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Magma
magma: G := TransitiveGroup(15, 50);
Group action invariants
| Degree $n$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $50$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_5\wr C_3$ | ||
| CHM label: | $[D(5)^{3}]3=D(5)wr3$ | ||
| 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: | (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (3,12)(6,9), (3,6,9,12,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $12$: $A_4$ $24$: $A_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: None
Low degree siblings
20T269, 30T408, 30T418, 30T419 x 2, 30T420 x 2, 30T421, 30T425, 30T426, 40T2384Siblings 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 |
| $1^{15}$ | $1$ | $1$ | $()$ | |
| $5,1^{10}$ | $6$ | $5$ | $( 3, 6, 9,12,15)$ | |
| $5,1^{10}$ | $6$ | $5$ | $( 3, 9,15, 6,12)$ | |
| $5^{2},1^{5}$ | $12$ | $5$ | $( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ | |
| $5^{2},1^{5}$ | $12$ | $5$ | $( 2, 5, 8,11,14)( 3, 9,15, 6,12)$ | |
| $5^{2},1^{5}$ | $12$ | $5$ | $( 2, 8,14, 5,11)( 3, 6, 9,12,15)$ | |
| $5^{2},1^{5}$ | $12$ | $5$ | $( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ | |
| $5^{3}$ | $8$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ | |
| $5^{3}$ | $24$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$ | |
| $5^{3}$ | $24$ | $5$ | $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ | |
| $5^{3}$ | $8$ | $5$ | $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ | |
| $2^{4},1^{7}$ | $75$ | $2$ | $( 4,13)( 6,15)( 7,10)( 9,12)$ | |
| $5,2^{4},1^{2}$ | $150$ | $10$ | $( 2, 5, 8,11,14)( 4,13)( 6,15)( 7,10)( 9,12)$ | |
| $5,2^{4},1^{2}$ | $150$ | $10$ | $( 2, 8,14, 5,11)( 4,13)( 6,15)( 7,10)( 9,12)$ | |
| $3^{5}$ | $100$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ | |
| $15$ | $200$ | $15$ | $( 1, 9,14, 4,12, 2, 7,15, 5,10, 3, 8,13, 6,11)$ | |
| $15$ | $200$ | $15$ | $( 1,12, 2, 7, 3, 8,13, 9,14, 4,15, 5,10, 6,11)$ | |
| $3^{5}$ | $100$ | $3$ | $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$ | |
| $15$ | $200$ | $15$ | $( 1,11, 9, 4,14,12, 7, 2,15,10, 5, 3,13, 8, 6)$ | |
| $15$ | $200$ | $15$ | $( 1,11,12, 7, 2, 3,13, 8, 9, 4,14,15,10, 5, 6)$ | |
| $2^{2},1^{11}$ | $15$ | $2$ | $( 6,15)( 9,12)$ | |
| $5,2^{2},1^{6}$ | $30$ | $10$ | $( 2, 5, 8,11,14)( 6,15)( 9,12)$ | |
| $5,2^{2},1^{6}$ | $30$ | $10$ | $( 2, 8,14, 5,11)( 6,15)( 9,12)$ | |
| $5,2^{2},1^{6}$ | $30$ | $10$ | $( 1, 4, 7,10,13)( 6,15)( 9,12)$ | |
| $5^{2},2^{2},1$ | $60$ | $10$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 6,15)( 9,12)$ | |
| $5^{2},2^{2},1$ | $60$ | $10$ | $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 6,15)( 9,12)$ | |
| $5,2^{2},1^{6}$ | $30$ | $10$ | $( 1, 7,13, 4,10)( 6,15)( 9,12)$ | |
| $5^{2},2^{2},1$ | $60$ | $10$ | $( 1, 7,13, 4,10)( 2, 5, 8,11,14)( 6,15)( 9,12)$ | |
| $5^{2},2^{2},1$ | $60$ | $10$ | $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 6,15)( 9,12)$ | |
| $2^{6},1^{3}$ | $125$ | $2$ | $( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)$ | |
| $6^{2},3$ | $500$ | $6$ | $( 1, 6, 5,10,15,11)( 2, 7,12,14, 4, 9)( 3, 8,13)$ | |
| $6^{2},3$ | $500$ | $6$ | $( 1,11, 6,10, 5,15)( 2,12, 4,14, 9, 7)( 3,13, 8)$ |
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.bv | magma: IdentifyGroup(G);
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| Character table: | 32 x 32 character table |
magma: CharacterTable(G);