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Properties

Label 18T36
Degree $18$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\SOPlus(4,2)$

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Show commands: Magma

magma: G := TransitiveGroup(18, 36);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $36$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\SOPlus(4,2)$
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:  (1,10)(2,9)(3,6)(4,5)(7,18)(8,17)(11,12)(13,14)(15,16), (3,10,18,11)(4,9,17,12)(5,8,15,14)(6,7,16,13)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: None

Degree 9: $S_3^2:C_2$

Low degree siblings

6T13 x 2, 9T16, 12T34 x 2, 12T35 x 2, 12T36 x 2, 18T34 x 2, 24T72 x 2, 36T53, 36T54 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 $ $18$ $4$ $( 3,10,18,11)( 4, 9,17,12)( 5, 8,15,14)( 6, 7,16,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $9$ $2$ $( 3,18)( 4,17)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12)(10,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,11)(10,12)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3, 9)( 4,10)( 5, 6)( 7,14)( 8,13)(11,17)(12,18)(15,16)$
$ 6, 6, 6 $ $12$ $6$ $( 1, 3, 6, 8,16,11)( 2, 4, 5, 7,15,12)( 9,18,13,10,17,14)$
$ 6, 6, 6 $ $12$ $6$ $( 1, 3,17, 2, 4,18)( 5,16,10,13, 8,12)( 6,15, 9,14, 7,11)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 4,17)( 2, 3,18)( 5, 8,10)( 6, 7, 9)(11,14,15)(12,13,16)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 6,16)( 2, 5,15)( 3, 8,11)( 4, 7,12)( 9,13,17)(10,14,18)$

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:  72.40
magma: IdentifyGroup(G);
 
Character table:    
     2  3  2  3  2  2  1  1  1  1
     3  2  .  .  1  1  1  1  2  2

       1a 4a 2a 2b 2c 6a 6b 3a 3b
    2P 1a 2a 1a 1a 1a 3b 3a 3a 3b
    3P 1a 4a 2a 2b 2c 2c 2b 1a 1a
    5P 1a 4a 2a 2b 2c 6a 6b 3a 3b

X.1     1  1  1  1  1  1  1  1  1
X.2     1 -1  1 -1  1  1 -1  1  1
X.3     1 -1  1  1 -1 -1  1  1  1
X.4     1  1  1 -1 -1 -1 -1  1  1
X.5     2  . -2  .  .  .  .  2  2
X.6     4  .  . -2  .  .  1  1 -2
X.7     4  .  .  . -2  1  . -2  1
X.8     4  .  .  .  2 -1  . -2  1
X.9     4  .  .  2  .  . -1  1 -2

magma: CharacterTable(G);