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Magma
magma: G := TransitiveGroup(12, 39);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_6.D_6$ | ||
CHM label: | $[3^{2}:2]4$ | ||
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: | (1,5)(2,10)(4,8)(7,11), (2,6,10)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$ x 2 $8$: $C_4\times C_2$ $12$: $D_{6}$ x 2 $24$: $S_3 \times C_4$ x 2 $36$: $S_3^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $C_4$
Degree 6: $S_3^2$
Low degree siblings
12T39, 24T75, 36T32 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 $ | $1$ | $1$ | $()$ |
$ 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 $ | $4$ | $3$ | $( 2, 6,10)( 4, 8,12)$ |
$ 12 $ | $6$ | $12$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$ |
$ 12 $ | $6$ | $12$ | $( 1, 2, 3,12, 5,10, 7, 8, 9, 6,11, 4)$ |
$ 4, 4, 4 $ | $3$ | $4$ | $( 1, 2, 7, 8)( 3, 4, 9,10)( 5, 6,11,12)$ |
$ 4, 4, 4 $ | $3$ | $4$ | $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$ |
$ 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 $ | $2$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2,12,10, 8, 6, 4)$ |
$ 12 $ | $6$ | $12$ | $( 1, 4,11, 6, 9, 8, 7,10, 5,12, 3, 2)$ |
$ 12 $ | $6$ | $12$ | $( 1, 4, 3, 6, 5, 8, 7,10, 9,12,11, 2)$ |
$ 4, 4, 4 $ | $3$ | $4$ | $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$ |
$ 4, 4, 4 $ | $3$ | $4$ | $( 1, 4, 7,10)( 2, 9, 8, 3)( 5,12,11, 6)$ |
$ 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: | 72.21 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 1 2 2 3 3 3 2 1 2 2 2 3 3 2 2 3 3 2 . 2 1 1 1 1 . 2 2 2 1 1 1 1 2 2 2 1a 2a 3a 12a 12b 4a 4b 2b 6a 6b 6c 12c 12d 4c 4d 3b 3c 2c 2P 1a 1a 3a 6a 6c 2c 2c 1a 3b 3a 3c 6c 6a 2c 2c 3b 3c 1a 3P 1a 2a 1a 4c 4d 4c 4d 2b 2c 2c 2c 4b 4a 4a 4b 1a 1a 2c 5P 1a 2a 3a 12a 12b 4a 4b 2b 6a 6b 6c 12c 12d 4c 4d 3b 3c 2c 7P 1a 2a 3a 12d 12c 4c 4d 2b 6a 6b 6c 12b 12a 4a 4b 3b 3c 2c 11P 1a 2a 3a 12d 12c 4c 4d 2b 6a 6b 6c 12b 12a 4a 4b 3b 3c 2c 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 A -A A -A 1 -1 -1 -1 A -A -A A 1 1 -1 X.6 1 -1 1 -A A -A A 1 -1 -1 -1 -A A A -A 1 1 -1 X.7 1 1 1 A A A A -1 -1 -1 -1 -A -A -A -A 1 1 -1 X.8 1 1 1 -A -A -A -A -1 -1 -1 -1 A A A A 1 1 -1 X.9 2 . -1 . -1 . 2 . 2 -1 -1 -1 . . 2 2 -1 2 X.10 2 . -1 . 1 . -2 . 2 -1 -1 1 . . -2 2 -1 2 X.11 2 . -1 -1 . 2 . . -1 -1 2 . -1 2 . -1 2 2 X.12 2 . -1 1 . -2 . . -1 -1 2 . 1 -2 . -1 2 2 X.13 2 . -1 . A . B . -2 1 1 -A . . -B 2 -1 -2 X.14 2 . -1 . -A . -B . -2 1 1 A . . B 2 -1 -2 X.15 2 . -1 A . B . . 1 1 -2 . -A -B . -1 2 -2 X.16 2 . -1 -A . -B . . 1 1 -2 . A B . -1 2 -2 X.17 4 . 1 . . . . . -2 1 -2 . . . . -2 -2 4 X.18 4 . 1 . . . . . 2 -1 2 . . . . -2 -2 -4 A = -E(4) = -Sqrt(-1) = -i B = 2*E(4) = 2*Sqrt(-1) = 2i |
magma: CharacterTable(G);