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Magma
magma: G := TransitiveGroup(15, 31);
Group action invariants
Degree $n$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $31$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5^2:D_{15}$ | ||
CHM label: | $1/2[5^{3}:2]S(3)$ | ||
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: | (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8), (3,6,9,12,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $10$: $D_{5}$ $30$: $D_{15}$ $150$: $(C_5^2 : C_3):C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 5: None
Low degree siblings
15T31 x 3, 30T186 x 4, 30T187 x 2Siblings 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, 5, 5 $ | $6$ | $5$ | $( 1,13,10, 7, 4)( 2, 8,14, 5,11)( 3,15,12, 9, 6)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1,10, 4,13, 7)( 2,14,11, 8, 5)( 3,12, 6,15, 9)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 2,11, 5,14, 8)( 3, 9,15, 6,12)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 1,13,10, 7, 4)( 3, 6, 9,12,15)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 1,10, 4,13, 7)( 3, 9,15, 6,12)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 1, 4, 7,10,13)( 3,15,12, 9, 6)$ |
$ 3, 3, 3, 3, 3 $ | $50$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ |
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 3, 6, 9,12,15)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 1,13,10, 7, 4)( 2, 8,14, 5,11)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1,10, 4,13, 7)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2,11, 5,14, 8)( 3, 9,15, 6,12)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 2,11, 5,14, 8)( 3,12, 6,15, 9)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 1,13,10, 7, 4)( 3, 9,15, 6,12)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,15,12, 9, 6)$ |
$ 5, 5, 5 $ | $2$ | $5$ | $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1,10, 4,13, 7)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$ |
$ 15 $ | $50$ | $15$ | $( 1, 6,14, 4, 9, 2, 7,12, 5,10,15, 8,13, 3,11)$ |
$ 15 $ | $50$ | $15$ | $( 1,11, 6, 4,14, 9, 7, 2,12,10, 5,15,13, 8, 3)$ |
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 3, 9,15, 6,12)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1,13,10, 7, 4)( 2, 8,14, 5,11)( 3, 6, 9,12,15)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 1,10, 4,13, 7)( 2,14,11, 8, 5)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1,10, 4,13, 7)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)$ |
$ 5, 5, 5 $ | $6$ | $5$ | $( 1, 7,13, 4,10)( 2,14,11, 8, 5)( 3, 6, 9,12,15)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $6$ | $5$ | $( 1,13,10, 7, 4)( 2,11, 5,14, 8)$ |
$ 5, 5, 5 $ | $2$ | $5$ | $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$ |
$ 15 $ | $50$ | $15$ | $( 1, 6, 2, 7,12, 8,13, 3,14, 4, 9, 5,10,15,11)$ |
$ 15 $ | $50$ | $15$ | $( 1,11, 6, 7, 2,12,13, 8, 3, 4,14, 9,10, 5,15)$ |
$ 10, 2, 2, 1 $ | $75$ | $10$ | $( 2,15,11, 6, 5,12,14, 3, 8, 9)( 4,13)( 7,10)$ |
$ 10, 2, 2, 1 $ | $75$ | $10$ | $( 1,13)( 2,12, 5, 9, 8, 6,11, 3,14,15)( 4,10)$ |
$ 10, 2, 2, 1 $ | $75$ | $10$ | $( 1,10)( 2, 9,14,12,11,15, 8, 3, 5, 6)( 4, 7)$ |
$ 2, 2, 2, 2, 2, 2, 2, 1 $ | $75$ | $2$ | $( 1, 4)( 2, 3)( 5,15)( 6,14)( 7,13)( 8,12)( 9,11)$ |
$ 10, 2, 2, 1 $ | $75$ | $10$ | $( 1, 7)( 2, 6, 8,15,14, 9, 5, 3,11,12)(10,13)$ |
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: | 750.27 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);