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Magma
magma: G := TransitiveGroup(15, 41);
Group action invariants
Degree $n$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $41$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^4:F_5$ | ||
CHM label: | $[3^{4}]F(5)$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $3$ | 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), (1,7,4,13)(2,14,8,11)(3,6,12,9) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $20$: $F_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 5: $F_5$
Low degree siblings
15T42, 30T295, 30T296, 30T298, 30T299, 30T302, 45T204, 45T210Siblings 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 $ | $20$ | $3$ | $( 1, 6,11)( 4,14, 9)$ |
$ 3, 3, 3, 3, 1, 1, 1 $ | $20$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3,13, 8)( 4,14, 9)$ |
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 1,11, 6)( 3,13, 8)( 4,14, 9)$ |
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 2, 7,12)( 3, 8,13)( 4, 9,14)$ |
$ 3, 3, 3, 3, 1, 1, 1 $ | $10$ | $3$ | $( 1,11, 6)( 2, 7,12)( 3, 8,13)( 4,14, 9)$ |
$ 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 $ | $5$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,15,10)$ |
$ 5, 5, 5 $ | $324$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $45$ | $2$ | $( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15)$ |
$ 6, 3, 2, 2, 2 $ | $90$ | $6$ | $( 1, 6,11)( 2, 5)( 3, 4,13,14, 8, 9)( 7,10)(12,15)$ |
$ 6, 3, 2, 2, 2 $ | $90$ | $6$ | $( 1,11, 6)( 2, 5)( 3,14, 8, 4,13, 9)( 7,10)(12,15)$ |
$ 6, 6, 1, 1, 1 $ | $90$ | $6$ | $( 2, 5, 7,10,12,15)( 3, 9,13, 4, 8,14)$ |
$ 6, 6, 3 $ | $45$ | $6$ | $( 1, 6,11)( 2, 5, 7,10,12,15)( 3, 4, 8, 9,13,14)$ |
$ 6, 6, 3 $ | $45$ | $6$ | $( 1,11, 6)( 2, 5,12,15, 7,10)( 3,14,13, 9, 8, 4)$ |
$ 4, 4, 4, 1, 1, 1 $ | $135$ | $4$ | $( 2, 8, 5,14)( 3,15, 9,12)( 4, 7,13,10)$ |
$ 12, 3 $ | $135$ | $12$ | $( 1, 6,11)( 2, 8, 5, 9,12, 3,15, 4, 7,13,10,14)$ |
$ 12, 3 $ | $135$ | $12$ | $( 1,11, 6)( 2, 8, 5, 4, 7,13,10, 9,12, 3,15,14)$ |
$ 4, 4, 4, 1, 1, 1 $ | $135$ | $4$ | $( 2,14, 5, 8)( 3,12, 9,15)( 4,10,13, 7)$ |
$ 12, 3 $ | $135$ | $12$ | $( 1, 6,11)( 2, 9,15, 3,12, 4,10,13, 7,14, 5, 8)$ |
$ 12, 3 $ | $135$ | $12$ | $( 1,11, 6)( 2, 4,10,13, 7, 9,15, 3,12,14, 5, 8)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $1620=2^{2} \cdot 3^{4} \cdot 5$ | 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: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 1620.421 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);