# Properties

 Label 22T22 Degree $22$ Order $7920$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $M_{11}$

# Related objects

## Group action invariants

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

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 11: $M_{11}$

## Low degree siblings

11T6, 12T272

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $165$ $2$ $( 1,17)( 2,18)( 5, 8)( 6, 7)( 9,21)(10,22)(11,20)(12,19)$ $3, 3, 3, 3, 3, 3, 1, 1, 1, 1$ $440$ $3$ $( 1, 5,21)( 2, 6,22)( 3,15,13)( 4,16,14)( 7,10,18)( 8, 9,17)$ $6, 6, 3, 3, 2, 2$ $1320$ $6$ $( 1, 9, 5,17,21, 8)( 2,10, 6,18,22, 7)( 3,13,15)( 4,14,16)(11,20)(12,19)$ $11, 11$ $720$ $11$ $( 1, 5,13,11,17, 9,21, 8,15, 3,20)( 2, 6,14,12,18,10,22, 7,16, 4,19)$ $11, 11$ $720$ $11$ $( 1,20, 3,15, 8,21, 9,17,11,13, 5)( 2,19, 4,16, 7,22,10,18,12,14, 6)$ $5, 5, 5, 5, 1, 1$ $1584$ $5$ $( 1, 4,21,14, 7)( 2, 3,22,13, 8)( 5,20,16,10,18)( 6,19,15, 9,17)$ $4, 4, 4, 4, 2, 2, 1, 1$ $990$ $4$ $( 3,20,11,18)( 4,19,12,17)( 5,14,22, 9)( 6,13,21,10)( 7, 8)(15,16)$ $8, 8, 4, 2$ $990$ $8$ $( 1, 2)( 3, 5,20,14,11,22,18, 9)( 4, 6,19,13,12,21,17,10)( 7,16, 8,15)$ $8, 8, 4, 2$ $990$ $8$ $( 1, 2)( 3, 9,18,22,11,14,20, 5)( 4,10,17,21,12,13,19, 6)( 7,15, 8,16)$

## Group invariants

 Order: $7920=2^{4} \cdot 3^{2} \cdot 5 \cdot 11$ Cyclic: no Abelian: no Solvable: no GAP id: not available
 Character table:  2 4 4 1 1 . 3 3 3 . . 3 2 1 2 1 . . . . . . 5 1 . . . 1 . . . . . 11 1 . . . . . . . 1 1 1a 2a 3a 6a 5a 4a 8a 8b 11a 11b 2P 1a 1a 3a 3a 5a 2a 4a 4a 11b 11a 3P 1a 2a 1a 2a 5a 4a 8a 8b 11a 11b 5P 1a 2a 3a 6a 1a 4a 8b 8a 11a 11b 7P 1a 2a 3a 6a 5a 4a 8b 8a 11b 11a 11P 1a 2a 3a 6a 5a 4a 8a 8b 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 X.2 10 2 1 -1 . 2 . . -1 -1 X.3 10 -2 1 1 . . A -A -1 -1 X.4 10 -2 1 1 . . -A A -1 -1 X.5 11 3 2 . 1 -1 -1 -1 . . X.6 16 . -2 . 1 . . . B /B X.7 16 . -2 . 1 . . . /B B X.8 44 4 -1 1 -1 . . . . . X.9 45 -3 . . . 1 -1 -1 1 1 X.10 55 -1 1 -1 . -1 1 1 . . A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 B = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10 = (-1-Sqrt(-11))/2 = -1-b11