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Magma
magma: G := TransitiveGroup(16, 32);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $Q_{16}:C_2$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,15)(2,16)(3,5)(4,6)(7,8)(13,14), (1,14,15,12,2,13,16,11)(3,8,5,9,4,7,6,10), (1,6)(2,5)(3,15)(4,16)(7,12)(8,11)(9,14)(10,13) | 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 $8$: $D_{4}$ x 2, $C_2^3$ $16$: $D_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4\times C_2$
Low degree siblings
16T50, 32T18Siblings 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, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $4$ | $2$ | $( 5, 6)( 7,10)( 8, 9)(11,13)(12,14)(15,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 3, 2, 4)( 5,15, 6,16)( 7,11, 8,12)( 9,14,10,13)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5,16, 6,15)( 7,13, 8,14)( 9,12,10,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,11)( 8,12)( 9,13)(10,14)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 7, 2, 8)( 3,14, 4,13)( 5,11, 6,12)( 9,16,10,15)$ |
$ 8, 8 $ | $4$ | $8$ | $( 1, 7,15,10, 2, 8,16, 9)( 3,14, 5,12, 4,13, 6,11)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,11, 2,12)( 3,10, 4, 9)( 5, 7, 6, 8)(13,16,14,15)$ |
$ 8, 8 $ | $4$ | $8$ | $( 1,11,16,13, 2,12,15,14)( 3,10, 6, 7, 4, 9, 5, 8)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,15, 2,16)( 3, 5, 4, 6)( 7,10, 8, 9)(11,14,12,13)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $32=2^{5}$ | 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: | $3$ | ||
Label: | 32.44 | magma: IdentifyGroup(G);
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Character table: |
2 5 3 5 3 4 4 3 3 3 3 4 1a 2a 2b 4a 4b 2c 4c 8a 4d 8b 4e 2P 1a 1a 1a 2b 2b 1a 2b 4e 2b 4e 2b 3P 1a 2a 2b 4a 4b 2c 4c 8a 4d 8b 4e 5P 1a 2a 2b 4a 4b 2c 4c 8a 4d 8b 4e 7P 1a 2a 2b 4a 4b 2c 4c 8a 4d 8b 4e X.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 X.3 1 -1 1 -1 1 1 1 -1 1 -1 1 X.4 1 -1 1 1 -1 -1 -1 1 1 -1 1 X.5 1 -1 1 1 -1 -1 1 -1 -1 1 1 X.6 1 1 1 -1 -1 -1 -1 -1 1 1 1 X.7 1 1 1 -1 -1 -1 1 1 -1 -1 1 X.8 1 1 1 1 1 1 -1 -1 -1 -1 1 X.9 2 . 2 . 2 -2 . . . . -2 X.10 2 . 2 . -2 2 . . . . -2 X.11 4 . -4 . . . . . . . . |
magma: CharacterTable(G);