Group action invariants
| Degree $n$ : | $22$ | |
| Transitive number $t$ : | $33$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,3,10,6,15)(2,4,9,5,16)(7,21,20,13,18,8,22,19,14,17)(11,12), (1,2)(3,12,7,9,19)(4,11,8,10,20)(5,22,14,18,15,6,21,13,17,16) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 5: $C_5$ 55: $C_{11}:C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 11: $C_{11}:C_5$
Low degree siblings
22T33 x 2, 44T288Siblings 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, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 1, 2)( 7, 8)(11,12)(15,16)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 3, 4)(13,14)(15,16)(17,18)(19,20)(21,22)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 5, 6)( 7, 8)(15,16)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)(11,12)(13,14)(15,16)(19,20)(21,22)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $11$ | $2$ | $( 7, 8)(11,12)(13,14)(17,18)(19,20)(21,22)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $(13,14)(17,18)(19,20)(21,22)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 1, 2)( 7, 8)(11,12)(13,14)(15,16)(21,22)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 3, 4)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)(11,12)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 9,10)(11,12)(15,16)(17,18)(19,20)(21,22)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 3, 4)( 9,10)(11,12)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)( 9,10)(13,14)(15,16)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 5, 6)( 7, 8)( 9,10)(11,12)(17,18)(21,22)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 9,10)(13,14)(17,18)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 1, 2)(15,16)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $11$ | $2$ | $( 1, 2)( 9,10)(11,12)(13,14)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)(19,20)(21,22)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 1, 2)( 7, 8)(11,12)(21,22)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 1, 2)( 7, 8)( 9,10)(13,14)(15,16)(21,22)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $11$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(19,20)(21,22)$ |
| $ 11, 11 $ | $5120$ | $11$ | $( 1,16, 7,22,14, 6,20,11, 3,17, 9)( 2,15, 8,21,13, 5,19,12, 4,18,10)$ |
| $ 11, 11 $ | $5120$ | $11$ | $( 1, 7,14,20, 3, 9,16,22, 6,11,17)( 2, 8,13,19, 4,10,15,21, 5,12,18)$ |
| $ 5, 5, 5, 5, 1, 1 $ | $2816$ | $5$ | $( 3,19, 9, 8,11)( 4,20,10, 7,12)( 5,16,17,13,21)( 6,15,18,14,22)$ |
| $ 10, 5, 5, 2 $ | $2816$ | $10$ | $( 1, 2)( 3,20,10, 8,12, 4,19, 9, 7,11)( 5,15,17,13,21)( 6,16,18,14,22)$ |
| $ 10, 10, 1, 1 $ | $2816$ | $10$ | $( 3,20,10, 8,12, 4,19, 9, 7,11)( 5,16,18,13,22, 6,15,17,14,21)$ |
| $ 10, 5, 5, 2 $ | $2816$ | $10$ | $( 1, 2)( 3,19, 9, 8,11)( 4,20,10, 7,12)( 5,15,18,13,22, 6,16,17,14,21)$ |
| $ 5, 5, 5, 5, 1, 1 $ | $2816$ | $5$ | $( 3, 9,11,19, 8)( 4,10,12,20, 7)( 5,17,21,16,13)( 6,18,22,15,14)$ |
| $ 10, 5, 5, 2 $ | $2816$ | $10$ | $( 1, 2)( 3, 9,12,19, 7, 4,10,11,20, 8)( 5,18,22,16,13)( 6,17,21,15,14)$ |
| $ 10, 10, 1, 1 $ | $2816$ | $10$ | $( 3, 9,12,19, 7, 4,10,11,20, 8)( 5,18,21,16,14, 6,17,22,15,13)$ |
| $ 10, 5, 5, 2 $ | $2816$ | $10$ | $( 1, 2)( 3, 9,11,19, 8)( 4,10,12,20, 7)( 5,17,22,16,14, 6,18,21,15,13)$ |
| $ 5, 5, 5, 5, 1, 1 $ | $2816$ | $5$ | $( 3,11, 8, 9,19)( 4,12, 7,10,20)( 5,21,13,17,16)( 6,22,14,18,15)$ |
| $ 10, 5, 5, 2 $ | $2816$ | $10$ | $( 1, 2)( 3,12, 8, 9,20, 4,11, 7,10,19)( 5,21,13,18,16)( 6,22,14,17,15)$ |
| $ 10, 10, 1, 1 $ | $2816$ | $10$ | $( 3,12, 8, 9,20, 4,11, 7,10,19)( 5,22,13,18,15, 6,21,14,17,16)$ |
| $ 10, 5, 5, 2 $ | $2816$ | $10$ | $( 1, 2)( 3,11, 8, 9,19)( 4,12, 7,10,20)( 5,22,13,17,15, 6,21,14,18,16)$ |
| $ 5, 5, 5, 5, 1, 1 $ | $2816$ | $5$ | $( 3, 8,19,11, 9)( 4, 7,20,12,10)( 5,13,16,21,17)( 6,14,15,22,18)$ |
| $ 10, 5, 5, 2 $ | $2816$ | $10$ | $( 1, 2)( 3, 7,19,12,10, 4, 8,20,11, 9)( 5,13,15,22,17)( 6,14,16,21,18)$ |
| $ 10, 10, 1, 1 $ | $2816$ | $10$ | $( 3, 7,19,12,10, 4, 8,20,11, 9)( 5,14,15,21,18, 6,13,16,22,17)$ |
| $ 10, 5, 5, 2 $ | $2816$ | $10$ | $( 1, 2)( 3, 8,19,11, 9)( 4, 7,20,12,10)( 5,14,16,22,18, 6,13,15,21,17)$ |
Group invariants
| Order: | $56320=2^{10} \cdot 5 \cdot 11$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |