Group action invariants
| Degree $n$: | $23$ | |
| Transitive number $t$: | $5$ | |
| Group: | $M_{23}$ | |
| Parity: | $1$ | |
| Primitive: | yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,7,12,16,19,21,6)(2,8,13,17,20,5,11)(3,9,14,18,4,10,15), (2,14,18,20,8)(3,7,12,13,19)(4,21,17,15,10)(5,11,16,6,9), (3,19)(4,14)(5,20)(6,10)(8,15)(11,18)(17,21)(22,23), (1,22)(2,10)(3,14)(4,17)(8,15)(9,11)(13,20)(19,21) |
Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $3795$ | $2$ | $( 2, 4)( 3, 8)( 5, 9)( 7,12)(13,17)(14,23)(15,18)(16,22)$ |
| $ 4, 4, 4, 4, 2, 2, 1, 1, 1 $ | $318780$ | $4$ | $( 1,10)( 2, 7, 4,12)( 3,23, 8,14)( 5,18, 9,15)( 6,20)(13,22,17,16)$ |
| $ 8, 8, 4, 2, 1 $ | $1275120$ | $8$ | $( 1,20,10, 6)( 2,16, 7,13, 4,22,12,17)( 3, 9,23,15, 8, 5,14,18)(11,21)$ |
| $ 7, 7, 7, 1, 1 $ | $728640$ | $7$ | $( 1, 8,11,14, 6, 2,10)( 4,16,19,12,17,22,20)( 7,18,13, 9,21,15,23)$ |
| $ 7, 7, 7, 1, 1 $ | $728640$ | $7$ | $( 1,10, 2, 6,14,11, 8)( 4,20,22,17,12,19,16)( 7,23,15,21, 9,13,18)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $56672$ | $3$ | $( 1,13,10)( 2,17,15)( 3, 4,20)( 6,18,16)( 8,14,22)( 9,21,19)$ |
| $ 5, 5, 5, 5, 1, 1, 1 $ | $680064$ | $5$ | $( 1,21,16,22, 2)( 5,12,23,11, 7)( 6, 8,17,13,19)( 9,18,14,15,10)$ |
| $ 15, 5, 3 $ | $680064$ | $15$ | $( 1,18,17,21,14,13,16,15,19,22,10, 6, 2, 9, 8)( 3,20, 4)( 5,23, 7,12,11)$ |
| $ 15, 5, 3 $ | $680064$ | $15$ | $( 1, 6,15,21, 8,10,16,17, 9,22,13,18, 2,19,14)( 3, 4,20)( 5,23, 7,12,11)$ |
| $ 11, 11, 1 $ | $927360$ | $11$ | $( 1,14,20,18, 2, 4,15,13,11,22, 3)( 6,17,16, 9,19, 7, 8,21,10,23,12)$ |
| $ 11, 11, 1 $ | $927360$ | $11$ | $( 1, 3,22,11,13,15, 4, 2,18,20,14)( 6,12,23,10,21, 8, 7,19, 9,16,17)$ |
| $ 6, 6, 3, 3, 2, 2, 1 $ | $850080$ | $6$ | $( 1,10,11)( 2, 7)( 3, 5,16,13,15,23)( 4,12)( 6,19,20)( 8, 9,22,17,18,14)$ |
| $ 14, 7, 2 $ | $728640$ | $14$ | $( 1,15,22,18, 2, 9,17)( 3,13,21,10,20, 4,14,12, 6,19, 8, 7,11,16)( 5,23)$ |
| $ 14, 7, 2 $ | $728640$ | $14$ | $( 1,18,17,22, 9,15, 2)( 3,10,14,19,11,13,20,12, 8,16,21, 4, 6, 7)( 5,23)$ |
| $ 23 $ | $443520$ | $23$ | $( 1, 5, 4, 7,10,12,20, 8,16,14, 3,13,15,19,17,21, 9,22,18,23,11, 6, 2)$ |
| $ 23 $ | $443520$ | $23$ | $( 1, 2, 6,11,23,18,22, 9,21,17,19,15,13, 3,14,16, 8,20,12,10, 7, 4, 5)$ |
Group invariants
| Order: | $10200960=2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 23$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | no | |
| GAP id: | not available |
| Character table: not available. |