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Magma
magma: G := TransitiveGroup(36, 14);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^2:C_4$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $36$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,16,6,30)(2,15,5,29)(3,13,8,32)(4,14,7,31)(9,27,21,19)(10,28,22,20)(11,26,24,18)(12,25,23,17)(33,35,34,36), (1,8,28,21)(2,7,27,22)(3,5,26,23)(4,6,25,24)(9,31,18,33)(10,32,17,34)(11,29,20,36)(12,30,19,35)(13,16,14,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $C_4$
Degree 6: $C_3^2:C_4$ x 2
Degree 9: $C_3^2:C_4$
Degree 12: $(C_3\times C_3):C_4$ x 2
Degree 18: $C_3^2 : C_4$
Low degree siblings
6T10 x 2, 9T9, 12T17 x 2, 18T10Siblings 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, 2, 2 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5,34)( 6,33)( 7,36)( 8,35)( 9,29)(10,30)(11,32)(12,31)(13,27) (14,28)(15,25)(16,26)(17,21)(18,22)(19,24)(20,23)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $9$ | $4$ | $( 1, 3, 2, 4)( 5,18,34,22)( 6,17,33,21)( 7,19,36,24)( 8,20,35,23)( 9,14,29,28) (10,13,30,27)(11,15,32,25)(12,16,31,26)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $9$ | $4$ | $( 1, 4, 2, 3)( 5,22,34,18)( 6,21,33,17)( 7,24,36,19)( 8,23,35,20)( 9,28,29,14) (10,27,30,13)(11,25,32,15)(12,26,31,16)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5,33)( 2, 6,34)( 3, 7,35)( 4, 8,36)( 9,15,17)(10,16,18)(11,13,19) (12,14,20)(21,25,29)(22,26,30)(23,28,31)(24,27,32)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,12,32)( 2,11,31)( 3,10,29)( 4, 9,30)( 5,14,24)( 6,13,23)( 7,16,21) ( 8,15,22)(17,26,36)(18,25,35)(19,28,34)(20,27,33)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $36=2^{2} \cdot 3^{2}$ | 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: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 36.9 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 . . 3 2 . . . 2 2 1a 2a 4a 4b 3a 3b 2P 1a 1a 2a 2a 3a 3b 3P 1a 2a 4b 4a 1a 1a X.1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 X.3 1 -1 A -A 1 1 X.4 1 -1 -A A 1 1 X.5 4 . . . 1 -2 X.6 4 . . . -2 1 A = -E(4) = -Sqrt(-1) = -i |
magma: CharacterTable(G);