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Magma
magma: G := TransitiveGroup(16, 33);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^3:C_4$ | ||
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: | (1,11,6,8)(2,12,5,7)(3,10,16,13)(4,9,15,14), (1,9,3,7)(2,10,4,8)(5,13,15,11)(6,14,16,12) | 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, $C_2^2$ $8$: $D_{4}$ 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 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4$, $C_2^3 : C_4 $, $C_2^3: C_4$
Low degree siblings
8T19 x 2, 8T20, 8T21, 16T33, 16T52, 16T53, 32T19Siblings 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 $ | $2$ | $2$ | $( 7,12)( 8,11)( 9,14)(10,13)$ |
$ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 2)( 3,15)( 4,16)( 5, 6)( 7,10,12,13)( 8, 9,11,14)$ |
$ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 2)( 3,15)( 4,16)( 5, 6)( 7,13,12,10)( 8,14,11, 9)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7, 9)( 8,10)(11,13)(12,14)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,14)( 8,13)( 9,12)(10,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 6)( 2, 5)( 3,16)( 4,15)( 7,12)( 8,11)( 9,14)(10,13)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 7, 3, 9)( 2, 8, 4,10)( 5,11,15,13)( 6,12,16,14)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 7,16,14)( 2, 8,15,13)( 3, 9, 6,12)( 4,10, 5,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 8)( 2, 7)( 3,13)( 4,14)( 5,12)( 6,11)( 9,15)(10,16)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 8, 6,11)( 2, 7, 5,12)( 3,13,16,10)( 4,14,15, 9)$ |
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: | 32.6 | magma: IdentifyGroup(G);
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Character table: |
2 5 4 3 3 4 4 5 3 3 3 3 1a 2a 4a 4b 2b 2c 2d 4c 4d 2e 4e 2P 1a 1a 2a 2a 1a 1a 1a 2b 2b 1a 2d 3P 1a 2a 4b 4a 2b 2c 2d 4d 4c 2e 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 A -A -1 1 1 A -A -1 1 X.6 1 -1 -A A -1 1 1 -A A -1 1 X.7 1 -1 A -A -1 1 1 -A A 1 -1 X.8 1 -1 -A A -1 1 1 A -A 1 -1 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);