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Magma
magma: G := TransitiveGroup(12, 115);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $115$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_4^2:D_6$ | ||
CHM label: | $[2^{3}]S_{4}(6)_{8}$ | ||
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: | (1,12)(2,3)(6,7)(8,9), (1,12)(2,3)(4,5), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (2,4)(3,5)(6,7)(8,10)(9,11) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ $24$: $S_4$ $48$: $S_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 4: None
Degree 6: $S_4$
Low degree siblings
12T112 x 2, 12T113 x 2, 12T114 x 2, 12T115, 16T431, 24T530 x 2, 24T531 x 2, 24T532 x 2, 24T533 x 2, 24T534 x 2, 24T535, 24T536 x 2, 24T537 x 2, 24T538 x 2, 24T539, 24T540, 24T541, 32T2145, 32T2146 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, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $(4,5)(6,7)(8,9)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 4, 5)( 8, 9)(10,11)$ |
$ 2, 2, 2, 2, 2, 1, 1 $ | $12$ | $2$ | $( 2, 4)( 3, 5)( 6, 7)( 8,10)( 9,11)$ |
$ 4, 4, 1, 1, 1, 1 $ | $12$ | $4$ | $( 2, 4, 3, 5)( 8,11, 9,10)$ |
$ 4, 4, 1, 1, 1, 1 $ | $6$ | $4$ | $( 2, 8, 3, 9)( 4,10, 5,11)$ |
$ 2, 2, 2, 2, 2, 1, 1 $ | $12$ | $2$ | $( 2, 8)( 3, 9)( 4,11)( 5,10)( 6, 7)$ |
$ 4, 4, 2, 1, 1 $ | $12$ | $4$ | $( 2,10, 3,11)( 4, 8, 5, 9)( 6, 7)$ |
$ 2, 2, 2, 2, 2, 2 $ | $12$ | $2$ | $( 1, 2)( 3,12)( 4, 5)( 6, 8)( 7, 9)(10,11)$ |
$ 3, 3, 3, 3 $ | $32$ | $3$ | $( 1, 2, 4)( 3, 5,12)( 6, 8,11)( 7, 9,10)$ |
$ 6, 3, 3 $ | $32$ | $6$ | $( 1, 2, 4,12, 3, 5)( 6, 9,11)( 7, 8,10)$ |
$ 8, 2, 2 $ | $24$ | $8$ | $( 1, 2, 6, 9,12, 3, 7, 8)( 4,10)( 5,11)$ |
$ 8, 4 $ | $24$ | $8$ | $( 1, 2, 6, 8,12, 3, 7, 9)( 4,11, 5,10)$ |
$ 4, 4, 2, 2 $ | $6$ | $4$ | $( 1, 6,12, 7)( 2, 3)( 4,11, 5,10)( 8, 9)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $192=2^{6} \cdot 3$ | 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: | 192.956 | magma: IdentifyGroup(G);
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Character table: |
2 6 4 6 4 4 5 4 4 4 1 1 3 3 5 3 1 1 . . . . . . . 1 1 . . . 1a 2a 2b 2c 4a 4b 2d 4c 2e 3a 6a 8a 8b 4d 2P 1a 1a 1a 1a 2b 2b 1a 2b 1a 3a 3a 4b 4d 2b 3P 1a 2a 2b 2c 4a 4b 2d 4c 2e 1a 2a 8a 8b 4d 5P 1a 2a 2b 2c 4a 4b 2d 4c 2e 3a 6a 8a 8b 4d 7P 1a 2a 2b 2c 4a 4b 2d 4c 2e 3a 6a 8a 8b 4d X.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 X.3 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 X.5 2 -2 2 . . 2 -2 . . -1 1 . . 2 X.6 2 2 2 . . 2 2 . . -1 -1 . . 2 X.7 3 -3 3 -1 1 -1 1 -1 1 . . 1 -1 -1 X.8 3 -3 3 1 -1 -1 1 1 -1 . . -1 1 -1 X.9 3 3 3 -1 -1 -1 -1 -1 -1 . . 1 1 -1 X.10 3 3 3 1 1 -1 -1 1 1 . . -1 -1 -1 X.11 6 . -2 -2 . -2 . 2 . . . . . 2 X.12 6 . -2 . -2 2 . . 2 . . . . -2 X.13 6 . -2 . 2 2 . . -2 . . . . -2 X.14 6 . -2 2 . -2 . -2 . . . . . 2 |
magma: CharacterTable(G);