Show commands:
Magma
magma: G := TransitiveGroup(12, 121);
Group action invariants
| Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $121$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_3^2:C_6$ | ||
| CHM label: | $1/2[3^{3}:2]dD(4)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (2,8)(4,10)(6,12), (1,7)(3,9)(5,11), (1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (2,6,10)(3,7,11)(4,8,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $8$: $D_{4}$ $12$: $C_6\times C_2$ $24$: $D_4 \times C_3$ $72$: $C_3^2:D_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $D_{4}$
Degree 6: None
Low degree siblings
12T121, 18T93 x 2, 24T561 x 2, 27T84, 36T258 x 2, 36T259 x 2, 36T260 x 2, 36T292 x 2Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 3, 7,11)( 4,12, 8)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 3,11, 7)( 4, 8,12)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 2, 4)( 6, 8)(10,12)$ |
| $ 6, 3, 1, 1, 1 $ | $12$ | $6$ | $( 2, 4,10,12, 6, 8)( 3, 7,11)$ |
| $ 6, 3, 1, 1, 1 $ | $12$ | $6$ | $( 2, 4, 6, 8,10,12)( 3,11, 7)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 2, 6,10)( 4,12, 8)$ |
| $ 3, 3, 3, 1, 1, 1 $ | $4$ | $3$ | $( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
| $ 3, 3, 3, 1, 1, 1 $ | $4$ | $3$ | $( 2,10, 6)( 3,11, 7)( 4,12, 8)$ |
| $ 6, 2, 2, 2 $ | $12$ | $6$ | $( 1, 2)( 3, 4, 7,12,11, 8)( 5,10)( 6, 9)$ |
| $ 6, 2, 2, 2 $ | $12$ | $6$ | $( 1, 2)( 3, 8,11,12, 7, 4)( 5,10)( 6, 9)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$ |
| $ 12 $ | $18$ | $12$ | $( 1, 2, 3, 4, 5,10, 7,12, 9, 6,11, 8)$ |
| $ 12 $ | $18$ | $12$ | $( 1, 2, 3, 8, 9, 6,11,12, 5,10, 7, 4)$ |
| $ 4, 4, 4 $ | $18$ | $4$ | $( 1, 2, 3,12)( 4, 9, 6,11)( 5,10, 7, 8)$ |
| $ 6, 6 $ | $6$ | $6$ | $( 1, 2, 5,10, 9, 6)( 3, 8, 7, 4,11,12)$ |
| $ 6, 6 $ | $12$ | $6$ | $( 1, 2, 5,10, 9, 6)( 3,12,11, 4, 7, 8)$ |
| $ 6, 6 $ | $6$ | $6$ | $( 1, 2, 9, 6, 5,10)( 3, 4,11, 8, 7,12)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$ |
| $ 6, 6 $ | $9$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4,10,12, 6, 8)$ |
| $ 6, 6 $ | $9$ | $6$ | $( 1, 3, 9,11, 5, 7)( 2, 4, 6, 8,10,12)$ |
| $ 3, 3, 2, 2, 2 $ | $12$ | $6$ | $( 1, 3)( 2, 6,10)( 4,12, 8)( 5, 7)( 9,11)$ |
| $ 6, 3, 3 $ | $6$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 6,10)( 4, 8,12)$ |
| $ 6, 3, 3 $ | $6$ | $6$ | $( 1, 3, 9,11, 5, 7)( 2,10, 6)( 4,12, 8)$ |
| $ 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$ |
| $ 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$ |
| $ 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 9, 5)( 2, 6,10)( 3,11, 7)( 4, 8,12)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $216=2^{3} \cdot 3^{3}$ | 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: | 216.157 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);