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Magma
magma: G := TransitiveGroup(36, 555);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $555$ | 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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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,13,36)(2,16,33)(3,17,34)(4,15,31)(5,18,32)(6,14,35)(7,26,19)(8,29,23)(9,30,20)(10,28,21)(11,25,22)(12,27,24), (1,4,2)(3,5,6)(7,9,10)(8,11,12)(13,34,22)(14,35,21)(15,36,24)(16,31,20)(17,32,19)(18,33,23)(25,30,29)(26,28,27) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: None
Degree 4: None
Degree 6: $A_6$ x 2
Degree 9: None
Degree 12: None
Degree 18: None
Low degree siblings
6T15 x 2, 10T26, 15T20 x 2, 20T89, 30T88 x 2, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $45$ | $2$ | $( 3, 5)( 4, 6)( 7,36)( 8,32)( 9,35)(10,33)(11,34)(12,31)(13,28)(14,27)(15,30) (16,26)(17,25)(18,29)(19,21)(22,23)$ |
$ 5, 5, 5, 5, 5, 5, 5, 1 $ | $72$ | $5$ | $( 2, 3, 6, 4, 5)( 7,29,13,31,22)( 8,30,15,32,20)( 9,26,16,35,24) (10,25,14,34,19)(11,27,17,33,21)(12,28,18,36,23)$ |
$ 5, 5, 5, 5, 5, 5, 5, 1 $ | $72$ | $5$ | $( 2, 4, 3, 5, 6)( 7,31,29,22,13)( 8,32,30,20,15)( 9,35,26,24,16) (10,34,25,19,14)(11,33,27,21,17)(12,36,28,23,18)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $40$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7,23,16)( 8,24,14)( 9,20,17)(10,19,18)(11,21,13) (12,22,15)(25,28,30)(26,27,29)(31,33,32)(34,35,36)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 2, 2 $ | $90$ | $4$ | $( 1, 7,17,31)( 2, 8,15,35)( 3,10,16,36)( 4, 9,14,34)( 5,12,18,33)( 6,11,13,32) (19,25)(20,29,21,26)(22,30)(23,28,24,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $40$ | $3$ | $( 1, 7,28)( 2,10,26)( 3, 8,29)( 4, 9,27)( 5,11,25)( 6,12,30)(13,33,19) (14,31,24)(15,35,20)(16,36,21)(17,34,23)(18,32,22)$ |
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 5a 5b 3a 4a 3b 2P 1a 1a 5b 5a 3a 2a 3b 3P 1a 2a 5b 5a 1a 4a 1a 5P 1a 2a 1a 1a 3a 4a 3b X.1 1 1 1 1 1 1 1 X.2 5 1 . . 2 -1 -1 X.3 5 1 . . -1 -1 2 X.4 8 . A *A -1 . -1 X.5 8 . *A A -1 . -1 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);