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Magma
magma: G := TransitiveGroup(16, 40);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_4.D_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,11,3,13,2,12,4,14)(5,15,7,10,6,16,8,9), (1,15,4,10,2,16,3,9)(5,12,8,14,6,11,7,13) | 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, $C_2^2$ $8$: $D_{4}$ x 2, $C_4\times C_2$ $16$: $C_2^2:C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 8: $C_4\times C_2$
Low degree siblings
32T25Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
$ 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 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,15,10,16)(11,13,12,14)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,16,10,15)(11,14,12,13)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,16,12,15)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,11,10,12)(13,15,14,16)$ |
$ 8, 8 $ | $4$ | $8$ | $( 1, 9, 3,16, 2,10, 4,15)( 5,13, 7,11, 6,14, 8,12)$ |
$ 8, 8 $ | $4$ | $8$ | $( 1,11, 3,13, 2,12, 4,14)( 5,15, 7,10, 6,16, 8, 9)$ |
$ 8, 8 $ | $4$ | $8$ | $( 1,13, 4,11, 2,14, 3,12)( 5,10, 8,15, 6, 9, 7,16)$ |
$ 8, 8 $ | $4$ | $8$ | $( 1,15, 4,10, 2,16, 3, 9)( 5,12, 8,14, 6,11, 7,13)$ |
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: | 32.8 | magma: IdentifyGroup(G);
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Character table: |
2 5 4 5 4 4 3 3 3 3 3 3 1a 2a 2b 4a 4b 4c 4d 8a 8b 8c 8d 2P 1a 1a 1a 2b 2b 2b 2b 4b 4a 4a 4b 3P 1a 2a 2b 4a 4b 4c 4d 8d 8c 8b 8a 5P 1a 2a 2b 4a 4b 4c 4d 8a 8b 8c 8d 7P 1a 2a 2b 4a 4b 4c 4d 8d 8c 8b 8a 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 A A -A -A X.6 1 1 1 -1 -1 -1 1 -A -A A A X.7 1 1 1 -1 -1 1 -1 A -A A -A X.8 1 1 1 -1 -1 1 -1 -A A -A A X.9 2 -2 2 -2 2 . . . . . . X.10 2 -2 2 2 -2 . . . . . . X.11 4 . -4 . . . . . . . . A = -E(4) = -Sqrt(-1) = -i |
magma: CharacterTable(G);