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Magma
magma: G := TransitiveGroup(36, 12);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3\times A_4$ | ||
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: | (1,33,15)(2,34,16)(3,35,13)(4,36,14)(5,28,19)(6,27,20)(7,26,18)(8,25,17)(9,29,21)(10,30,22)(11,32,23)(12,31,24), (1,28,23)(2,27,24)(3,26,21)(4,25,22)(5,29,14)(6,30,13)(7,32,16)(8,31,15)(9,34,17)(10,33,18)(11,36,20)(12,35,19) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $9$: $C_3^2$ $12$: $A_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$ x 4
Degree 4: $A_4$
Degree 6: $A_4$
Degree 9: $C_3^2$
Degree 12: $A_4$, $C_3\times A_4$ x 3
Degree 18: $A_4 \times C_3$
Low degree siblings
12T20 x 3, 18T8Siblings 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 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23) (22,24)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)$ |
$ 6, 6, 6, 6, 6, 6 $ | $3$ | $6$ | $( 1, 5, 9, 2, 6,10)( 3, 7,11, 4, 8,12)(13,20,23,15,17,21)(14,19,24,16,18,22) (25,30,35,28,32,34)(26,29,36,27,31,33)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 6, 9)( 2, 5,10)( 3, 8,11)( 4, 7,12)(13,17,23)(14,18,24)(15,20,21) (16,19,22)(25,32,35)(26,31,36)(27,29,33)(28,30,34)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 9, 6)( 2,10, 5)( 3,11, 8)( 4,12, 7)(13,23,17)(14,24,18)(15,21,20) (16,22,19)(25,35,32)(26,36,31)(27,33,29)(28,34,30)$ |
$ 6, 6, 6, 6, 6, 6 $ | $3$ | $6$ | $( 1,10, 6, 2, 9, 5)( 3,12, 8, 4,11, 7)(13,21,17,15,23,20)(14,22,18,16,24,19) (25,34,32,28,35,30)(26,33,31,27,36,29)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,13,34)( 2,14,33)( 3,15,36)( 4,16,35)( 5,18,27)( 6,17,28)( 7,19,25) ( 8,20,26)( 9,23,30)(10,24,29)(11,21,31)(12,22,32)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,17,30)( 2,18,29)( 3,20,31)( 4,19,32)( 5,24,33)( 6,23,34)( 7,22,35) ( 8,21,36)( 9,13,28)(10,14,27)(11,15,26)(12,16,25)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,21,27)( 2,22,28)( 3,23,25)( 4,24,26)( 5,16,30)( 6,15,29)( 7,14,31) ( 8,13,32)( 9,20,33)(10,19,34)(11,17,35)(12,18,36)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,25,24)( 2,26,23)( 3,27,22)( 4,28,21)( 5,31,13)( 6,32,14)( 7,30,15) ( 8,29,16)( 9,35,18)(10,36,17)(11,33,19)(12,34,20)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,29,20)( 2,30,19)( 3,32,17)( 4,31,18)( 5,34,22)( 6,33,21)( 7,36,24) ( 8,35,23)( 9,27,15)(10,28,16)(11,25,13)(12,26,14)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,33,15)( 2,34,16)( 3,35,13)( 4,36,14)( 5,28,19)( 6,27,20)( 7,26,18) ( 8,25,17)( 9,29,21)(10,30,22)(11,32,23)(12,31,24)$ |
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: | 36.11 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 2 2 . . . . . . 3 2 1 1 2 2 1 2 2 2 2 2 2 1a 2a 6a 3a 3b 6b 3c 3d 3e 3f 3g 3h 2P 1a 1a 3b 3b 3a 3a 3h 3g 3f 3e 3d 3c 3P 1a 2a 2a 1a 1a 2a 1a 1a 1a 1a 1a 1a 5P 1a 2a 6b 3b 3a 6a 3h 3g 3f 3e 3d 3c X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 A A A /A /A /A X.3 1 1 1 1 1 1 /A /A /A A A A X.4 1 1 A A /A /A 1 A /A A /A 1 X.5 1 1 /A /A A A 1 /A A /A A 1 X.6 1 1 A A /A /A A /A 1 1 A /A X.7 1 1 /A /A A A /A A 1 1 /A A X.8 1 1 A A /A /A /A 1 A /A 1 A X.9 1 1 /A /A A A A 1 /A A 1 /A X.10 3 -1 -1 3 3 -1 . . . . . . X.11 3 -1 -/A B /B -A . . . . . . X.12 3 -1 -A /B B -/A . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 3*E(3) = (-3+3*Sqrt(-3))/2 = 3b3 |
magma: CharacterTable(G);