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Properties

Label 18T35
Degree $18$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\PSU(3,2)$

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Show commands: Magma

magma: G := TransitiveGroup(18, 35);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $35$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSU(3,2)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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)
magma: Generators(G);
 

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, 24T82, 36T55

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  72.41
magma: IdentifyGroup(G);
 
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

magma: CharacterTable(G);