Show commands:
Magma
magma: G := TransitiveGroup(16, 32);
Group invariants
Abstract group: | $Q_{16}:C_2$ | magma: IdentifyGroup(G);
| |
Order: | $32=2^{5}$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | $3$ | magma: NilpotencyClass(G);
|
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Parity: | $1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| |
$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
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);
|
Low degree resolvents
$\card{(G/N)}$ 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
Label | Cycle Type | Size | Order | Index | Representative |
1A | $1^{16}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{8}$ | $1$ | $2$ | $8$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
2B | $2^{8}$ | $2$ | $2$ | $8$ | $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,12)( 8,11)( 9,14)(10,13)$ |
2C | $2^{6},1^{4}$ | $4$ | $2$ | $6$ | $( 1,16)( 2,15)( 3, 6)( 4, 5)( 9,10)(11,12)$ |
4A | $4^{4}$ | $2$ | $4$ | $12$ | $( 1,16, 2,15)( 3, 6, 4, 5)( 7, 9, 8,10)(11,13,12,14)$ |
4B | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 4, 2, 3)( 5,15, 6,16)( 7,14, 8,13)( 9,11,10,12)$ |
4C | $4^{4}$ | $4$ | $4$ | $12$ | $( 1, 9, 2,10)( 3,11, 4,12)( 5,13, 6,14)( 7,15, 8,16)$ |
4D | $4^{4}$ | $4$ | $4$ | $12$ | $( 1,14, 2,13)( 3, 8, 4, 7)( 5,10, 6, 9)(11,16,12,15)$ |
4E | $4^{4}$ | $4$ | $4$ | $12$ | $( 1, 3, 2, 4)( 5,15, 6,16)( 7,11, 8,12)( 9,14,10,13)$ |
8A | $8^{2}$ | $4$ | $8$ | $14$ | $( 1, 9,16, 8, 2,10,15, 7)( 3,11, 6,13, 4,12, 5,14)$ |
8B | $8^{2}$ | $4$ | $8$ | $14$ | $( 1,13,15,11, 2,14,16,12)( 3, 7, 5,10, 4, 8, 6, 9)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 8A | 8B | ||
Size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 4A | 4A | |
Type | ||||||||||||
32.44.1a | R | |||||||||||
32.44.1b | R | |||||||||||
32.44.1c | R | |||||||||||
32.44.1d | R | |||||||||||
32.44.1e | R | |||||||||||
32.44.1f | R | |||||||||||
32.44.1g | R | |||||||||||
32.44.1h | R | |||||||||||
32.44.2a | R | |||||||||||
32.44.2b | R | |||||||||||
32.44.4a | S |
magma: CharacterTable(G);
Regular extensions
Data not computed