Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1496$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,13,3,9,16,7,2,14,4,10,15,8)(5,11,6,12), (1,2)(3,4)(5,16)(6,15)(7,10,12)(8,9,11) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_4$ x 2, $C_2^2$ 8: $D_{4}$ x 2, $C_4\times C_2$ 16: $C_2^2:C_4$ 72: $C_3^2:D_4$ 144: 12T79 1152: $S_4\wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 8: $S_4\wr C_2$
Low degree siblings
12T237 x 2, 12T238 x 2, 16T1496, 16T1497 x 2, 24T5093 x 2, 24T5117 x 2, 24T5118 x 2, 24T5119, 24T5120 x 2, 24T5121 x 2, 24T5122 x 2, 24T5123 x 2, 24T5124, 24T5125 x 2, 24T5126 x 2, 24T5127 x 2, 24T5128 x 2, 32T205436 x 2, 32T205437 x 2, 32T205438, 32T205439, 36T3213 x 2, 36T3215 x 2, 36T3216 x 2, 36T3218, 36T3224, 36T3449 x 2, 36T3450 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, 2, 2 $ | $9$ | $2$ | $( 1, 4)( 2, 3)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11)(10,12)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 7,10)( 8, 9)(11,14)(12,13)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1 $ | $64$ | $3$ | $( 3,15, 5)( 4,16, 6)( 9,11,14)(10,12,13)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $3$ | $( 9,14,11)(10,13,12)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1 $ | $48$ | $6$ | $( 1, 4)( 2, 3)( 5,15)( 6,16)( 7,13,10)( 8,14, 9)$ |
| $ 6, 6, 2, 2 $ | $64$ | $6$ | $( 1, 3,16, 2, 4,15)( 5, 6)( 7,14,10, 8,13, 9)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1,15)( 2,16)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,13)(12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1,15)( 2,16)( 3, 6)( 4, 5)( 7, 8)( 9,10)(11,12)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 6, 2, 2, 2, 2, 2 $ | $48$ | $6$ | $( 1,15)( 2,16)( 3, 6)( 4, 5)( 7,14,10, 8,13, 9)(11,12)$ |
| $ 6, 2, 2, 2, 2, 2 $ | $16$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,11,13, 8,12,14)( 9,10)(15,16)$ |
| $ 12, 4 $ | $192$ | $12$ | $( 1,13, 3, 9,16, 7, 2,14, 4,10,15, 8)( 5,11, 6,12)$ |
| $ 4, 4, 4, 4 $ | $72$ | $4$ | $( 1,12,15, 8)( 2,11,16, 7)( 3,14, 6,10)( 4,13, 5, 9)$ |
| $ 4, 4, 4, 4 $ | $24$ | $4$ | $( 1,10, 2, 9)( 3, 8, 4, 7)( 5,11, 6,12)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $24$ | $4$ | $( 1, 9, 2,10)( 3, 7, 4, 8)( 5,12, 6,11)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $72$ | $4$ | $( 1,11,15, 7)( 2,12,16, 8)( 3,13, 6, 9)( 4,14, 5,10)$ |
| $ 12, 4 $ | $192$ | $12$ | $( 1,11, 3, 7,16, 9, 2,12, 4, 8,15,10)( 5,13, 6,14)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $2$ | $( 5,15)( 6,16)(11,14)(12,13)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $72$ | $4$ | $( 1, 4)( 2, 3)( 7,13,10,12)( 8,14, 9,11)$ |
| $ 4, 4, 4, 4 $ | $36$ | $4$ | $( 1, 6, 4,16)( 2, 5, 3,15)( 7,12,10,13)( 8,11, 9,14)$ |
| $ 4, 4, 4, 4 $ | $36$ | $4$ | $( 1, 3,16, 5)( 2, 4,15, 6)( 7,14,12, 9)( 8,13,11,10)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $36$ | $2$ | $( 1, 2)( 3, 6)( 4, 5)( 7, 8)( 9,13)(10,14)(11,12)(15,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $72$ | $4$ | $( 1, 3,16, 5)( 2, 4,15, 6)( 7,11)( 8,12)( 9,10)(13,14)$ |
| $ 6, 2, 2, 2, 1, 1, 1, 1 $ | $96$ | $6$ | $( 3, 5)( 4, 6)( 7,14,12, 8,13,11)( 9,10)$ |
| $ 6, 4, 4, 2 $ | $96$ | $12$ | $( 1, 4,16, 6)( 2, 3,15, 5)( 7, 8)( 9,12,14,10,11,13)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $36$ | $2$ | $( 3,15)( 4,16)( 7,11)( 8,12)( 9,13)(10,14)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $36$ | $4$ | $( 1, 4, 6,16)( 2, 3, 5,15)( 7, 9)( 8,10)(11,13)(12,14)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $12$ | $4$ | $( 1, 4, 6,16)( 2, 3, 5,15)( 7, 8)( 9,10)(11,12)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $12$ | $2$ | $( 3,15)( 4,16)( 7, 8)( 9,10)(11,12)(13,14)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $36$ | $4$ | $( 1, 3, 6,15)( 2, 4, 5,16)( 7,10)( 8, 9)(11,14)(12,13)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $36$ | $2$ | $( 1, 2)( 3,16)( 4,15)( 5, 6)( 7,12)( 8,11)( 9,14)(10,13)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $4$ | $( 1, 3, 6,15)( 2, 4, 5,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $2$ | $( 1, 2)( 3,16)( 4,15)( 5, 6)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1 $ | $96$ | $6$ | $( 1,15)( 2,16)( 3, 4)( 5, 6)( 7,12,10)( 8,11, 9)$ |
| $ 4, 4, 3, 3, 1, 1 $ | $96$ | $12$ | $( 1, 5,16, 3)( 2, 6,15, 4)( 7,10,13)( 8, 9,14)$ |
| $ 8, 8 $ | $144$ | $8$ | $( 1,13, 6,10,16, 7, 4,12)( 2,14, 5, 9,15, 8, 3,11)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $144$ | $4$ | $( 1, 7)( 2, 8)( 3, 9, 5,11)( 4,10, 6,12)(13,16)(14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $144$ | $4$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5, 7,15,13)( 6, 8,16,14)$ |
| $ 8, 8 $ | $144$ | $8$ | $( 1, 8,16,11, 4,14, 6, 9)( 2, 7,15,12, 3,13, 5,10)$ |
Group invariants
| Order: | $2304=2^{8} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |