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Magma
magma: G := TransitiveGroup(10, 11);
Group action invariants
Degree $n$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $11$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $A_5\times C_2$ | ||
CHM label: | $A(5)[x]2$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (2,4,10)(5,7,9), (1,6)(2,7)(3,8)(4,9)(5,10), (1,3,5,7,9)(2,4,6,8,10) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $60$: $A_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 5: $A_5$
Low degree siblings
12T75, 12T76, 20T31, 20T36, 24T203, 30T29, 30T30, 40T61Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 3, 3, 1, 1, 1, 1 $ | $20$ | $3$ | $( 3, 5, 9)( 4, 8,10)$ | |
$ 2, 2, 2, 2, 1, 1 $ | $15$ | $2$ | $( 2, 4)( 3, 5)( 7, 9)( 8,10)$ | |
$ 2, 2, 2, 2, 2 $ | $15$ | $2$ | $( 1, 2)( 3, 4)( 5,10)( 6, 7)( 8, 9)$ | |
$ 10 $ | $12$ | $10$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$ | |
$ 6, 2, 2 $ | $20$ | $6$ | $( 1, 2, 3, 6, 7, 8)( 4, 9)( 5,10)$ | |
$ 10 $ | $12$ | $10$ | $( 1, 2, 3,10, 9, 6, 7, 8, 5, 4)$ | |
$ 5, 5 $ | $12$ | $5$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ | |
$ 5, 5 $ | $12$ | $5$ | $( 1, 3, 5, 9, 7)( 2, 6, 8,10, 4)$ | |
$ 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $120=2^{3} \cdot 3 \cdot 5$ | 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: | 120.35 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A | 5A1 | 5A2 | 6A | 10A1 | 10A3 | ||
Size | 1 | 1 | 15 | 15 | 20 | 12 | 12 | 20 | 12 | 12 | |
2 P | 1A | 1A | 1A | 1A | 3A | 5A2 | 5A1 | 3A | 5A1 | 5A2 | |
3 P | 1A | 2A | 2B | 2C | 1A | 5A2 | 5A1 | 2A | 10A3 | 10A1 | |
5 P | 1A | 2A | 2B | 2C | 3A | 1A | 1A | 6A | 2A | 2A | |
Type | |||||||||||
120.35.1a | R | ||||||||||
120.35.1b | R | ||||||||||
120.35.3a1 | R | ||||||||||
120.35.3a2 | R | ||||||||||
120.35.3b1 | R | ||||||||||
120.35.3b2 | R | ||||||||||
120.35.4a | R | ||||||||||
120.35.4b | R | ||||||||||
120.35.5a | R | ||||||||||
120.35.5b | R |
magma: CharacterTable(G);