# Properties

 Label 22T38 Degree $22$ Order $443520$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $M_{22}$

# Related objects

## Group action invariants

 Degree $n$: $22$ Transitive number $t$: $38$ Group: $M_{22}$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,17,9,5,10,22,8)(3,20,15,12,19,11,14)(4,21,16,13,7,18,6), (1,5,10)(2,17,12)(3,8,4)(6,16,19)(9,18,21)(14,20,22)

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 11: 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 5, 5, 1, 1$ $88704$ $5$ $( 1,21, 9,12, 6)( 2,16,11,22,13)( 3, 8,17,20,10)( 5,15,19, 7,14)$ $7, 7, 7, 1$ $63360$ $7$ $( 1,14,19,18,13,16, 7)( 2, 4,22,20, 8,12,10)( 3, 9, 6,11,17,15,21)$ $7, 7, 7, 1$ $63360$ $7$ $( 1, 7,16,13,18,19,14)( 2,10,12, 8,20,22, 4)( 3,21,15,17,11, 6, 9)$ $11, 11$ $40320$ $11$ $( 1, 2,17,22,20, 7,16, 8, 3, 5, 4)( 6,18,14,19,11, 9,13,15,12,21,10)$ $11, 11$ $40320$ $11$ $( 1, 4, 5, 3, 8,16, 7,20,22,17, 2)( 6,10,21,12,15,13, 9,11,19,14,18)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $1155$ $2$ $( 2, 9)( 3,22)( 5,10)( 6,15)( 7,13)( 8,21)(14,18)(17,20)$ $4, 4, 4, 4, 2, 2, 1, 1$ $13860$ $4$ $( 2,14, 9,18)( 3, 5,22,10)( 4,16)( 6, 8,15,21)( 7,17,13,20)(12,19)$ $8, 8, 4, 2$ $55440$ $8$ $( 1,11)( 2,17,14,13, 9,20,18, 7)( 3,15, 5,21,22, 6,10, 8)( 4,12,16,19)$ $3, 3, 3, 3, 3, 3, 1, 1, 1, 1$ $12320$ $3$ $( 1,21, 6)( 2,22,18)( 4,15, 8)( 5,17,12)( 7,14, 9)(10,19,20)$ $6, 6, 3, 3, 2, 2$ $36960$ $6$ $( 1, 9,21, 7, 6,14)( 2,15,22, 8,18, 4)( 3,16)( 5,12,17)(10,20,19)(11,13)$ $4, 4, 4, 4, 2, 2, 1, 1$ $27720$ $4$ $( 2,22,21,20)( 3, 8,17, 9)( 4,16)( 5,14,13, 6)( 7,15,10,18)(12,19)$

## Group invariants

 Order: $443520=2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$ Cyclic: no Abelian: no Solvable: no GAP id: not available
 Character table:  2 7 . . . . 7 2 2 5 3 . 4 3 2 . . . . 1 2 1 . . . . 5 1 . . . . . . . . . 1 . 7 1 1 1 . . . . . . . . . 11 1 . . 1 1 . . . . . . . 1a 7a 7b 11a 11b 2a 3a 6a 4a 8a 5a 4b 2P 1a 7a 7b 11b 11a 1a 3a 3a 2a 4a 5a 2a 3P 1a 7b 7a 11a 11b 2a 1a 2a 4a 8a 5a 4b 5P 1a 7b 7a 11a 11b 2a 3a 6a 4a 8a 1a 4b 7P 1a 1a 1a 11b 11a 2a 3a 6a 4a 8a 5a 4b 11P 1a 7a 7b 1a 1a 2a 3a 6a 4a 8a 5a 4b X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 21 . . -1 -1 5 3 -1 1 -1 1 1 X.3 45 A /A 1 1 -3 . . 1 -1 . 1 X.4 45 /A A 1 1 -3 . . 1 -1 . 1 X.5 55 -1 -1 . . 7 1 1 3 1 . -1 X.6 99 1 1 . . 3 . . 3 -1 -1 -1 X.7 154 . . . . 10 1 1 -2 . -1 2 X.8 210 . . 1 1 2 3 -1 -2 . . -2 X.9 231 . . . . 7 -3 1 -1 -1 1 -1 X.10 280 . . B /B -8 1 1 . . . . X.11 280 . . /B B -8 1 1 . . . . X.12 385 . . . . 1 -2 -2 1 1 . 1 A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 B = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10 = (-1-Sqrt(-11))/2 = -1-b11