Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1692$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,5,12,10,14,4)(2,6,11,9,13,3)(7,15)(8,16), (1,13,7,12,9,6,16,4,2,14,8,11,10,5,15,3), (1,9)(2,10)(3,11,4,12)(5,14)(6,13)(7,15,8,16) | |
| $|\Aut(F/K)|$: | $2$ |
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: $\textrm{GL(2,3)}$
Low degree siblings
16T1691 x 4, 16T1692 x 3, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)(13,14)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $24$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(11,12)(15,16)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $2$ | $( 3, 4)( 5, 6)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $4$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $2$ | $( 1, 2)( 5, 6)( 9,10)(15,16)$ |
| $ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 3, 4)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,16)( 8,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $96$ | $4$ | $( 1, 9, 2,10)( 3,12, 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 2,10)( 3,12, 4,11)( 5,13, 6,14)( 7,16, 8,15)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1,10, 2, 9)( 3,12, 4,11)( 5,14, 6,13)( 7,15, 8,16)$ |
| $ 4, 4, 4, 4 $ | $384$ | $4$ | $( 1, 6, 9,14)( 2, 5,10,13)( 3, 8,12,15)( 4, 7,11,16)$ |
| $ 8, 8 $ | $192$ | $8$ | $( 1, 5,10,14, 2, 6, 9,13)( 3, 8,12,15, 4, 7,11,16)$ |
| $ 8, 8 $ | $192$ | $8$ | $( 1, 6, 9,13, 2, 5,10,14)( 3, 7,12,16, 4, 8,11,15)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1 $ | $128$ | $3$ | $( 3,15,14)( 4,16,13)( 5,11, 7)( 6,12, 8)$ |
| $ 6, 3, 3, 2, 1, 1 $ | $256$ | $6$ | $( 1, 2)( 3,15,13)( 4,16,14)( 5,11, 7, 6,12, 8)$ |
| $ 6, 6, 2, 2 $ | $128$ | $6$ | $( 1, 2)( 3,15,14, 4,16,13)( 5,12, 8, 6,11, 7)( 9,10)$ |
| $ 6, 3, 3, 2, 1, 1 $ | $256$ | $6$ | $( 3,15,13)( 4,16,14)( 5,12, 8, 6,11, 7)( 9,10)$ |
| $ 3, 3, 3, 3, 2, 2 $ | $128$ | $6$ | $( 1, 2)( 3,15,14)( 4,16,13)( 5,12, 8)( 6,11, 7)( 9,10)$ |
| $ 6, 6, 1, 1, 1, 1 $ | $128$ | $6$ | $( 3,15,14, 4,16,13)( 5,11, 7, 6,12, 8)$ |
| $ 6, 6, 2, 2 $ | $256$ | $6$ | $( 1, 9)( 2,10)( 3, 8,14,12,15, 6)( 4, 7,13,11,16, 5)$ |
| $ 12, 4 $ | $256$ | $12$ | $( 1, 9, 2,10)( 3, 8,13,11,16, 6, 4, 7,14,12,15, 5)$ |
| $ 12, 4 $ | $256$ | $12$ | $( 1,10, 2, 9)( 3, 8,13,12,15, 6, 4, 7,14,11,16, 5)$ |
| $ 6, 6, 2, 2 $ | $256$ | $6$ | $( 1,10)( 2, 9)( 3, 8,14,11,16, 6)( 4, 7,13,12,15, 5)$ |
| $ 4, 4, 4, 1, 1, 1, 1 $ | $96$ | $4$ | $( 3, 5, 4, 6)( 7,15, 8,16)(11,14,12,13)$ |
| $ 4, 4, 2, 2, 2, 1, 1 $ | $192$ | $4$ | $( 1, 2)( 3, 6, 4, 5)( 7,15, 8,16)(11,13)(12,14)$ |
| $ 4, 2, 2, 2, 2, 2, 2 $ | $96$ | $4$ | $( 1, 2)( 3, 5)( 4, 6)( 7,15, 8,16)( 9,10)(11,14)(12,13)$ |
| $ 4, 2, 2, 2, 2, 1, 1, 1, 1 $ | $192$ | $4$ | $( 3, 5, 4, 6)( 7,16)( 8,15)(11,14)(12,13)$ |
| $ 4, 4, 2, 2, 2, 1, 1 $ | $192$ | $4$ | $( 1, 2)( 3, 6, 4, 5)( 7,16)( 8,15)(11,13,12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $192$ | $2$ | $( 3, 6)( 4, 5)( 7,16)( 8,15)( 9,10)(11,13)(12,14)$ |
| $ 4, 2, 2, 2, 2, 2, 2 $ | $192$ | $4$ | $( 1, 2)( 3, 5)( 4, 6)( 7,16)( 8,15)( 9,10)(11,14,12,13)$ |
| $ 4, 4, 2, 2, 2, 1, 1 $ | $192$ | $4$ | $( 3, 5, 4, 6)( 7,15, 8,16)( 9,10)(11,13)(12,14)$ |
| $ 4, 4, 4, 2, 2 $ | $96$ | $4$ | $( 1, 2)( 3, 6, 4, 5)( 7,15, 8,16)( 9,10)(11,14,12,13)$ |
| $ 4, 2, 2, 2, 2, 1, 1, 1, 1 $ | $96$ | $4$ | $( 3, 6)( 4, 5)( 7,15, 8,16)(11,14)(12,13)$ |
| $ 16 $ | $384$ | $16$ | $( 1, 6, 8, 4, 9,14,15,12, 2, 5, 7, 3,10,13,16,11)$ |
| $ 16 $ | $384$ | $16$ | $( 1, 5, 7, 4, 9,13,16,11, 2, 6, 8, 3,10,14,15,12)$ |
| $ 16 $ | $384$ | $16$ | $( 1,14, 8,11, 9, 6,15, 3, 2,13, 7,12,10, 5,16, 4)$ |
| $ 16 $ | $384$ | $16$ | $( 1,13, 7,12,10, 6,15, 4, 2,14, 8,11, 9, 5,16, 3)$ |
Group invariants
| Order: | $6144=2^{11} \cdot 3$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |