Show commands:
Magma
magma: G := TransitiveGroup(16, 45);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $D_8:C_2$ | ||
Parity: | $1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (1,9)(2,10)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15), (1,4,2,3)(5,16,6,15)(7,11,8,12)(9,14,10,13), (1,11)(2,12)(3,9)(4,10)(5,8)(6,7)(13,15)(14,16) | magma: Generators(G);
|
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
8T15 x 2, 16T35, 16T38 x 2, 32T21Siblings 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 $ | $4$ | $4$ | $( 1, 3, 2, 4)( 5,15, 6,16)( 7,12, 8,11)( 9,13,10,14)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5,16, 6,15)( 7,14, 8,13)( 9,11,10,12)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,12)( 8,11)( 9,14)(10,13)$ |
$ 8, 8 $ | $4$ | $8$ | $( 1, 7,16, 9, 2, 8,15,10)( 3,13, 6,12, 4,14, 5,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 7)( 2, 8)( 3,13)( 4,14)( 5,12)( 6,11)( 9,15)(10,16)$ |
$ 8, 8 $ | $4$ | $8$ | $( 1,11,15,14, 2,12,16,13)( 3, 9, 5, 7, 4,10, 6, 8)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1,11)( 2,12)( 3, 9)( 4,10)( 5, 8)( 6, 7)(13,15)(14,16)$ |
$ 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);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | $3$ | ||
Label: | 32.43 | magma: IdentifyGroup(G);
|
Character table: |
2 5 3 5 3 4 4 3 3 3 3 4 1a 2a 2b 4a 4b 2c 8a 2d 8b 2e 4c 2P 1a 1a 1a 2b 2b 1a 4c 1a 4c 1a 2b 3P 1a 2a 2b 4a 4b 2c 8a 2d 8b 2e 4c 5P 1a 2a 2b 4a 4b 2c 8a 2d 8b 2e 4c 7P 1a 2a 2b 4a 4b 2c 8a 2d 8b 2e 4c 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);