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Properties

Label 18T26
Degree $18$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2:C_{18}$

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Show commands: Magma

magma: G := TransitiveGroup(18, 26);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $26$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2:C_{18}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7,16,6,11,13,3,10,17)(2,8,15,5,12,14,4,9,18), (1,8,15,5,11,13,3,9,18,2,7,16,6,12,14,4,10,17)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$9$:  $C_9$
$12$:  $A_4$
$18$:  $C_{18}$
$24$:  $A_4\times C_2$
$36$:  $C_2^2 : C_9$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4\times C_2$

Degree 9: $C_9$

Low degree siblings

36T16, 36T30

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

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:  72.16
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);