Group action invariants
| Degree $n$ : | $46$ | |
| Transitive number $t$ : | $27$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,34,26)(2,33,25)(3,15,11,24,7,36)(4,16,12,23,8,35)(5,27,14)(6,28,13)(9,44,42,20,39,37)(10,43,41,19,40,38)(17,31)(18,32)(21,45)(22,46), (1,12,46,13,17,2,11,45,14,18)(3,8,5,10,29,4,7,6,9,30)(15,19,44,21,27,16,20,43,22,28)(23,42,32,36,25,24,41,31,35,26)(33,34)(37,38)(39,40) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 10200960: $M_{23}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 23: $M_{23}$
Low degree siblings
There are no siblings 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, 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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44) (45,46)$ |
| $ 22, 22, 2 $ | $927360$ | $22$ | $( 1,25, 9,33,37,19, 5,12,15,43,29, 2,26,10,34,38,20, 6,11,16,44,30) ( 3,45,22,32,42,35,39,13,24, 8,17, 4,46,21,31,41,36,40,14,23, 7,18)(27,28)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $927360$ | $11$ | $( 1,26, 9,34,37,20, 5,11,15,44,29)( 2,25,10,33,38,19, 6,12,16,43,30) ( 3,46,22,31,42,36,39,14,24, 7,17)( 4,45,21,32,41,35,40,13,23, 8,18)$ |
| $ 22, 22, 2 $ | $927360$ | $22$ | $( 1,30,44,16,11, 6,20,38,34,10,26, 2,29,43,15,12, 5,19,37,33, 9,25) ( 3,18, 7,23,14,40,36,41,31,21,46, 4,17, 8,24,13,39,35,42,32,22,45)(27,28)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $927360$ | $11$ | $( 1,29,44,15,11, 5,20,37,34, 9,26)( 2,30,43,16,12, 6,19,38,33,10,25) ( 3,17, 7,24,14,39,36,42,31,22,46)( 4,18, 8,23,13,40,35,41,32,21,45)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3795$ | $2$ | $( 1, 9)( 2,10)( 5,37)( 6,38)( 7,26)( 8,25)(11,42)(12,41)(15,17)(16,18)(19,43) (20,44)(21,23)(22,24)(33,45)(34,46)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3795$ | $2$ | $( 1,10)( 2, 9)( 3, 4)( 5,38)( 6,37)( 7,25)( 8,26)(11,41)(12,42)(13,14)(15,18) (16,17)(19,44)(20,43)(21,24)(22,23)(27,28)(29,30)(31,32)(33,46)(34,45)(35,36) (39,40)$ |
| $ 7, 7, 7, 7, 7, 7, 1, 1, 1, 1 $ | $728640$ | $7$ | $( 3,29,36,14,31,27,39)( 4,30,35,13,32,28,40)( 5,46,44,24,42, 7,15) ( 6,45,43,23,41, 8,16)(11,26,17,37,34,20,22)(12,25,18,38,33,19,21)$ |
| $ 14, 14, 14, 2, 2 $ | $728640$ | $14$ | $( 1, 2)( 3,30,36,13,31,28,39, 4,29,35,14,32,27,40)( 5,45,44,23,42, 8,15, 6,46, 43,24,41, 7,16)( 9,10)(11,25,17,38,34,19,22,12,26,18,37,33,20,21)$ |
| $ 14, 14, 14, 2, 2 $ | $728640$ | $14$ | $( 1, 2)( 3,40,27,32,14,35,29, 4,39,28,31,13,36,30)( 5,16, 7,41,24,43,46, 6,15, 8,42,23,44,45)( 9,10)(11,21,20,33,37,18,26,12,22,19,34,38,17,25)$ |
| $ 7, 7, 7, 7, 7, 7, 1, 1, 1, 1 $ | $728640$ | $7$ | $( 3,39,27,31,14,36,29)( 4,40,28,32,13,35,30)( 5,15, 7,42,24,44,46) ( 6,16, 8,41,23,43,45)(11,22,20,34,37,17,26)(12,21,19,33,38,18,25)$ |
| $ 14, 14, 7, 7, 2, 2 $ | $728640$ | $14$ | $( 1, 9)( 2,10)( 3,31,29,27,36,39,14)( 4,32,30,28,35,40,13)( 5,11,46,26,44,17, 24,37,42,34, 7,20,15,22)( 6,12,45,25,43,18,23,38,41,33, 8,19,16,21)$ |
| $ 14, 14, 14, 2, 2 $ | $728640$ | $14$ | $( 1,10)( 2, 9)( 3,32,29,28,36,40,14, 4,31,30,27,35,39,13)( 5,12,46,25,44,18, 24,38,42,33, 7,19,15,21)( 6,11,45,26,43,17,23,37,41,34, 8,20,16,22)$ |
| $ 14, 14, 7, 7, 2, 2 $ | $728640$ | $14$ | $( 1, 9)( 2,10)( 3,27,14,29,39,31,36)( 4,28,13,30,40,32,35)( 5,26,24,34,15,11, 44,37, 7,22,46,17,42,20)( 6,25,23,33,16,12,43,38, 8,21,45,18,41,19)$ |
| $ 14, 14, 14, 2, 2 $ | $728640$ | $14$ | $( 1,10)( 2, 9)( 3,28,14,30,39,32,36, 4,27,13,29,40,31,35)( 5,25,24,33,15,12, 44,38, 7,21,46,18,42,19)( 6,26,23,34,16,11,43,37, 8,22,45,17,41,20)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $56672$ | $3$ | $( 3,14,29)( 4,13,30)( 5,26,46)( 6,25,45)( 7,34,37)( 8,33,38)(15,20,24) (16,19,23)(17,44,22)(18,43,21)(27,39,36)(28,40,35)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2, 2, 2 $ | $56672$ | $6$ | $( 1, 2)( 3,13,29, 4,14,30)( 5,25,46, 6,26,45)( 7,33,37, 8,34,38)( 9,10)(11,12) (15,19,24,16,20,23)(17,43,22,18,44,21)(27,40,36,28,39,35)(31,32)(41,42)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2, 2, 2 $ | $850080$ | $6$ | $( 1,41)( 2,42)( 3,30,14, 4,29,13)( 5,23,26,16,46,19)( 6,24,25,15,45,20) ( 7,18,34,43,37,21)( 8,17,33,44,38,22)( 9,12)(10,11)(27,35,39,28,36,40)(31,32)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1 $ | $850080$ | $6$ | $( 1,42)( 2,41)( 3,29,14)( 4,30,13)( 5,24,26,15,46,20)( 6,23,25,16,45,19) ( 7,17,34,44,37,22)( 8,18,33,43,38,21)( 9,11)(10,12)(27,36,39)(28,35,40)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $318780$ | $4$ | $( 1,24,15,26)( 2,23,16,25)( 3,29,39,31)( 4,30,40,32)( 5,42,37,11)( 6,41,38,12) ( 7, 9,22,17)( 8,10,21,18)(13,28)(14,27)(33,45)(34,46)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2 $ | $318780$ | $4$ | $( 1,23,15,25)( 2,24,16,26)( 3,30,39,32)( 4,29,40,31)( 5,41,37,12)( 6,42,38,11) ( 7,10,22,18)( 8, 9,21,17)(13,27)(14,28)(19,20)(33,46)(34,45)(35,36)(43,44)$ |
| $ 8, 8, 8, 8, 4, 4, 2, 2, 1, 1 $ | $1275120$ | $8$ | $( 1,39,26,29,15, 3,24,31)( 2,40,25,30,16, 4,23,32)( 5,17,11,22,37, 9,42, 7) ( 6,18,12,21,38,10,41, 8)(13,33,28,45)(14,34,27,46)(19,43)(20,44)$ |
| $ 8, 8, 8, 8, 4, 4, 2, 2, 2 $ | $1275120$ | $8$ | $( 1,40,26,30,15, 4,24,32)( 2,39,25,29,16, 3,23,31)( 5,18,11,21,37,10,42, 8) ( 6,17,12,22,38, 9,41, 7)(13,34,28,46)(14,33,27,45)(19,44)(20,43)(35,36)$ |
| $ 10, 10, 10, 10, 2, 2, 2 $ | $680064$ | $10$ | $( 1,12, 9,38,27, 2,11,10,37,28)( 3,19,29,43, 7, 4,20,30,44, 8)( 5,35,17,33,39, 6,36,18,34,40)(13,14)(15,45,22,32,42,16,46,21,31,41)(23,24)(25,26)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1 $ | $680064$ | $5$ | $( 1,11, 9,37,27)( 2,12,10,38,28)( 3,20,29,44, 7)( 4,19,30,43, 8) ( 5,36,17,34,39)( 6,35,18,33,40)(15,46,22,31,42)(16,45,21,32,41)$ |
| $ 30, 10, 6 $ | $680064$ | $30$ | $( 1, 8, 5,10,20,18,27,43,39,12, 3,35,37,30,34, 2, 7, 6, 9,19,17,28,44,40,11, 4,36,38,29,33)(13,24,25,14,23,26)(15,41,31,21,46,16,42,32,22,45)$ |
| $ 15, 15, 5, 5, 3, 3 $ | $680064$ | $15$ | $( 1, 7, 5, 9,20,17,27,44,39,11, 3,36,37,29,34)( 2, 8, 6,10,19,18,28,43,40,12, 4,35,38,30,33)(13,23,25)(14,24,26)(15,42,31,22,46)(16,41,32,21,45)$ |
| $ 30, 10, 6 $ | $680064$ | $30$ | $( 1,35,44,10,34, 4,27, 6,29,12,17, 8,37,40,20, 2,36,43, 9,33, 3,28, 5,30,11, 18, 7,38,39,19)(13,26,23,14,25,24)(15,41,31,21,46,16,42,32,22,45)$ |
| $ 15, 15, 5, 5, 3, 3 $ | $680064$ | $15$ | $( 1,36,44, 9,34, 3,27, 5,29,11,17, 7,37,39,20)( 2,35,43,10,33, 4,28, 6,30,12, 18, 8,38,40,19)(13,25,23)(14,26,24)(15,42,31,22,46)(16,41,32,21,45)$ |
| $ 23, 23 $ | $443520$ | $23$ | $( 1,31, 3,27,17, 9,20,22,11,39, 7,37,36,14,34,44,26,46,15,29,42, 5,24) ( 2,32, 4,28,18,10,19,21,12,40, 8,38,35,13,33,43,25,45,16,30,41, 6,23)$ |
| $ 46 $ | $443520$ | $46$ | $( 1,32, 3,28,17,10,20,21,11,40, 7,38,36,13,34,43,26,45,15,30,42, 6,24, 2,31, 4,27,18, 9,19,22,12,39, 8,37,35,14,33,44,25,46,16,29,41, 5,23)$ |
| $ 46 $ | $443520$ | $46$ | $( 1,23, 5,41,29,16,46,25,44,33,14,35,37, 8,39,12,22,19, 9,18,27, 4,31, 2,24, 6,42,30,15,45,26,43,34,13,36,38, 7,40,11,21,20,10,17,28, 3,32)$ |
| $ 23, 23 $ | $443520$ | $23$ | $( 1,24, 5,42,29,15,46,26,44,34,14,36,37, 7,39,11,22,20, 9,17,27, 3,31) ( 2,23, 6,41,30,16,45,25,43,33,13,35,38, 8,40,12,21,19,10,18,28, 4,32)$ |
Group invariants
| Order: | $20401920=2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 23$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |