# Properties

 Label 22T14 Order $$1320$$ n $$22$$ Cyclic No Abelian No Solvable No Primitive No $p$-group No

# Related objects

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 11: None

## Low degree siblings

12T218, 24T2949

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $66$ $2$ $( 1,16)( 2,20)( 3,12)( 4,21)( 5,17)( 6,19)( 7,22)( 8,18)( 9,15)(10,14)(11,13)$ $5, 5, 5, 5, 1, 1$ $132$ $5$ $( 1, 6, 5, 9, 7)( 2, 3, 4,10, 8)(12,21,14,18,20)(15,22,16,19,17)$ $5, 5, 5, 5, 1, 1$ $132$ $5$ $( 1, 9, 6, 7, 5)( 2,10, 3, 8, 4)(12,18,21,20,14)(15,19,22,17,16)$ $10, 10, 2$ $132$ $10$ $( 1,15, 6,22, 5,16, 9,19, 7,17)( 2,14, 3,18, 4,20,10,12, 8,21)(11,13)$ $10, 10, 2$ $132$ $10$ $( 1,19, 5,15, 7,16, 6,17, 9,22)( 2,12, 4,14, 8,20, 3,21,10,18)(11,13)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $55$ $2$ $( 1, 7)( 2, 4)( 6, 9)( 8,10)(14,18)(15,19)(16,22)(20,21)$ $3, 3, 3, 3, 3, 3, 1, 1, 1, 1$ $110$ $3$ $( 1, 4, 9)( 2, 6, 7)( 3, 5,11)(12,13,17)(15,21,16)(19,20,22)$ $6, 6, 3, 3, 2, 2$ $110$ $6$ $( 1, 6, 4, 7, 9, 2)( 3,11, 5)( 8,10)(12,17,13)(14,18)(15,22,21,19,16,20)$ $4, 4, 4, 4, 2, 2, 2$ $110$ $4$ $( 1,22, 7,16)( 2,15, 4,19)( 3,12)( 5,13)( 6,21, 9,20)( 8,18,10,14)(11,17)$ $12, 6, 4$ $110$ $12$ $( 1,15, 6,22, 4,21, 7,19, 9,16, 2,20)( 3,13,11,12, 5,17)( 8,14,10,18)$ $12, 6, 4$ $110$ $12$ $( 1,19, 6,16, 4,20, 7,15, 9,22, 2,21)( 3,13,11,12, 5,17)( 8,18,10,14)$ $11, 11$ $120$ $11$ $( 1, 6,11, 4, 2, 9,10, 8, 7, 3, 5)(12,16,14,18,19,21,20,13,15,22,17)$

## Group invariants

 Order: $1320=2^{3} \cdot 3 \cdot 5 \cdot 11$ Cyclic: No Abelian: No Solvable: No GAP id: [1320, 133]
 Character table:  2 3 1 1 2 1 1 3 2 2 2 2 2 . 3 1 . . . . . 1 1 1 1 1 1 . 5 1 1 1 1 1 1 . . . . . . . 11 1 . . . . . . . . . . . 1 1a 5a 5b 2a 10a 10b 2b 3a 6a 4a 12a 12b 11a 2P 1a 5b 5a 1a 5b 5a 1a 3a 3a 2b 6a 6a 11a 3P 1a 5b 5a 2a 10b 10a 2b 1a 2b 4a 4a 4a 11a 5P 1a 1a 1a 2a 2a 2a 2b 3a 6a 4a 12b 12a 11a 7P 1a 5b 5a 2a 10b 10a 2b 3a 6a 4a 12b 12a 11a 11P 1a 5a 5b 2a 10a 10b 2b 3a 6a 4a 12a 12b 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 X.3 10 . . . . . -2 1 1 -2 1 1 -1 X.4 10 . . . . . -2 1 1 2 -1 -1 -1 X.5 10 . . . . . 2 -2 2 . . . -1 X.6 10 . . . . . 2 1 -1 . B -B -1 X.7 10 . . . . . 2 1 -1 . -B B -1 X.8 11 1 1 -1 -1 -1 -1 -1 -1 1 1 1 . X.9 11 1 1 1 1 1 -1 -1 -1 -1 -1 -1 . X.10 12 A *A -2 -A -*A . . . . . . 1 X.11 12 *A A -2 -*A -A . . . . . . 1 X.12 12 A *A 2 A *A . . . . . . 1 X.13 12 *A A 2 *A A . . . . . . 1 A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 B = -E(12)^7+E(12)^11 = Sqrt(3) = r3