Properties

 Label 22T34 Order $$112640$$ n $$22$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No

Related objects

Group action invariants

 Degree $n$ : $22$ Transitive number $t$ : $34$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (3,16,12,6,7,22,10,13,20,17)(4,15,11,5,8,21,9,14,19,18), (1,15,14,7,12,2,16,13,8,11)(3,21,9,18,20)(4,22,10,17,19)(5,6) $|\Aut(F/K)|$: $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
10:  $C_{10}$
110:  $F_{11}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 11: $F_{11}$

Low degree siblings

22T35, 44T312, 44T313, 44T316

Siblings 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $11$ $2$ $( 1, 2)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$ $2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $55$ $2$ $( 3, 4)(15,16)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $110$ $2$ $( 1, 2)( 5, 6)( 7, 8)( 9,10)(13,14)(17,18)(19,20)(21,22)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $55$ $2$ $( 1, 2)( 3, 4)(11,12)(15,16)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $110$ $2$ $( 7, 8)( 9,10)(13,14)(17,18)(19,20)(21,22)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $110$ $2$ $( 1, 2)( 3, 4)( 5, 6)(11,12)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $55$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $110$ $2$ $( 1, 2)( 3, 4)( 5, 6)(15,16)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $110$ $2$ $( 9,10)(13,14)(15,16)(17,18)(19,20)(21,22)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $110$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(11,12)(15,16)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $55$ $2$ $( 9,10)(11,12)(13,14)(17,18)(19,20)(21,22)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $55$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $55$ $2$ $( 5, 6)( 7, 8)( 9,10)(17,18)(19,20)(21,22)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $22$ $2$ $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)$ $11, 11$ $10240$ $11$ $( 1,19,15,11, 8, 3,21,18,13,10, 5)( 2,20,16,12, 7, 4,22,17,14, 9, 6)$ $5, 5, 5, 5, 1, 1$ $2816$ $5$ $( 3,12, 7,10,20)( 4,11, 8, 9,19)( 5,21,14,18,15)( 6,22,13,17,16)$ $10, 5, 5, 2$ $5632$ $10$ $( 1, 2)( 3,11, 7, 9,20)( 4,12, 8,10,19)( 5,22,14,17,15, 6,21,13,18,16)$ $10, 10, 1, 1$ $2816$ $10$ $( 3,12, 7,10,20, 4,11, 8, 9,19)( 5,21,14,18,16, 6,22,13,17,15)$ $5, 5, 5, 5, 1, 1$ $2816$ $5$ $( 3, 7,20,12,10)( 4, 8,19,11, 9)( 5,14,15,21,18)( 6,13,16,22,17)$ $10, 5, 5, 2$ $5632$ $10$ $( 1, 2)( 3, 8,20,11,10)( 4, 7,19,12, 9)( 5,13,15,22,18, 6,14,16,21,17)$ $10, 10, 1, 1$ $2816$ $10$ $( 3, 7,20,12,10, 4, 8,19,11, 9)( 5,14,16,22,17, 6,13,15,21,18)$ $5, 5, 5, 5, 1, 1$ $2816$ $5$ $( 3,20,10, 7,12)( 4,19, 9, 8,11)( 5,15,18,14,21)( 6,16,17,13,22)$ $10, 5, 5, 2$ $5632$ $10$ $( 1, 2)( 3,19,10, 8,12)( 4,20, 9, 7,11)( 5,16,18,13,21, 6,15,17,14,22)$ $10, 10, 1, 1$ $2816$ $10$ $( 3,20,10, 7,12, 4,19, 9, 8,11)( 5,16,17,13,22, 6,15,18,14,21)$ $5, 5, 5, 5, 1, 1$ $2816$ $5$ $( 3,10,12,20, 7)( 4, 9,11,19, 8)( 5,18,21,15,14)( 6,17,22,16,13)$ $10, 5, 5, 2$ $5632$ $10$ $( 1, 2)( 3, 9,12,19, 7)( 4,10,11,20, 8)( 5,17,21,16,14, 6,18,22,15,13)$ $10, 10, 1, 1$ $2816$ $10$ $( 3,10,12,20, 7, 4, 9,11,19, 8)( 5,18,21,16,13, 6,17,22,15,14)$ $10, 10, 1, 1$ $5632$ $10$ $( 3,16,12, 6, 7,22,10,13,20,17)( 4,15,11, 5, 8,21, 9,14,19,18)$ $20, 2$ $5632$ $20$ $( 1, 2)( 3,15,12, 5, 7,21,10,14,20,18, 4,16,11, 6, 8,22, 9,13,19,17)$ $10, 10, 1, 1$ $5632$ $10$ $( 3, 6,10,17,12,22,20,16, 7,13)( 4, 5, 9,18,11,21,19,15, 8,14)$ $20, 2$ $5632$ $20$ $( 1, 2)( 3, 5,10,18,12,21,20,15, 7,14, 4, 6, 9,17,11,22,19,16, 8,13)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $352$ $2$ $( 3,22)( 4,21)( 5,19)( 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,14)(12,13)$ $4, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1760$ $4$ $( 1, 2)( 3,21, 4,22)( 5,20)( 6,19)( 7,18)( 8,17)( 9,16)(10,15)(11,13)(12,14)$ $4, 4, 2, 2, 2, 2, 2, 2, 1, 1$ $1760$ $4$ $( 3,22, 4,21)( 5,19)( 6,20)( 7,17)( 8,18)( 9,16,10,15)(11,14)(12,13)$ $4, 4, 4, 2, 2, 2, 2, 2$ $1760$ $4$ $( 1, 2)( 3,21, 4,22)( 5,20)( 6,19)( 7,18)( 8,17)( 9,15,10,16)(11,13,12,14)$ $4, 4, 2, 2, 2, 2, 2, 2, 1, 1$ $1760$ $4$ $( 3,22, 4,21)( 5,19)( 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,14,12,13)$ $4, 4, 4, 4, 2, 2, 1, 1$ $1760$ $4$ $( 3,21, 4,22)( 5,20, 6,19)( 7,18)( 8,17)( 9,15,10,16)(11,13,12,14)$ $4, 4, 4, 2, 2, 2, 2, 2$ $1760$ $4$ $( 1, 2)( 3,22, 4,21)( 5,19, 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,14,12,13)$ $4, 4, 4, 4, 4, 2$ $352$ $4$ $( 1, 2)( 3,22, 4,21)( 5,19, 6,20)( 7,17, 8,18)( 9,16,10,15)(11,14,12,13)$ $10, 10, 1, 1$ $5632$ $10$ $( 3,17,20,13,10,22, 7, 6,12,16)( 4,18,19,14, 9,21, 8, 5,11,15)$ $20, 2$ $5632$ $20$ $( 1, 2)( 3,18,20,14,10,21, 7, 5,12,15, 4,17,19,13, 9,22, 8, 6,11,16)$ $10, 10, 1, 1$ $5632$ $10$ $( 3,13, 7,16,20,22,12,17,10, 6)( 4,14, 8,15,19,21,11,18, 9, 5)$ $20, 2$ $5632$ $20$ $( 1, 2)( 3,14, 7,15,20,21,12,18,10, 5, 4,13, 8,16,19,22,11,17, 9, 6)$

Group invariants

 Order: $112640=2^{11} \cdot 5 \cdot 11$ Cyclic: No Abelian: No Solvable: Yes GAP id: Data not available
 Character table: Data not available.