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Magma
magma: G := TransitiveGroup(12, 119);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $119$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^2:C_4\times S_3$ | ||
CHM label: | $[3^{3}:2]4$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,5)(2,10)(4,8)(7,11), (2,6,10)(3,7,11)(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, $C_2^2$ $6$: $S_3$ $8$: $C_4\times C_2$ $12$: $D_{6}$ $24$: $S_3 \times C_4$ $36$: $C_3^2:C_4$ $72$: 12T40 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $C_4$
Degree 6: None
Low degree siblings
12T119, 18T95 x 2, 24T559 x 2, 27T85, 36T261 x 2, 36T262 x 2, 36T263 x 2, 36T295 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 $ | $27$ | $2$ | $( 4, 8)( 5, 9)( 6,10)( 7,11)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 3, 7,11)( 4,12, 8)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 2, 6,10)( 4,12, 8)$ |
$ 3, 3, 3, 1, 1, 1 $ | $8$ | $3$ | $( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
$ 4, 4, 4 $ | $27$ | $4$ | $( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10,11,12)$ |
$ 12 $ | $18$ | $12$ | $( 1, 2, 3, 4, 5,10, 7,12, 9, 6,11, 8)$ |
$ 4, 4, 4 $ | $9$ | $4$ | $( 1, 2, 3,12)( 4, 9, 6,11)( 5,10, 7, 8)$ |
$ 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$ |
$ 6, 2, 2, 2 $ | $12$ | $6$ | $( 1, 3)( 2, 4, 6,12,10, 8)( 5,11)( 7, 9)$ |
$ 6, 6 $ | $18$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4,10,12, 6, 8)$ |
$ 6, 6 $ | $12$ | $6$ | $( 1, 3, 9, 7, 5,11)( 2, 4,10, 8, 6,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 3)( 2,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$ |
$ 4, 4, 4 $ | $27$ | $4$ | $( 1, 4, 3, 2)( 5, 8, 7, 6)( 9,12,11,10)$ |
$ 12 $ | $18$ | $12$ | $( 1, 4, 7,10, 5,12,11, 6, 9, 8, 3, 2)$ |
$ 4, 4, 4 $ | $9$ | $4$ | $( 1, 4,11, 2)( 3,10, 5,12)( 6, 9, 8, 7)$ |
$ 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$ |
$ 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$ |
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: | 216.156 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 . 1 . 3 2 3 3 1 2 1 3 3 2 3 1 2 3 3 . 3 3 3 . 1 1 1 2 1 2 2 . 1 1 3 3 1a 2a 3a 3b 3c 4a 12a 4b 2b 6a 6b 6c 2c 4c 12b 4d 3d 3e 2P 1a 1a 3a 3b 3c 2b 6b 2b 1a 3b 3e 3d 1a 2b 6b 2b 3d 3e 3P 1a 2a 1a 1a 1a 4c 4d 4d 2b 2c 2b 2c 2c 4a 4b 4b 1a 1a 5P 1a 2a 3a 3b 3c 4a 12a 4b 2b 6a 6b 6c 2c 4c 12b 4d 3d 3e 7P 1a 2a 3a 3b 3c 4c 12b 4d 2b 6a 6b 6c 2c 4a 12a 4b 3d 3e 11P 1a 2a 3a 3b 3c 4c 12b 4d 2b 6a 6b 6c 2c 4a 12a 4b 3d 3e 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 A -A -A -1 1 -1 1 1 -A A A 1 1 X.6 1 -1 1 1 1 -A A A -1 1 -1 1 1 A -A -A 1 1 X.7 1 1 1 1 1 A A A -1 -1 -1 -1 -1 -A -A -A 1 1 X.8 1 1 1 1 1 -A -A -A -1 -1 -1 -1 -1 A A A 1 1 X.9 2 . -1 2 -1 . -1 2 2 . -1 . . . -1 2 2 -1 X.10 2 . -1 2 -1 . 1 -2 2 . -1 . . . 1 -2 2 -1 X.11 2 . -1 2 -1 . A B -2 . 1 . . . -A -B 2 -1 X.12 2 . -1 2 -1 . -A -B -2 . 1 . . . A B 2 -1 X.13 4 . -2 1 1 . . . . -1 . 2 -4 . . . -2 4 X.14 4 . -2 1 1 . . . . 1 . -2 4 . . . -2 4 X.15 4 . 1 -2 -2 . . . . -2 . 1 4 . . . 1 4 X.16 4 . 1 -2 -2 . . . . 2 . -1 -4 . . . 1 4 X.17 8 . 2 2 -1 . . . . . . . . . . . -4 -4 X.18 8 . -1 -4 2 . . . . . . . . . . . 2 -4 A = -E(4) = -Sqrt(-1) = -i B = 2*E(4) = 2*Sqrt(-1) = 2i |
magma: CharacterTable(G);