Show commands:
Magma
magma: G := TransitiveGroup(10, 31);
Group action invariants
| Degree $n$: | $10$ | 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: | $M_{10}$ | ||
| CHM label: | $M(10)=L(10)'2$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | yes | 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: | (1,2)(4,7)(5,8)(9,10), (1,4,2,8)(3,7,6,5), (1,2,10)(3,4,5)(6,7,8), (1,3,2,6)(4,5,8,7) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 5: None
Low degree siblings
12T181, 20T148, 20T150 x 2, 30T162, 36T1253, 40T591, 45T109Siblings 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 $ | $1$ | $1$ | $()$ | |
| $ 4, 4, 1, 1 $ | $180$ | $4$ | $( 3, 4, 6, 5)( 7,10, 8, 9)$ | |
| $ 2, 2, 2, 2, 1, 1 $ | $45$ | $2$ | $( 3, 6)( 4, 5)( 7, 8)( 9,10)$ | |
| $ 4, 4, 1, 1 $ | $90$ | $4$ | $( 3, 9, 6,10)( 4, 8, 5, 7)$ | |
| $ 3, 3, 3, 1 $ | $80$ | $3$ | $( 2, 3, 6)( 4, 9, 7)( 5, 8,10)$ | |
| $ 8, 2 $ | $90$ | $8$ | $( 1, 2)( 3, 4,10, 7, 6, 5, 9, 8)$ | |
| $ 8, 2 $ | $90$ | $8$ | $( 1, 2)( 3, 5,10, 8, 6, 4, 9, 7)$ | |
| $ 5, 5 $ | $144$ | $5$ | $( 1, 2, 3, 4, 9)( 5, 7,10, 6, 8)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $720=2^{4} \cdot 3^{2} \cdot 5$ | 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: | 720.765 | magma: IdentifyGroup(G);
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| Character table: |
| 1A | 2A | 3A | 4A | 4B | 5A | 8A1 | 8A-1 | ||
| Size | 1 | 45 | 80 | 90 | 180 | 144 | 90 | 90 | |
| 2 P | 1A | 1A | 3A | 2A | 2A | 5A | 4A | 4A | |
| 3 P | 1A | 2A | 1A | 4A | 4B | 5A | 8A1 | 8A-1 | |
| 5 P | 1A | 2A | 3A | 4A | 4B | 1A | 8A-1 | 8A1 | |
| Type | |||||||||
| 720.765.1a | R | ||||||||
| 720.765.1b | R | ||||||||
| 720.765.9a | R | ||||||||
| 720.765.9b | R | ||||||||
| 720.765.10a | R | ||||||||
| 720.765.10b1 | C | ||||||||
| 720.765.10b2 | C | ||||||||
| 720.765.16a | R |
magma: CharacterTable(G);