Show commands:
Magma
magma: G := TransitiveGroup(10, 1);
Group action invariants
Degree $n$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $1$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{10}$ | ||
CHM label: | $C(10)=5[x]2$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $10$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,2,3,4,5,6,7,8,9,10) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 5: $C_5$
Low degree siblings
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Index | Representative |
1A | $1^{10}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{5}$ | $1$ | $2$ | $5$ | $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ |
5A1 | $5^{2}$ | $1$ | $5$ | $8$ | $( 1, 9, 7, 5, 3)( 2,10, 8, 6, 4)$ |
5A-1 | $5^{2}$ | $1$ | $5$ | $8$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ |
5A2 | $5^{2}$ | $1$ | $5$ | $8$ | $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$ |
5A-2 | $5^{2}$ | $1$ | $5$ | $8$ | $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$ |
10A1 | $10$ | $1$ | $10$ | $9$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$ |
10A-1 | $10$ | $1$ | $10$ | $9$ | $( 1,10, 9, 8, 7, 6, 5, 4, 3, 2)$ |
10A3 | $10$ | $1$ | $10$ | $9$ | $( 1, 4, 7,10, 3, 6, 9, 2, 5, 8)$ |
10A-3 | $10$ | $1$ | $10$ | $9$ | $( 1, 8, 5, 2, 9, 6, 3,10, 7, 4)$ |
Malle's constant $a(G)$: $1/5$
magma: ConjugacyClasses(G);
Group invariants
Order: | $10=2 \cdot 5$ | magma: Order(G);
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Cyclic: | yes | magma: IsCyclic(G);
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Abelian: | yes | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | $1$ | ||
Label: | 10.2 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 5A1 | 5A-1 | 5A2 | 5A-2 | 10A1 | 10A-1 | 10A3 | 10A-3 | ||
Size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
2 P | 1A | 1A | 5A-2 | 5A2 | 5A1 | 5A-1 | 5A1 | 5A-1 | 5A-2 | 5A2 | |
5 P | 1A | 2A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | |
Type | |||||||||||
10.2.1a | R | ||||||||||
10.2.1b | R | ||||||||||
10.2.1c1 | C | ||||||||||
10.2.1c2 | C | ||||||||||
10.2.1c3 | C | ||||||||||
10.2.1c4 | C | ||||||||||
10.2.1d1 | C | ||||||||||
10.2.1d2 | C | ||||||||||
10.2.1d3 | C | ||||||||||
10.2.1d4 | C |
magma: CharacterTable(G);