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Magma
magma: G := TransitiveGroup(25, 49);
Group action invariants
Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $49$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5^3.C_5$ | ||
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: | (1,15,17,8,21,2,11,18,9,22,3,12,19,10,23,4,13,20,6,24,5,14,16,7,25), (1,25,7,19,11,5,24,6,18,15,4,23,10,17,14,3,22,9,16,13,2,21,8,20,12) | 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 $125$: 25T14 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $C_5$
Low degree siblings
25T49 x 4Siblings 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, 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 $ | $1$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$ |
$ 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 $ | $1$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 6,10, 9, 8, 7)(11,14,12,15,13)(16,17,18,19,20)(21,23,25,22,24)$ |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 6, 9, 7,10, 8)(11,12,13,14,15)(16,18,20,17,19)(21,25,24,23,22)$ |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 6, 7, 8, 9,10)(11,13,15,12,14)(16,20,19,18,17)(21,24,22,25,23)$ |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 6, 8,10, 7, 9)(11,15,14,13,12)(16,19,17,20,18)(21,22,23,24,25)$ |
$ 25 $ | $25$ | $25$ | $( 1,15,17, 8,21, 2,11,18, 9,22, 3,12,19,10,23, 4,13,20, 6,24, 5,14,16, 7,25)$ |
$ 25 $ | $25$ | $25$ | $( 1,17,21,11, 9, 3,19,23,13, 6, 5,16,25,15, 8, 2,18,22,12,10, 4,20,24,14, 7)$ |
$ 25 $ | $25$ | $25$ | $( 1,21, 9,19,13, 5,25, 8,18,12, 4,24, 7,17,11, 3,23, 6,16,15, 2,22,10,20,14)$ |
$ 25 $ | $25$ | $25$ | $( 1, 8,11,22,19, 4, 6,14,25,17, 2, 9,12,23,20, 5, 7,15,21,18, 3,10,13,24,16)$ |
$ 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $(11,13,15,12,14)(16,18,20,17,19)(21,22,23,24,25)$ |
$ 5, 5, 5, 5, 5 $ | $5$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,15,14,13,12)(16,20,19,18,17) (21,24,22,25,23)$ |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,12,13,14,15)(16,17,18,19,20)$ |
$ 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(21,25,24,23,22)$ |
$ 5, 5, 5, 5, 5 $ | $5$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,14,12,15,13)(16,19,17,20,18) (21,23,25,22,24)$ |
$ 25 $ | $25$ | $25$ | $( 1,15,19, 7,25, 2,11,20, 8,21, 3,12,16, 9,22, 4,13,17,10,23, 5,14,18, 6,24)$ |
$ 25 $ | $25$ | $25$ | $( 1,17,23,14, 9, 3,19,25,11, 6, 5,16,22,13, 8, 2,18,24,15,10, 4,20,21,12, 7)$ |
$ 25 $ | $25$ | $25$ | $( 1,21,10,20,11, 5,25, 9,19,15, 4,24, 8,18,14, 3,23, 7,17,13, 2,22, 6,16,12)$ |
$ 25 $ | $25$ | $25$ | $( 1, 8,11,24,17, 4, 6,14,22,20, 2, 9,12,25,18, 5, 7,15,23,16, 3,10,13,21,19)$ |
$ 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $(11,15,14,13,12)(16,20,19,18,17)(21,23,25,22,24)$ |
$ 5, 5, 5, 5, 5 $ | $5$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,12,13,14,15)(16,17,18,19,20) (21,25,24,23,22)$ |
$ 5, 5, 5, 5, 5 $ | $5$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,14,12,15,13)(16,19,17,20,18) (21,22,23,24,25)$ |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,13,15,12,14)(16,18,20,17,19)$ |
$ 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(21,24,22,25,23)$ |
$ 25 $ | $25$ | $25$ | $( 1,15,16, 6,24, 2,11,17, 7,25, 3,12,18, 8,21, 4,13,19, 9,22, 5,14,20,10,23)$ |
$ 25 $ | $25$ | $25$ | $( 1,17,25,12, 9, 3,19,22,14, 6, 5,16,24,11, 8, 2,18,21,13,10, 4,20,23,15, 7)$ |
$ 25 $ | $25$ | $25$ | $( 1,21, 6,16,14, 5,25,10,20,13, 4,24, 9,19,12, 3,23, 8,18,11, 2,22, 7,17,15)$ |
$ 25 $ | $25$ | $25$ | $( 1, 8,11,21,20, 4, 6,14,24,18, 2, 9,12,22,16, 5, 7,15,25,19, 3,10,13,23,17)$ |
$ 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $(11,14,12,15,13)(16,19,17,20,18)(21,25,24,23,22)$ |
$ 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(21,22,23,24,25)$ |
$ 5, 5, 5, 5, 5 $ | $5$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,13,15,12,14)(16,18,20,17,19) (21,24,22,25,23)$ |
$ 5, 5, 5, 5, 5 $ | $5$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,12,13,14,15)(16,17,18,19,20) (21,23,25,22,24)$ |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,15,14,13,12)(16,20,19,18,17)$ |
$ 25 $ | $25$ | $25$ | $( 1,15,20, 9,22, 2,11,16,10,23, 3,12,17, 6,24, 4,13,18, 7,25, 5,14,19, 8,21)$ |
$ 25 $ | $25$ | $25$ | $( 1,17,24,13, 9, 3,19,21,15, 6, 5,16,23,12, 8, 2,18,25,14,10, 4,20,22,11, 7)$ |
$ 25 $ | $25$ | $25$ | $( 1,21, 8,18,15, 5,25, 7,17,14, 4,24, 6,16,13, 3,23,10,20,12, 2,22, 9,19,11)$ |
$ 25 $ | $25$ | $25$ | $( 1, 8,11,25,16, 4, 6,14,23,19, 2, 9,12,21,17, 5, 7,15,24,20, 3,10,13,22,18)$ |
$ 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $(11,12,13,14,15)(16,17,18,19,20)(21,24,22,25,23)$ |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,14,12,15,13)(16,19,17,20,18)$ |
$ 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(21,23,25,22,24)$ |
$ 5, 5, 5, 5, 5 $ | $5$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,15,14,13,12)(16,20,19,18,17) (21,22,23,24,25)$ |
$ 5, 5, 5, 5, 5 $ | $5$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,13,15,12,14)(16,18,20,17,19) (21,25,24,23,22)$ |
$ 25 $ | $25$ | $25$ | $( 1,15,18,10,23, 2,11,19, 6,24, 3,12,20, 7,25, 4,13,16, 8,21, 5,14,17, 9,22)$ |
$ 25 $ | $25$ | $25$ | $( 1,17,22,15, 9, 3,19,24,12, 6, 5,16,21,14, 8, 2,18,23,11,10, 4,20,25,13, 7)$ |
$ 25 $ | $25$ | $25$ | $( 1,21, 7,17,12, 5,25, 6,16,11, 4,24,10,20,15, 3,23, 9,19,14, 2,22, 8,18,13)$ |
$ 25 $ | $25$ | $25$ | $( 1, 8,11,23,18, 4, 6,14,21,16, 2, 9,12,24,19, 5, 7,15,22,17, 3,10,13,25,20)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $625=5^{4}$ | 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: | $3$ | ||
Label: | 625.8 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);