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Magma
magma: G := TransitiveGroup(15, 30);
Group action invariants
Degree $n$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5^3:C_6$ | ||
CHM label: | $[5^{3}:2]3$ | ||
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: | (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (3,6,9,12,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $10$: $D_{5}$ $30$: $D_5\times C_3$ $150$: $(C_5^2 : C_3):C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: None
Low degree siblings
15T30 x 7, 30T188 x 8Siblings 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 $ | $1$ | $1$ | $()$ |
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 3, 6, 9,12,15)$ |
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 3, 9,15, 6,12)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 2, 5, 8,11,14)( 3, 9,15, 6,12)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 2, 5, 8,11,14)( 3,12, 6,15, 9)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 2, 8,14, 5,11)( 3, 6, 9,12,15)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 2, 8,14, 5,11)( 3,15,12, 9, 6)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 2, 8,14, 5,11)( 3,12, 6,15, 9)$ |
$ 5, 5, 5 $ | $2$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,15,12, 9, 6)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3,15,12, 9, 6)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3,12, 6,15, 9)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2,14,11, 8, 5)( 3, 9,15, 6,12)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2,11, 5,14, 8)( 3, 9,15, 6,12)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$ |
$ 5, 5, 5 $ | $2$ | $5$ | $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3,12, 6,15, 9)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $125$ | $2$ | $( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)$ |
$ 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ |
$ 15 $ | $50$ | $15$ | $( 1, 9,14, 4,12, 2, 7,15, 5,10, 3, 8,13, 6,11)$ |
$ 15 $ | $50$ | $15$ | $( 1,12, 2, 7, 3, 8,13, 9,14, 4,15, 5,10, 6,11)$ |
$ 6, 6, 3 $ | $125$ | $6$ | $( 1,15,14,13, 3,11)( 2,10, 6, 8, 4,12)( 5, 7, 9)$ |
$ 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$ |
$ 15 $ | $50$ | $15$ | $( 1,11, 9, 4,14,12, 7, 2,15,10, 5, 3,13, 8, 6)$ |
$ 15 $ | $50$ | $15$ | $( 1,11,12, 7, 2, 3,13, 8, 9, 4,14,15,10, 5, 6)$ |
$ 6, 6, 3 $ | $125$ | $6$ | $( 1, 8, 3, 4, 5, 6)( 2, 9,13,11,15, 7)(10,14,12)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $750=2 \cdot 3 \cdot 5^{3}$ | 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: | 750.30 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);