Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1543$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,7,4,6)(2,8,3,5)(9,15,11,14)(10,16,12,13), (1,15,12,2,16,11)(3,14,10)(4,13,9)(7,8), (1,15,9,2,16,10)(3,13,12,4,14,11)(5,6)(7,8) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 3: $C_3$ 12: $A_4$ x 5 24: $\SL(2,3)$ 48: $C_2^4:C_3$ 96: $((C_2 \times D_4): C_2):C_3$ x 6, 24T88 192: 16T437 x 4, 16T438 x 3 384: 32T9460 1536: 48T? Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $A_4$
Degree 8: $((C_2 \times D_4): C_2):C_3$
Low degree siblings
16T1543 x 3, 32T205693 x 2, 32T205694 x 4Siblings 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, 2, 2 $ | $8$ | $2$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)$ |
| $ 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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 4, 4 $ | $192$ | $4$ | $( 1, 7, 4, 6)( 2, 8, 3, 5)( 9,15,11,14)(10,16,12,13)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $4$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)( 7, 8)(11,12)(15,16)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)( 7, 8)(11,12)(13,14)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $4$ | $( 9,11,10,12)(13,16,14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $96$ | $2$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)(11,12)(15,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $24$ | $4$ | $( 5, 6)( 7, 8)( 9,11,10,12)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $24$ | $4$ | $( 1, 2)( 3, 4)( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $24$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,10,11)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $192$ | $4$ | $( 1, 7, 4, 6)( 2, 8, 3, 5)( 9,14,11,16)(10,13,12,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $192$ | $4$ | $( 1,10)( 2, 9)( 3,11, 4,12)( 5,15, 6,16)( 7,13)( 8,14)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $192$ | $4$ | $( 1,16, 2,15)( 3,14)( 4,13)( 5,11, 6,12)( 7,10)( 8, 9)$ |
| $ 6, 3, 3, 2, 1, 1 $ | $128$ | $6$ | $( 3, 4)( 5,15, 9, 6,16,10)( 7,13,12)( 8,14,11)$ |
| $ 12, 4 $ | $128$ | $12$ | $( 1, 4, 2, 3)( 5,14, 9, 7,16,12, 6,13,10, 8,15,11)$ |
| $ 6, 6, 1, 1, 1, 1 $ | $64$ | $6$ | $( 1, 7,10, 2, 8, 9)( 3, 6,12, 4, 5,11)$ |
| $ 6, 6, 2, 2 $ | $128$ | $6$ | $( 1, 6,10, 3, 7,12)( 2, 5, 9, 4, 8,11)(13,16)(14,15)$ |
| $ 3, 3, 3, 3, 2, 2 $ | $64$ | $6$ | $( 1, 8, 9)( 2, 7,10)( 3, 5,11)( 4, 6,12)(13,14)(15,16)$ |
| $ 6, 6, 2, 2 $ | $128$ | $6$ | $( 1,10,15, 3,12,13)( 2, 9,16, 4,11,14)( 5, 8)( 6, 7)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1 $ | $64$ | $3$ | $( 1,12,16)( 2,11,15)( 3,10,14)( 4, 9,13)$ |
| $ 6, 6, 2, 2 $ | $64$ | $6$ | $( 1,12,15, 2,11,16)( 3,10,13, 4, 9,14)( 5, 6)( 7, 8)$ |
| $ 12, 4 $ | $128$ | $12$ | $( 1,16, 6, 4,14, 8, 2,15, 5, 3,13, 7)( 9,12,10,11)$ |
| $ 6, 3, 3, 2, 1, 1 $ | $128$ | $6$ | $( 1,13, 6)( 2,14, 5)( 3,16, 7, 4,15, 8)( 9,10)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1 $ | $64$ | $3$ | $( 5, 9,16)( 6,10,15)( 7,12,13)( 8,11,14)$ |
| $ 6, 6, 2, 2 $ | $128$ | $6$ | $( 1, 4)( 2, 3)( 5,11,15, 7,10,14)( 6,12,16, 8, 9,13)$ |
| $ 6, 6, 2, 2 $ | $64$ | $6$ | $( 1, 2)( 3, 4)( 5,10,15, 6, 9,16)( 7,11,14, 8,12,13)$ |
| $ 12, 4 $ | $128$ | $12$ | $( 1, 7,13, 4, 6,16, 2, 8,14, 3, 5,15)( 9,12,10,11)$ |
| $ 6, 3, 3, 2, 1, 1 $ | $128$ | $6$ | $( 1, 6,13)( 2, 5,14)( 3, 8,15, 4, 7,16)( 9,10)$ |
| $ 6, 3, 3, 2, 1, 1 $ | $128$ | $6$ | $( 1,10, 8, 2, 9, 7)( 3,11, 5)( 4,12, 6)(15,16)$ |
| $ 12, 4 $ | $128$ | $12$ | $( 1,12, 7, 4,10, 5, 2,11, 8, 3, 9, 6)(13,16,14,15)$ |
| $ 6, 6, 1, 1, 1, 1 $ | $64$ | $6$ | $( 1,16,11, 2,15,12)( 3,14, 9, 4,13,10)$ |
| $ 6, 6, 2, 2 $ | $128$ | $6$ | $( 1,13,12, 4,16, 9)( 2,14,11, 3,15,10)( 5, 8)( 6, 7)$ |
| $ 3, 3, 3, 3, 2, 2 $ | $64$ | $6$ | $( 1,15,12)( 2,16,11)( 3,13,10)( 4,14, 9)( 5, 6)( 7, 8)$ |
Group invariants
| Order: | $3072=2^{10} \cdot 3$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |