Group action invariants
| Degree $n$ : | $32$ | |
| Transitive number $t$ : | $22195$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $5$ | |
| Generators: | (1,9,3,19,2,10,4,20)(5,21,6,22)(7,16,8,15)(11,25,18,27,12,26,17,28)(13,14)(29,32)(30,31), (1,18)(2,17)(3,11)(4,12)(5,29,6,30)(7,32,8,31)(9,19)(10,20)(13,21,14,22)(15,23,16,24)(27,28), (1,29,19,21,11,24,26,7,3,31,9,15,18,13,27,6,2,30,20,22,12,23,25,8,4,32,10,16,17,14,28,5) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_4$ x 4, $C_2^2$ x 7 8: $D_{4}$ x 2, $C_8$ x 4, $C_4\times C_2$ x 6, $C_2^3$ 16: $D_4\times C_2$, $Q_8:C_2$, $C_4\times C_2^2$, $C_8\times C_2$ x 6 32: $Z_8 : Z_8^\times$ x 2, $(C_8:C_2):C_2$, $C_4 \times D_4$, 32T37 64: 16T113, 32T195, 32T245 128: 16T256 x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 8: $Z_8 : Z_8^\times$
Degree 16: 16T256
Low degree siblings
32T22195Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(17,18)(19,20)(25,26)(27,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$ |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 1, 3, 2, 4)( 5, 6)( 7, 8)( 9,19,10,20)(11,18,12,17)(15,16)(21,22) (25,27,26,28)$ |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 1, 4, 2, 3)( 5, 6)( 7, 8)( 9,20,10,19)(11,17,12,18)(15,16)(21,22) (25,28,26,27)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,20,10,19)(11,18,12,17)(13,23,14,24)(15,22,16,21) (25,28,26,27)(29,31,30,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,19,10,20)(11,17,12,18)(13,23,14,24)(15,22,16,21) (25,27,26,28)(29,31,30,32)$ |
| $ 8, 8, 8, 8 $ | $4$ | $8$ | $( 1,11, 3,18, 2,12, 4,17)( 5,22, 7,16, 6,21, 8,15)( 9,27,20,25,10,28,19,26) (13,29,23,31,14,30,24,32)$ |
| $ 8, 8, 8, 8 $ | $4$ | $8$ | $( 1,12, 3,17, 2,11, 4,18)( 5,22, 7,16, 6,21, 8,15)( 9,28,20,26,10,27,19,25) (13,29,23,31,14,30,24,32)$ |
| $ 8, 8, 8, 8 $ | $4$ | $8$ | $( 1,18, 4,11, 2,17, 3,12)( 5,21, 7,15, 6,22, 8,16)( 9,26,19,28,10,25,20,27) (13,29,23,31,14,30,24,32)$ |
| $ 8, 8, 8, 8 $ | $4$ | $8$ | $( 1,17, 4,12, 2,18, 3,11)( 5,21, 7,15, 6,22, 8,16)( 9,25,19,27,10,26,20,28) (13,29,23,31,14,30,24,32)$ |
| $ 8, 8, 4, 4, 2, 2, 2, 1, 1 $ | $16$ | $8$ | $( 1, 9, 3,19, 2,10, 4,20)( 5,21, 6,22)( 7,16, 8,15)(11,25,18,27,12,26,17,28) (13,14)(29,32)(30,31)$ |
| $ 8, 8, 4, 4, 2, 2, 2, 1, 1 $ | $16$ | $8$ | $( 1,10, 3,20, 2, 9, 4,19)( 5,21, 6,22)( 7,16, 8,15)(11,26,18,28,12,25,17,27) (13,14)(29,32)(30,31)$ |
| $ 8, 8, 4, 4, 2, 2, 2, 1, 1 $ | $16$ | $8$ | $( 1,19, 4, 9, 2,20, 3,10)( 5,22, 6,21)( 7,15, 8,16)(11,27,17,25,12,28,18,26) (13,14)(29,32)(30,31)$ |
| $ 8, 8, 4, 4, 2, 2, 2, 1, 1 $ | $16$ | $8$ | $( 1,20, 4,10, 2,19, 3, 9)( 5,22, 6,21)( 7,15, 8,16)(11,28,17,26,12,27,18,25) (13,14)(29,32)(30,31)$ |
| $ 16, 16 $ | $8$ | $16$ | $( 1, 9,11,27, 3,20,18,25, 2,10,12,28, 4,19,17,26)( 5,29,22,23, 7,31,16,14, 6, 30,21,24, 8,32,15,13)$ |
| $ 16, 16 $ | $8$ | $16$ | $( 1,10,11,28, 3,19,18,26, 2, 9,12,27, 4,20,17,25)( 5,29,22,23, 7,31,16,14, 6, 30,21,24, 8,32,15,13)$ |
| $ 16, 16 $ | $8$ | $16$ | $( 1,19,11,26, 3, 9,18,27, 2,20,12,25, 4,10,17,28)( 5,29,21,24, 7,31,15,13, 6, 30,22,23, 8,32,16,14)$ |
| $ 16, 16 $ | $8$ | $16$ | $( 1,20,11,25, 3,10,18,28, 2,19,12,26, 4, 9,17,27)( 5,29,21,24, 7,31,15,13, 6, 30,22,23, 8,32,16,14)$ |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $16$ | $4$ | $( 3, 4)( 5,23, 6,24)( 7,13, 8,14)( 9,25)(10,26)(11,17)(12,18)(15,31,16,32) (19,28)(20,27)(21,30,22,29)$ |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $16$ | $4$ | $( 1, 2)( 5,23, 6,24)( 7,13, 8,14)( 9,26)(10,25)(11,18)(12,17)(15,31,16,32) (19,27)(20,28)(21,30,22,29)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $16$ | $2$ | $( 1, 3)( 2, 4)( 5,23)( 6,24)( 7,13)( 8,14)( 9,27)(10,28)(15,31)(16,32)(17,18) (19,25)(20,26)(21,30)(22,29)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $16$ | $2$ | $( 1, 4)( 2, 3)( 5,23)( 6,24)( 7,13)( 8,14)( 9,28)(10,27)(11,12)(15,31)(16,32) (19,26)(20,25)(21,30)(22,29)$ |
| $ 16, 16 $ | $8$ | $16$ | $( 1, 9,18,26, 4,19,11,28, 2,10,17,25, 3,20,12,27)( 5,23,22,32, 7,14,16,29, 6, 24,21,31, 8,13,15,30)$ |
| $ 16, 16 $ | $8$ | $16$ | $( 1,10,18,25, 4,20,11,27, 2, 9,17,26, 3,19,12,28)( 5,23,22,32, 7,14,16,29, 6, 24,21,31, 8,13,15,30)$ |
| $ 16, 16 $ | $8$ | $16$ | $( 1,19,18,28, 4,10,11,25, 2,20,17,27, 3, 9,12,26)( 5,23,21,31, 7,14,15,30, 6, 24,22,32, 8,13,16,29)$ |
| $ 16, 16 $ | $8$ | $16$ | $( 1,20,18,27, 4, 9,11,26, 2,19,17,28, 3,10,12,25)( 5,23,21,31, 7,14,15,30, 6, 24,22,32, 8,13,16,29)$ |
| $ 8, 8, 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $8$ | $( 5,15, 7,22, 6,16, 8,21)( 9,19,10,20)(11,12)(13,29,24,32,14,30,23,31)(17,18) (25,28,26,27)$ |
| $ 8, 8, 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $8$ | $( 1, 2)( 3, 4)( 5,15, 7,22, 6,16, 8,21)( 9,20,10,19)(13,29,24,32,14,30,23,31) (25,27,26,28)$ |
| $ 8, 8, 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $8$ | $( 1, 3, 2, 4)( 5,16, 7,21, 6,15, 8,22)( 9,10)(11,17,12,18)(13,29,24,32,14,30, 23,31)(19,20)$ |
| $ 8, 8, 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $8$ | $( 1, 4, 2, 3)( 5,16, 7,21, 6,15, 8,22)(11,18,12,17)(13,29,24,32,14,30,23,31) (25,26)(27,28)$ |
| $ 32 $ | $16$ | $32$ | $( 1,29,19,21,11,24,26, 7, 3,31, 9,15,18,13,27, 6, 2,30,20,22,12,23,25, 8, 4, 32,10,16,17,14,28, 5)$ |
| $ 32 $ | $16$ | $32$ | $( 1,29,20,22,11,24,25, 8, 3,31,10,16,18,13,28, 5, 2,30,19,21,12,23,26, 7, 4, 32, 9,15,17,14,27, 6)$ |
| $ 32 $ | $16$ | $32$ | $( 1,29,10,15,12,23,27, 5, 3,31,19,22,17,14,25, 7, 2,30, 9,16,11,24,28, 6, 4, 32,20,21,18,13,26, 8)$ |
| $ 32 $ | $16$ | $32$ | $( 1,29, 9,16,12,23,28, 6, 3,31,20,21,17,14,26, 8, 2,30,10,15,11,24,27, 5, 4, 32,19,22,18,13,25, 7)$ |
| $ 8, 8, 8, 8 $ | $16$ | $8$ | $( 1,32, 4,29, 2,31, 3,30)( 5, 9, 7,19, 6,10, 8,20)(11,24,17,13,12,23,18,14) (15,27,22,26,16,28,21,25)$ |
| $ 8, 8, 8, 8 $ | $16$ | $8$ | $( 1,32, 3,30, 2,31, 4,29)( 5,10, 8,19, 6, 9, 7,20)(11,24,18,14,12,23,17,13) (15,28,21,26,16,27,22,25)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1,32)( 2,31)( 3,30)( 4,29)( 5,19)( 6,20)( 7,10)( 8, 9)(11,24)(12,23)(13,17) (14,18)(15,26)(16,25)(21,27)(22,28)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,32, 2,31)( 3,30, 4,29)( 5,20, 6,19)( 7, 9, 8,10)(11,24,12,23)(13,18,14,17) (15,25,16,26)(21,28,22,27)$ |
| $ 16, 16 $ | $16$ | $16$ | $( 1,21,17, 7, 4,15,12, 6, 2,22,18, 8, 3,16,11, 5)( 9,31,25,14,19,30,27,24,10, 32,26,13,20,29,28,23)$ |
| $ 16, 16 $ | $16$ | $16$ | $( 1,21,18, 8, 4,15,11, 5, 2,22,17, 7, 3,16,12, 6)( 9,31,26,13,19,30,28,23,10, 32,25,14,20,29,27,24)$ |
| $ 16, 16 $ | $16$ | $16$ | $( 1,22,12, 5, 3,15,17, 8, 2,21,11, 6, 4,16,18, 7)( 9,31,27,24,20,29,25,14,10, 32,28,23,19,30,26,13)$ |
| $ 16, 16 $ | $16$ | $16$ | $( 1,22,11, 6, 3,15,18, 7, 2,21,12, 5, 4,16,17, 8)( 9,31,28,23,20,29,26,13,10, 32,27,24,19,30,25,14)$ |
| $ 32 $ | $16$ | $32$ | $( 1, 6,10,29,11,21,28,23, 3, 8,19,31,18,15,26,14, 2, 5, 9,30,12,22,27,24, 4, 7,20,32,17,16,25,13)$ |
| $ 32 $ | $16$ | $32$ | $( 1, 6, 9,30,11,21,27,24, 3, 8,20,32,18,15,25,13, 2, 5,10,29,12,22,28,23, 4, 7,19,31,17,16,26,14)$ |
| $ 32 $ | $16$ | $32$ | $( 1, 5,19,31,12,21,25,13, 3, 7, 9,30,17,15,28,23, 2, 6,20,32,11,22,26,14, 4, 8,10,29,18,16,27,24)$ |
| $ 32 $ | $16$ | $32$ | $( 1, 5,20,32,12,21,26,14, 3, 7,10,29,17,15,27,24, 2, 6,19,31,11,22,25,13, 4, 8, 9,30,18,16,28,23)$ |
Group invariants
| Order: | $512=2^{9}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |