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Magma
magma: G := TransitiveGroup(12, 37);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $37$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_3\times D_6$ | ||
CHM label: | $[3^{2}:2]E(4)$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (1,5)(2,10)(4,8)(7,11), (2,6,10)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $6$: $S_3$ x 2 $8$: $C_2^3$ $12$: $D_{6}$ x 6 $24$: $S_3 \times C_2^2$ x 2 $36$: $S_3^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: None
Degree 4: $C_2^2$
Degree 6: $S_3^2$
Low degree siblings
12T37, 18T29 x 4, 24T73, 36T34 x 2, 36T40 x 4Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3,11)( 4,12)( 5, 9)( 6,10)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 2, 6,10)( 4, 8,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$ |
$ 6, 6 $ | $6$ | $6$ | $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$ |
$ 6, 6 $ | $6$ | $6$ | $( 1, 2, 5,10, 9, 6)( 3,12, 7, 8,11, 4)$ |
$ 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 3)( 2, 4)( 5,11)( 6,12)( 7, 9)( 8,10)$ |
$ 6, 6 $ | $2$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ |
$ 6, 2, 2, 2 $ | $4$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 8)( 4,10)( 6,12)$ |
$ 6, 6 $ | $2$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2,12,10, 8, 6, 4)$ |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)$ |
$ 6, 6 $ | $6$ | $6$ | $( 1, 4, 5, 8, 9,12)( 2, 3, 6, 7,10,11)$ |
$ 6, 6 $ | $6$ | $6$ | $( 1, 4, 5,12, 9, 8)( 2, 7,10,11, 6, 3)$ |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ |
$ 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
$ 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | 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: | 72.46 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 1 3 3 2 2 3 2 1 2 3 2 2 3 2 2 3 3 2 . 2 1 1 1 1 . 2 2 2 1 1 1 1 2 2 2 1a 2a 3a 2b 2c 6a 6b 2d 6c 6d 6e 2e 6f 6g 2f 3b 3c 2g 2P 1a 1a 3a 1a 1a 3b 3c 1a 3b 3a 3c 1a 3b 3c 1a 3b 3c 1a 3P 1a 2a 1a 2b 2c 2b 2c 2d 2g 2g 2g 2e 2f 2e 2f 1a 1a 2g 5P 1a 2a 3a 2b 2c 6a 6b 2d 6c 6d 6e 2e 6f 6g 2f 3b 3c 2g X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 -1 1 -1 1 1 1 1 -1 1 -1 1 1 1 X.3 1 -1 1 -1 1 -1 1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 X.4 1 -1 1 1 -1 1 -1 -1 1 1 1 -1 1 -1 1 1 1 1 X.5 1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 1 -1 X.6 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 X.7 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 X.8 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 X.9 2 . -1 . -2 . 1 . -2 1 1 2 . -1 . 2 -1 -2 X.10 2 . -1 . -2 . 1 . 2 -1 -1 -2 . 1 . 2 -1 2 X.11 2 . -1 . 2 . -1 . -2 1 1 -2 . 1 . 2 -1 -2 X.12 2 . -1 . 2 . -1 . 2 -1 -1 2 . -1 . 2 -1 2 X.13 2 . -1 -2 . 1 . . -1 -1 2 . 1 . -2 -1 2 2 X.14 2 . -1 -2 . 1 . . 1 1 -2 . -1 . 2 -1 2 -2 X.15 2 . -1 2 . -1 . . -1 -1 2 . -1 . 2 -1 2 2 X.16 2 . -1 2 . -1 . . 1 1 -2 . 1 . -2 -1 2 -2 X.17 4 . 1 . . . . . -2 1 -2 . . . . -2 -2 4 X.18 4 . 1 . . . . . 2 -1 2 . . . . -2 -2 -4 |
magma: CharacterTable(G);