Show commands:
Magma
magma: G := TransitiveGroup(15, 53);
Group action invariants
| Degree $n$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $53$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^4:A_5$ | ||
| CHM label: | $[3^{4}]A(5)$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,6,11)(4,14,9), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15), (1,13)(2,14)(3,6)(4,7)(8,11)(9,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $60$: $A_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 5: $A_5$
Low degree siblings
15T53 x 2, 30T558, 30T562 x 3, 45T363, 45T364Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 4, 9,14)( 5,15,10)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 2, 7,12)( 4, 9,14)( 5,10,15)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 2,12, 7)( 4,14, 9)( 5,15,10)$ |
| $ 3, 3, 3, 3, 1, 1, 1 $ | $30$ | $3$ | $( 2, 7,12)( 3,13, 8)( 4, 9,14)( 5,15,10)$ |
| $ 3, 3, 3, 3, 3 $ | $5$ | $3$ | $( 1,11, 6)( 2, 7,12)( 3,13, 8)( 4,14, 9)( 5,15,10)$ |
| $ 3, 3, 3, 3, 3 $ | $5$ | $3$ | $( 1,11, 6)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $135$ | $2$ | $( 1, 2)( 3, 4)( 6, 7)( 8, 9)(11,12)(13,14)$ |
| $ 6, 3, 2, 2, 2 $ | $270$ | $6$ | $( 1, 2)( 3, 9, 8,14,13, 4)( 5,15,10)( 6, 7)(11,12)$ |
| $ 6, 3, 2, 2, 2 $ | $270$ | $6$ | $( 1, 2)( 3,14,13, 9, 8, 4)( 5,10,15)( 6, 7)(11,12)$ |
| $ 6, 6, 3 $ | $135$ | $6$ | $( 1, 7, 6,12,11, 2)( 3, 9, 8,14,13, 4)( 5,10,15)$ |
| $ 6, 6, 1, 1, 1 $ | $270$ | $6$ | $( 1, 7, 6,12,11, 2)( 3,14,13, 9, 8, 4)$ |
| $ 6, 6, 3 $ | $135$ | $6$ | $( 1,12,11, 7, 6, 2)( 3,14,13, 9, 8, 4)( 5,15,10)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $180$ | $3$ | $( 1, 2, 3)( 6, 7, 8)(11,12,13)$ |
| $ 3, 3, 3, 3, 3 $ | $180$ | $3$ | $( 1, 2, 3)( 4, 9,14)( 5,15,10)( 6, 7, 8)(11,12,13)$ |
| $ 3, 3, 3, 3, 3 $ | $180$ | $3$ | $( 1, 2, 3)( 4,14, 9)( 5,10,15)( 6, 7, 8)(11,12,13)$ |
| $ 9, 3, 1, 1, 1 $ | $180$ | $9$ | $( 1, 7, 8, 6,12,13,11, 2, 3)( 5,15,10)$ |
| $ 9, 3, 3 $ | $180$ | $9$ | $( 1, 7, 8, 6,12,13,11, 2, 3)( 4, 9,14)( 5,10,15)$ |
| $ 9, 3, 1, 1, 1 $ | $180$ | $9$ | $( 1, 7, 8, 6,12,13,11, 2, 3)( 4,14, 9)$ |
| $ 9, 3, 1, 1, 1 $ | $180$ | $9$ | $( 1,12,13,11, 7, 8, 6, 2, 3)( 5,10,15)$ |
| $ 9, 3, 1, 1, 1 $ | $180$ | $9$ | $( 1,12,13,11, 7, 8, 6, 2, 3)( 4, 9,14)$ |
| $ 9, 3, 3 $ | $180$ | $9$ | $( 1,12,13,11, 7, 8, 6, 2, 3)( 4,14, 9)( 5,15,10)$ |
| $ 5, 5, 5 $ | $972$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)$ |
| $ 5, 5, 5 $ | $972$ | $5$ | $( 1, 2, 3, 5, 4)( 6, 7, 8,10, 9)(11,12,13,15,14)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $4860=2^{2} \cdot 3^{5} \cdot 5$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 4860.j | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);