Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1691$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,16,3,13)(2,15,4,14)(5,8,6,7)(9,11)(10,12), (1,16,11,8,3,14,9,5,2,15,12,7,4,13,10,6) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 6: $S_3$ 24: $S_4$ x 3 48: $\textrm{GL(2,3)}$ 96: $V_4^2:S_3$ 192: $C_2^3:S_4$, $Q_8:S_4$, 24T314 384: 16T770, 16T771 x 2 768: 32T34981 3072: 32T205578 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $S_4$
Degree 8: $Q_8:S_4$
Low degree siblings
16T1691 x 3, 16T1692 x 4, 32T397505 x 4, 32T397506 x 2, 32T397507 x 2, 32T397508 x 2, 32T397509 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, 2, 2, 2, 2, 1, 1, 1, 1 $ | $24$ | $2$ | $( 1, 3)( 2, 4)( 5, 6)( 7, 8)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $24$ | $2$ | $( 1, 4)( 2, 3)( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $24$ | $2$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,10)(11,12)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $2$ | $( 1, 3)( 2, 4)( 9,12)(10,11)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $4$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 1, 2)( 3, 4)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 5, 6)( 7, 8)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $2$ | $( 3, 4)( 7, 8)(11,12)(15,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $96$ | $4$ | $( 1, 3, 2, 4)( 5, 6)(11,12)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$ |
| $ 8, 8 $ | $192$ | $8$ | $( 1, 8, 4, 5, 2, 7, 3, 6)( 9,13,12,15,10,14,11,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $384$ | $4$ | $( 1, 7)( 2, 8)( 3, 5, 4, 6)( 9,16)(10,15)(11,13,12,14)$ |
| $ 8, 8 $ | $192$ | $8$ | $( 1, 8, 3, 6, 2, 7, 4, 5)( 9,15,11,14,10,16,12,13)$ |
| $ 6, 6, 1, 1, 1, 1 $ | $128$ | $6$ | $( 5,12,16, 6,11,15)( 7, 9,14, 8,10,13)$ |
| $ 6, 6, 2, 2 $ | $256$ | $6$ | $( 1, 3)( 2, 4)( 5,12,13, 8,10,16)( 6,11,14, 7, 9,15)$ |
| $ 6, 6, 2, 2 $ | $128$ | $6$ | $( 1, 2)( 3, 4)( 5, 9,16, 6,10,15)( 7,12,14, 8,11,13)$ |
| $ 6, 6, 2, 2 $ | $256$ | $6$ | $( 1, 4)( 2, 3)( 5, 9,13, 8,11,16)( 6,10,14, 7,12,15)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1 $ | $128$ | $3$ | $( 5,12,15)( 6,11,16)( 7, 9,13)( 8,10,14)$ |
| $ 3, 3, 3, 3, 2, 2 $ | $128$ | $6$ | $( 1, 2)( 3, 4)( 5, 9,15)( 6,10,16)( 7,12,13)( 8,11,14)$ |
| $ 6, 3, 3, 2, 1, 1 $ | $256$ | $6$ | $( 3, 4)( 5,11,16, 6,12,15)( 7, 9,14)( 8,10,13)$ |
| $ 12, 4 $ | $256$ | $12$ | $( 1, 3, 2, 4)( 5,11,13, 7, 9,15, 6,12,14, 8,10,16)$ |
| $ 6, 3, 3, 2, 1, 1 $ | $256$ | $6$ | $( 1, 2)( 5,10,15)( 6, 9,16)( 7,12,13, 8,11,14)$ |
| $ 12, 4 $ | $256$ | $12$ | $( 1, 4, 2, 3)( 5,10,14, 8,11,15, 6, 9,13, 7,12,16)$ |
| $ 4, 4, 2, 1, 1, 1, 1, 1, 1 $ | $192$ | $4$ | $( 7, 8)( 9,14,11,16)(10,13,12,15)$ |
| $ 4, 4, 2, 2, 2, 1, 1 $ | $192$ | $4$ | $( 1, 3)( 2, 4)( 5, 6)( 9,15,10,16)(11,13,12,14)$ |
| $ 4, 4, 2, 2, 2, 1, 1 $ | $192$ | $4$ | $( 1, 2)( 3, 4)( 7, 8)( 9,16,12,13)(10,15,11,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $192$ | $2$ | $( 1, 4)( 2, 3)( 5, 6)( 9,13)(10,14)(11,15)(12,16)$ |
| $ 4, 4, 4, 1, 1, 1, 1 $ | $96$ | $4$ | $( 5, 7, 6, 8)( 9,13,10,14)(11,15,12,16)$ |
| $ 4, 4, 4, 2, 2 $ | $192$ | $4$ | $( 1, 3)( 2, 4)( 5, 8, 6, 7)( 9,16,11,14)(10,15,12,13)$ |
| $ 4, 2, 2, 2, 2, 2, 2 $ | $96$ | $4$ | $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,15)(10,16)(11,13)(12,14)$ |
| $ 4, 2, 2, 2, 2, 1, 1, 1, 1 $ | $96$ | $4$ | $( 5, 7, 6, 8)( 9,15)(10,16)(11,13)(12,14)$ |
| $ 4, 4, 4, 2, 2 $ | $192$ | $4$ | $( 1, 3)( 2, 4)( 5, 8, 6, 7)( 9,14,12,15)(10,13,11,16)$ |
| $ 4, 4, 4, 2, 2 $ | $96$ | $4$ | $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,13,10,14)(11,15,12,16)$ |
| $ 16 $ | $384$ | $16$ | $( 1,10, 7,15, 4,12, 5,13, 2, 9, 8,16, 3,11, 6,14)$ |
| $ 16 $ | $384$ | $16$ | $( 1,10, 8,13, 4,12, 6,15, 2, 9, 7,14, 3,11, 5,16)$ |
| $ 16 $ | $384$ | $16$ | $( 1,10, 8,15, 3,12, 5,13, 2, 9, 7,16, 4,11, 6,14)$ |
| $ 16 $ | $384$ | $16$ | $( 1,10, 7,13, 4,11, 5,16, 2, 9, 8,14, 3,12, 6,15)$ |
Group invariants
| Order: | $6144=2^{11} \cdot 3$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |