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Magma
magma: G := TransitiveGroup(12, 50);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $50$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\GL(2,\mathbb{Z}/4)$ | ||
CHM label: | $1/2e[1/16.D(4)^{3}]S(3)$ | ||
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: | (3,9)(6,12), (1,7)(3,9)(5,11), (1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (1,5,9)(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 $4$: $C_2^2$ $6$: $S_3$ $8$: $D_{4}$ $12$: $D_{6}$ $24$: $S_4$, $(C_6\times C_2):C_2$ $48$: $S_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: None
Degree 6: $S_3$
Low degree siblings
12T49 x 2, 12T52, 16T186, 16T193, 24T153, 24T154, 24T155 x 2, 24T156, 24T157, 24T158, 24T159, 24T165, 24T166, 32T392Siblings 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 $ | $6$ | $2$ | $( 4,10)( 5,11)( 6,12)$ |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3, 9)( 6,12)$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 2, 8)( 4,10)( 6,12)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 8)( 3, 9)( 5,11)( 6,12)$ |
$ 4, 2, 2, 2, 2 $ | $12$ | $4$ | $( 1, 2)( 3, 6, 9,12)( 4, 5)( 7, 8)(10,11)$ |
$ 2, 2, 2, 2, 2, 2 $ | $12$ | $2$ | $( 1, 2)( 3, 6)( 4,11)( 5,10)( 7, 8)( 9,12)$ |
$ 4, 4, 4 $ | $12$ | $4$ | $( 1, 2, 7, 8)( 3, 6, 9,12)( 4, 5,10,11)$ |
$ 4, 4, 2, 2 $ | $12$ | $4$ | $( 1, 2, 7, 8)( 3, 6)( 4,11,10, 5)( 9,12)$ |
$ 6, 6 $ | $8$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ |
$ 6, 3, 3 $ | $8$ | $6$ | $( 1, 3, 5)( 2, 4,12, 8,10, 6)( 7, 9,11)$ |
$ 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3,11)( 2, 4, 6)( 5, 7, 9)( 8,10,12)$ |
$ 6, 3, 3 $ | $8$ | $6$ | $( 1, 3,11, 7, 9, 5)( 2, 4,12)( 6, 8,10)$ |
$ 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: | $96=2^{5} \cdot 3$ | 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: | 96.195 | magma: IdentifyGroup(G);
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Character table: |
2 5 4 5 4 5 3 3 3 3 2 2 2 2 5 3 1 . . 1 . . . . . 1 1 1 1 1 1a 2a 2b 2c 2d 4a 2e 4b 4c 6a 6b 3a 6c 2f 2P 1a 1a 1a 1a 1a 2b 1a 2f 2d 3a 3a 3a 3a 1a 3P 1a 2a 2b 2c 2d 4a 2e 4b 4c 2f 2c 1a 2c 2f 5P 1a 2a 2b 2c 2d 4a 2e 4b 4c 6a 6c 3a 6b 2f 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 -1 1 2 X.6 2 2 2 2 2 . . . . -1 -1 -1 -1 2 X.7 2 . -2 . 2 . . . . -2 . 2 . -2 X.8 2 . -2 . 2 . . . . 1 A -1 -A -2 X.9 2 . -2 . 2 . . . . 1 -A -1 A -2 X.10 3 -1 -1 3 -1 -1 1 1 -1 . . . . 3 X.11 3 -1 -1 3 -1 1 -1 -1 1 . . . . 3 X.12 3 1 -1 -3 -1 -1 -1 1 1 . . . . 3 X.13 3 1 -1 -3 -1 1 1 -1 -1 . . . . 3 X.14 6 . 2 . -2 . . . . . . . . -6 A = -E(3)+E(3)^2 = -Sqrt(-3) = -i3 |
magma: CharacterTable(G);