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Magma
magma: G := TransitiveGroup(10, 26);
Group action invariants
Degree $n$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\PSL(2,9)$ | ||
CHM label: | $L(10)=PSL(2,9)$ | ||
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), (1,2,10)(3,4,5)(6,7,8), (1,3,2,6)(4,5,8,7) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 5: None
Low degree siblings
6T15 x 2, 15T20 x 2, 20T89, 30T88 x 2, 36T555, 40T304, 45T49Siblings 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 $ | $1$ | $1$ | $()$ |
$ 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 $ | $40$ | $3$ | $( 2, 3, 6)( 4, 9, 7)( 5, 8,10)$ |
$ 3, 3, 3, 1 $ | $40$ | $3$ | $( 2, 4, 5)( 3, 9, 8)( 6, 7,10)$ |
$ 5, 5 $ | $72$ | $5$ | $( 1, 2, 3, 4, 9)( 5, 7,10, 6, 8)$ |
$ 5, 5 $ | $72$ | $5$ | $( 1, 2, 3, 8,10)( 4, 7, 5, 9, 6)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $360=2^{3} \cdot 3^{2} \cdot 5$ | 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: | 360.118 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 2 . . . . 3 2 . . 2 2 . . 5 1 . . . . 1 1 1a 2a 4a 3a 3b 5a 5b 2P 1a 1a 2a 3a 3b 5b 5a 3P 1a 2a 4a 1a 1a 5b 5a 5P 1a 2a 4a 3a 3b 1a 1a X.1 1 1 1 1 1 1 1 X.2 5 1 -1 2 -1 . . X.3 5 1 -1 -1 2 . . X.4 8 . . -1 -1 A *A X.5 8 . . -1 -1 *A A X.6 9 1 1 . . -1 -1 X.7 10 -2 . 1 1 . . A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 |
magma: CharacterTable(G);