Show commands:
Magma
magma: G := TransitiveGroup(16, 396);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $396$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $C_2\wr D_4$ | ||
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: | (7,15)(8,16), (1,5)(2,6)(3,4)(7,8)(9,13)(10,14)(11,12)(15,16), (1,4,6,15)(2,3,5,16)(7,9,12,14)(8,10,11,13) | 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 6, $C_2^3$ $16$: $D_4\times C_2$ x 3 $32$: $C_2^2 \wr C_2$ $64$: $(((C_4 \times C_2): C_2):C_2):C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4$, $C_2 \wr C_2\wr C_2$ x 2
Low degree siblings
8T35 x 8, 16T376 x 4, 16T388 x 4, 16T390 x 4, 16T391 x 4, 16T393 x 4, 16T395 x 4, 16T396 x 3, 16T401 x 4, 32T852 x 4, 32T853 x 2, 32T854 x 2, 32T872 x 2, 32T876 x 4, 32T877 x 2, 32T880 x 2, 32T882 x 2, 32T883 x 4, 32T884 x 2, 32T885 x 2Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 7,15)( 8,16)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 5,13)( 6,14)( 7,15)( 8,16)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 3,11)( 4,12)( 7,15)( 8,16)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $4$ | $2$ | $( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,10)(11,15)(12,16)(13,14)$ |
$ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 2)( 3, 7,11,15)( 4, 8,12,16)( 5, 6)( 9,10)(13,14)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 2)( 3, 7)( 4, 8)( 5,14)( 6,13)( 9,10)(11,15)(12,16)$ |
$ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 2)( 3, 7,11,15)( 4, 8,12,16)( 5,14)( 6,13)( 9,10)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$ |
$ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 3)( 2, 4)( 5, 7,13,15)( 6, 8,14,16)( 9,11)(10,12)$ |
$ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 3, 9,11)( 2, 4,10,12)( 5, 7,13,15)( 6, 8,14,16)$ |
$ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1, 4, 6, 7)( 2, 3, 5, 8)( 9,12,14,15)(10,11,13,16)$ |
$ 8, 8 $ | $16$ | $8$ | $( 1, 4, 6, 7, 9,12,14,15)( 2, 3, 5, 8,10,11,13,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 5)( 2, 6)( 3,12)( 4,11)( 7,16)( 8,15)( 9,13)(10,14)$ |
$ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 5, 9,13)( 2, 6,10,14)( 3,12)( 4,11)( 7,16)( 8,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,14)(10,13)(11,16)(12,15)$ |
$ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 6)( 2, 5)( 3, 8,11,16)( 4, 7,12,15)( 9,14)(10,13)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 6, 9,14)( 2, 5,10,13)( 3, 8,11,16)( 4, 7,12,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $128=2^{7}$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | $4$ | ||
Label: | 128.928 | magma: IdentifyGroup(G);
|
Character table: |
2 7 5 5 6 5 5 5 4 4 4 3 4 3 3 5 5 5 4 5 7 1a 2a 2b 2c 2d 2e 4a 2f 4b 2g 4c 4d 4e 8a 2h 4f 2i 4g 4h 2j 2P 1a 1a 1a 1a 1a 1a 2c 1a 2c 1a 2b 2j 2i 4h 1a 2c 1a 2c 2j 1a 3P 1a 2a 2b 2c 2d 2e 4a 2f 4b 2g 4c 4d 4e 8a 2h 4f 2i 4g 4h 2j 5P 1a 2a 2b 2c 2d 2e 4a 2f 4b 2g 4c 4d 4e 8a 2h 4f 2i 4g 4h 2j 7P 1a 2a 2b 2c 2d 2e 4a 2f 4b 2g 4c 4d 4e 8a 2h 4f 2i 4g 4h 2j X.1 1 1 1 1 1 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 -1 1 1 -1 1 1 X.3 1 -1 1 1 -1 -1 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 1 -1 1 -1 1 1 X.5 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 1 X.6 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 X.7 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 X.8 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 X.9 2 2 2 2 2 . . . . . . . . . . . -2 -2 -2 2 X.10 2 -2 2 2 -2 . . . . . . . . . . . -2 2 -2 2 X.11 2 . -2 2 . 2 . . -2 . . . . . 2 . 2 . -2 2 X.12 2 . -2 2 . -2 . . 2 . . . . . -2 . 2 . -2 2 X.13 2 . -2 2 . . -2 2 . . . . . . . -2 -2 . 2 2 X.14 2 . -2 2 . . 2 -2 . . . . . . . 2 -2 . 2 2 X.15 4 . . -4 . . . . . -2 . 2 . . . . . . . 4 X.16 4 . . -4 . . . . . 2 . -2 . . . . . . . 4 X.17 4 -2 . . 2 -2 2 . . . . . . . 2 -2 . . . -4 X.18 4 -2 . . 2 2 -2 . . . . . . . -2 2 . . . -4 X.19 4 2 . . -2 -2 -2 . . . . . . . 2 2 . . . -4 X.20 4 2 . . -2 2 2 . . . . . . . -2 -2 . . . -4 |
magma: CharacterTable(G);