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Magma
magma: G := TransitiveGroup(15, 49);
Group action invariants
Degree $n$: | $15$ | 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: | $C_9^2\times C_{54}$ | ||
CHM label: | $[5^{3}:4]S(3)$ | ||
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: | (1,11)(2,7)(4,14)(5,10)(8,13), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (1,7,4,13)(2,14,8,11)(3,6,12,9), (3,6,9,12,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$ $8$: $C_4\times C_2$ $12$: $D_{6}$ $20$: $F_5$ $24$: $S_3 \times C_4$ $40$: $F_{5}\times C_2$ $120$: $F_5 \times S_3$ $600$: 15T27 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 5: None
Low degree siblings
15T49 x 3, 30T433 x 4, 30T435 x 2, 30T438 x 4, 30T442 x 4Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 5, 5, 5 $ | $12$ | $5$ | $( 1, 7,13, 4,10)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$ | |
$ 5, 5, 1, 1, 1, 1, 1 $ | $12$ | $5$ | $( 2,14,11, 8, 5)( 3, 6, 9,12,15)$ | |
$ 3, 3, 3, 3, 3 $ | $50$ | $3$ | $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$ | |
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 2,12)( 3, 8)( 5,15)( 6,11)( 9,14)$ | |
$ 10, 5 $ | $60$ | $10$ | $( 1, 7,13, 4,10)( 2, 9,11, 3, 5,12,14, 6, 8,15)$ | |
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $5$ | $( 3, 6, 9,12,15)$ | |
$ 5, 5, 1, 1, 1, 1, 1 $ | $24$ | $5$ | $( 1, 7,13, 4,10)( 2,14,11, 8, 5)$ | |
$ 5, 5, 5 $ | $12$ | $5$ | $( 1,13,10, 7, 4)( 2,11, 5,14, 8)( 3,15,12, 9, 6)$ | |
$ 5, 5, 5 $ | $24$ | $5$ | $( 1,10, 4,13, 7)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$ | |
$ 5, 5, 1, 1, 1, 1, 1 $ | $12$ | $5$ | $( 1,10, 4,13, 7)( 3,12, 6,15, 9)$ | |
$ 5, 5, 5 $ | $12$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,15,12, 9, 6)$ | |
$ 5, 5, 5 $ | $4$ | $5$ | $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ | |
$ 15 $ | $200$ | $15$ | $( 1,11, 6, 4,14, 9, 7, 2,12,10, 5,15,13, 8, 3)$ | |
$ 10, 1, 1, 1, 1, 1 $ | $60$ | $10$ | $( 2,12, 5,15, 8, 3,11, 6,14, 9)$ | |
$ 10, 5 $ | $60$ | $10$ | $( 1, 7,13, 4,10)( 2, 9,14, 6,11, 3, 8,15, 5,12)$ | |
$ 10, 5 $ | $60$ | $10$ | $( 1,13,10, 7, 4)( 2, 6, 8,12,14, 3, 5, 9,11,15)$ | |
$ 10, 5 $ | $60$ | $10$ | $( 1,10, 4,13, 7)( 2,15,11, 9, 5, 3,14,12, 8, 6)$ | |
$ 5, 2, 2, 2, 2, 2 $ | $60$ | $10$ | $( 1, 4, 7,10,13)( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $125$ | $2$ | $( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)$ | |
$ 6, 6, 3 $ | $250$ | $6$ | $( 1,11, 3,13,14,15)( 2,12, 4, 8, 6,10)( 5, 9, 7)$ | |
$ 10, 2, 2, 1 $ | $300$ | $10$ | $( 2,12,14,15,11, 3, 8, 6, 5, 9)( 4,13)( 7,10)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 1 $ | $75$ | $2$ | $( 2, 3)( 4,13)( 5,15)( 6,14)( 7,10)( 8,12)( 9,11)$ | |
$ 4, 4, 4, 1, 1, 1 $ | $125$ | $4$ | $( 4, 7,13,10)( 5, 8,14,11)( 6, 9,15,12)$ | |
$ 12, 3 $ | $250$ | $12$ | $( 1,11,15, 7, 8, 9,10,14, 6, 4, 2,12)( 3,13, 5)$ | |
$ 4, 4, 4, 2, 1 $ | $375$ | $4$ | $( 2,12,11,15)( 3, 8, 9, 5)( 4, 7,13,10)( 6,14)$ | |
$ 4, 4, 4, 1, 1, 1 $ | $125$ | $4$ | $( 4,10,13, 7)( 5,11,14, 8)( 6,12,15, 9)$ | |
$ 12, 3 $ | $250$ | $12$ | $( 1,11, 9)( 2,12,10, 8,15, 4, 5, 6, 7,14, 3,13)$ | |
$ 4, 4, 4, 2, 1 $ | $375$ | $4$ | $( 2,12, 5, 6)( 3, 8,15,14)( 4,10,13, 7)( 9,11)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $3000=2^{3} \cdot 3 \cdot 5^{3}$ | 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: | 3000.bf | magma: IdentifyGroup(G);
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Character table: |
Size | |
2 P | |
3 P | |
5 P | |
Type |
magma: CharacterTable(G);