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Magma
magma: G := TransitiveGroup(20, 89);
Group action invariants
Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $89$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $A_6$ | ||
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: | (1,5,14)(2,6,13)(3,15,18)(4,16,17)(7,12,20)(8,11,19), (1,19,6)(2,20,5)(3,17,12)(4,18,11)(9,16,14)(10,15,13) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 5: None
Degree 10: $\PSL(2,9)$
Low degree siblings
6T15 x 2, 10T26, 15T20 x 2, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $45$ | $2$ | $( 5,11)( 6,12)( 7, 9)( 8,10)(13,15)(14,16)(17,19)(18,20)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1 $ | $40$ | $3$ | $( 3, 5,11)( 4, 6,12)( 7,17,13)( 8,18,14)( 9,15,19)(10,16,20)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1 $ | $40$ | $3$ | $( 3, 8,10)( 4, 7, 9)( 5,18,16)( 6,17,15)(11,14,20)(12,13,19)$ |
$ 4, 4, 4, 4, 2, 2 $ | $90$ | $4$ | $( 1, 2)( 3, 4)( 5,17,11,19)( 6,18,12,20)( 7,16, 9,14)( 8,15,10,13)$ |
$ 5, 5, 5, 5 $ | $72$ | $5$ | $( 1, 3, 7,17,16)( 2, 4, 8,18,15)( 5,13,10,11,20)( 6,14, 9,12,19)$ |
$ 5, 5, 5, 5 $ | $72$ | $5$ | $( 1, 3, 7,12,14)( 2, 4, 8,11,13)( 5,20,15,10,18)( 6,19,16, 9,17)$ |
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 3a 3b 4a 5a 5b 2P 1a 1a 3a 3b 2a 5b 5a 3P 1a 2a 1a 1a 4a 5b 5a 5P 1a 2a 3a 3b 4a 1a 1a X.1 1 1 1 1 1 1 1 X.2 5 1 2 -1 -1 . . X.3 5 1 -1 2 -1 . . 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);