Properties

Label 22T38
Order \(443520\)
n \(22\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $M_{22}$

Related objects

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Group action invariants

Degree $n$ :  $22$
Transitive number $t$ :  $38$
Group :  $M_{22}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

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

Group invariants

Order:  $443520=2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:   
      2   7   .   .  .  .  7  5  3  .  2  2  4
      3   2   .   .  .  .  1  .  .  .  2  1  .
      5   1   .   .  .  .  .  .  .  1  .  .  .
      7   1   .   .  1  1  .  .  .  .  .  .  .
     11   1   1   1  .  .  .  .  .  .  .  .  .

         1a 11a 11b 7a 7b 2a 4a 8a 5a 3a 6a 4b
     2P  1a 11b 11a 7a 7b 1a 2a 4a 5a 3a 3a 2a
     3P  1a 11a 11b 7b 7a 2a 4a 8a 5a 1a 2a 4b
     5P  1a 11a 11b 7b 7a 2a 4a 8a 1a 3a 6a 4b
     7P  1a 11b 11a 1a 1a 2a 4a 8a 5a 3a 6a 4b
    11P  1a  1a  1a 7a 7b 2a 4a 8a 5a 3a 6a 4b

X.1       1   1   1  1  1  1  1  1  1  1  1  1
X.2      21  -1  -1  .  .  5  1 -1  1  3 -1  1
X.3      45   1   1  B /B -3  1 -1  .  .  .  1
X.4      45   1   1 /B  B -3  1 -1  .  .  .  1
X.5      55   .   . -1 -1  7  3  1  .  1  1 -1
X.6      99   .   .  1  1  3  3 -1 -1  .  . -1
X.7     154   .   .  .  . 10 -2  . -1  1  1  2
X.8     210   1   1  .  .  2 -2  .  .  3 -1 -2
X.9     231   .   .  .  .  7 -1 -1  1 -3  1 -1
X.10    280   A  /A  .  . -8  .  .  .  1  1  .
X.11    280  /A   A  .  . -8  .  .  .  1  1  .
X.12    385   .   .  .  .  1  1  1  . -2 -2  1

A = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10
  = (-1-Sqrt(-11))/2 = -1-b11
B = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7