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Magma
magma: G := TransitiveGroup(40, 304);
Group action invariants
Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $304$ | 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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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,22,34,7,27)(2,21,33,8,28)(3,23,36,5,26)(4,24,35,6,25)(9,32,38,19,15)(10,31,37,20,16)(11,30,39,18,13)(12,29,40,17,14), (1,3,2,4)(5,14,10,29)(6,13,9,30)(7,16,11,31)(8,15,12,32)(17,40,34,27)(18,39,33,28)(19,38,36,26)(20,37,35,25)(21,23,22,24) | 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 8: None
Degree 10: $\PSL(2,9)$
Degree 20: $A_6$
Low degree siblings
6T15 x 2, 10T26, 15T20 x 2, 20T89, 30T88 x 2, 36T555, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $40$ | $3$ | $( 5, 9,21)( 6,10,22)( 7,12,23)( 8,11,24)(13,33,25)(14,34,26)(15,36,28) (16,35,27)(17,29,38)(18,30,37)(19,32,39)(20,31,40)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $40$ | $3$ | $( 5,15,20)( 6,16,19)( 7,14,18)( 8,13,17)( 9,36,31)(10,35,32)(11,33,29) (12,34,30)(21,28,40)(22,27,39)(23,26,37)(24,25,38)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $45$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,22)(10,21)(11,23)(12,24)(13,18)(14,17)(15,19) (16,20)(25,30)(26,29)(27,31)(28,32)(33,37)(34,38)(35,40)(36,39)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $90$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,34,22,38)(10,33,21,37)(11,36,23,39)(12,35,24,40) (13,31,18,27)(14,32,17,28)(15,30,19,25)(16,29,20,26)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5 $ | $72$ | $5$ | $( 1, 5,13,33,31)( 2, 6,14,34,32)( 3, 8,16,35,29)( 4, 7,15,36,30) ( 9,26,20,22,39)(10,25,19,21,40)(11,28,17,23,37)(12,27,18,24,38)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5 $ | $72$ | $5$ | $( 1, 5,26,34,20)( 2, 6,25,33,19)( 3, 8,28,36,17)( 4, 7,27,35,18) ( 9,40,22,14,32)(10,39,21,13,31)(11,38,23,15,30)(12,37,24,16,29)$ |
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 3a 3b 2a 4a 5a 5b 2P 1a 3a 3b 1a 2a 5b 5a 3P 1a 1a 1a 2a 4a 5b 5a 5P 1a 3a 3b 2a 4a 1a 1a X.1 1 1 1 1 1 1 1 X.2 5 2 -1 1 -1 . . X.3 5 -1 2 1 -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 1 1 -2 . . . A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 |
magma: CharacterTable(G);