Show commands:
Magma
magma: G := TransitiveGroup(10, 37);
Group action invariants
| Degree $n$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $37$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $(C_2^4:A_5) : C_2$ | ||
| CHM label: | $[2^{4}]S(5)$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (2,10)(5,7), (1,3,5,7,9)(2,4,6,8,10), (2,7)(5,10) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $120$: $S_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 5: $S_5$
Low degree siblings
10T38, 16T1328, 20T218, 20T219, 20T222, 20T223, 20T226, 30T329, 30T332, 30T333, 30T341, 32T97736, 40T1581, 40T1582, 40T1583, 40T1584, 40T1587, 40T1588, 40T1595, 40T1596, 40T1658, 40T1659, 40T1676, 40T1677, 40T1678Siblings 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 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 1, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $( 4, 9)( 5,10)$ | |
| $ 2, 2, 2, 2, 1, 1 $ | $5$ | $2$ | $( 1, 6)( 2, 7)( 4, 9)( 5,10)$ | |
| $ 2, 2, 1, 1, 1, 1, 1, 1 $ | $20$ | $2$ | $( 4,10)( 5, 9)$ | |
| $ 4, 2, 1, 1, 1, 1 $ | $60$ | $4$ | $( 2, 7)( 4, 5, 9,10)$ | |
| $ 2, 2, 2, 2, 1, 1 $ | $60$ | $2$ | $( 1, 6)( 2, 7)( 4,10)( 5, 9)$ | |
| $ 4, 2, 2, 2 $ | $20$ | $4$ | $( 1, 6)( 2, 7)( 3, 8)( 4, 5, 9,10)$ | |
| $ 3, 3, 1, 1, 1, 1 $ | $80$ | $3$ | $( 3, 9, 5)( 4,10, 8)$ | |
| $ 6, 2, 1, 1 $ | $160$ | $6$ | $( 2, 7)( 3, 9,10, 8, 4, 5)$ | |
| $ 3, 3, 2, 2 $ | $80$ | $6$ | $( 1, 6)( 2, 7)( 3, 9, 5)( 4,10, 8)$ | |
| $ 2, 2, 2, 2, 1, 1 $ | $60$ | $2$ | $( 2, 8)( 3, 7)( 4,10)( 5, 9)$ | |
| $ 4, 4, 1, 1 $ | $60$ | $4$ | $( 2, 8, 7, 3)( 4, 5, 9,10)$ | |
| $ 4, 2, 2, 2 $ | $120$ | $4$ | $( 1, 6)( 2, 8)( 3, 7)( 4, 5, 9,10)$ | |
| $ 4, 4, 1, 1 $ | $240$ | $4$ | $( 2, 8, 4,10)( 3, 9, 5, 7)$ | |
| $ 8, 2 $ | $240$ | $8$ | $( 1, 6)( 2, 8, 4, 5, 7, 3, 9,10)$ | |
| $ 3, 3, 2, 2 $ | $160$ | $6$ | $( 1, 7)( 2, 6)( 3, 9, 5)( 4,10, 8)$ | |
| $ 6, 4 $ | $160$ | $12$ | $( 1, 2, 6, 7)( 3, 9,10, 8, 4, 5)$ | |
| $ 5, 5 $ | $384$ | $5$ | $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $1920=2^{7} \cdot 3 \cdot 5$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 1920.240996 | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| Type |
magma: CharacterTable(G);