,10,19,11,25)( 3, 9,16,14,21)( 4, 8,18,12,22) ( 5, 7,20,15,23)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1,11, 6,21,16)( 2,12, 7,24,18, 3,13, 8,22,20, 5,15,10,23,19, 4,14, 9,25,17)$ |
$ 10, 10, 5 $ | $25$ | $10$ | $( 1,11, 6,21,16)( 2,13,10,25,18, 5,14, 7,22,19)( 3,15, 9,24,20, 4,12, 8,23,17)$ |
$ 5, 5, 5, 5, 5 $ | $5$ | $5$ | $( 1,11, 6,21,16)( 2,14,10,22,18)( 3,12, 9,23,20)( 4,15, 8,24,17) ( 5,13, 7,25,19)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1,11, 8,25,17)( 2,14, 7,21,19)( 3,12, 6,22,16)( 4,15,10,23,18) ( 5,13, 9,24,20)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1,11, 6,21,16)( 2,15, 7,23,18, 4,13, 9,22,17, 5,12,10,24,19, 3,14, 8,25,20)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1,16,21, 6,11)( 2,17,25, 9,14, 4,19,23,10,15, 5,20,22, 8,13, 3,18,24, 7,12)$ |
$ 5, 5, 5, 5, 5 $ | $5$ | $5$ | $( 1,16,21, 6,11)( 2,18,22,10,14)( 3,20,23, 9,12)( 4,17,24, 8,15) ( 5,19,25, 7,13)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1,16,25, 9,13)( 2,18,21, 8,11)( 3,20,22, 7,14)( 4,17,23, 6,12) ( 5,19,24,10,15)$ |
$ 10, 10, 5 $ | $25$ | $10$ | $( 1,16,21, 6,11)( 2,19,22, 7,14, 5,18,25,10,13)( 3,17,23, 8,12, 4,20,24, 9,15)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1,16,21, 6,11)( 2,20,25, 8,14, 3,19,24,10,12, 5,17,22, 9,13, 4,18,23, 7,15)$ |
$ 5, 5, 5, 5, 5 $ | $5$ | $5$ | $( 1,21,11,16, 6)( 2,22,14,18,10)( 3,23,12,20, 9)( 4,24,15,17, 8) ( 5,25,13,19, 7)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1,21,13,20, 7)( 2,22,11,17, 6)( 3,23,14,19,10)( 4,24,12,16, 9) ( 5,25,15,18, 8)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1,21,11,16, 6)( 2,23,13,17,10, 3,25,15,18, 9, 5,24,14,20, 7, 4,22,12,19, 8)$ |
$ 20, 5 $ | $25$ | $20$ | $( 1,21,11,16, 6)( 2,24,13,20,10, 4,25,12,18, 8, 5,23,14,17, 7, 3,22,15,19, 9)$ |
$ 10, 10, 5 $ | $25$ | $10$ | $( 1,21,11,16, 6)( 2,25,14,19,10, 5,22,13,18, 7)( 3,24,12,17, 9, 4,23,15,20, 8)$ |
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);