Properties

Label 18T35
Order \(72\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $PSU(3,2)$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $35$
Group :  $PSU(3,2)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,15,17,5)(4,16,18,6)(7,11,13,9)(8,12,14,10), (1,7,12,6)(2,8,11,5)(3,18,15,13)(4,17,16,14)(9,10)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: None

Degree 9: $C_3^2:Q_8$

Low degree siblings

9T14, 12T47, 18T35 x 2

Siblings are shown 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 $ $1$ $1$ $()$
$ 4, 4, 4, 4, 1, 1 $ $18$ $4$ $( 3, 5,17,15)( 4, 6,18,16)( 7, 9,13,11)( 8,10,14,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $9$ $2$ $( 3,17)( 4,18)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11)(10,12)$
$ 4, 4, 4, 4, 2 $ $18$ $4$ $( 1, 2)( 3, 7,17,13)( 4, 8,18,14)( 5,11,15, 9)( 6,12,16,10)$
$ 4, 4, 4, 4, 2 $ $18$ $4$ $( 1, 2)( 3, 9,17,11)( 4,10,18,12)( 5, 7,15,13)( 6, 8,16,14)$
$ 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1, 3,17)( 2, 4,18)( 5, 8,10)( 6, 7, 9)(11,13,16)(12,14,15)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 41]
Character table:   
     2  3  2  3  2  2  .
     3  2  .  .  .  .  2

       1a 4a 2a 4b 4c 3a
    2P 1a 2a 1a 2a 2a 3a
    3P 1a 4a 2a 4b 4c 1a

X.1     1  1  1  1  1  1
X.2     1 -1  1 -1  1  1
X.3     1 -1  1  1 -1  1
X.4     1  1  1 -1 -1  1
X.5     2  . -2  .  .  2
X.6     8  .  .  .  . -1