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Properties

Label 18T31
Degree $18$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(18, 31);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times A_4$
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:  (1,13,9,5,15,8)(2,14,10,6,16,7)(3,17,11)(4,18,12), (1,17,7)(2,18,8)(3,13,9)(4,14,10)(5,16,11)(6,15,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$12$:  $A_4$
$18$:  $S_3\times C_3$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: $A_4$

Degree 9: $S_3\times C_3$

Low degree siblings

12T43, 18T32, 24T78, 24T83, 36T21, 36T50, 36T51

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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:  72.44
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B1 3B-1 3C1 3C-1 6A 6B1 6B-1
Size 1 3 3 9 2 4 4 8 8 6 12 12
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3C-1 3C1 3A 3B1 3B-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2A 2B 2B
Type
72.44.1a R 1 1 1 1 1 1 1 1 1 1 1 1
72.44.1b R 1 1 −1 −1 1 1 1 1 1 1 −1 −1
72.44.1c1 C 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1
72.44.1c2 C 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3
72.44.1d1 C 1 1 −1 −1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 −ζ3 −ζ3−1
72.44.1d2 C 1 1 −1 −1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 −ζ3−1 −ζ3
72.44.2a R 2 2 0 0 −1 2 2 −1 −1 −1 0 0
72.44.2b1 C 2 2 0 0 −1 2ζ3−1 2ζ3 −ζ3−1 −ζ3 −1 0 0
72.44.2b2 C 2 2 0 0 −1 2ζ3 2ζ3−1 −ζ3 −ζ3−1 −1 0 0
72.44.3a R 3 −1 3 −1 3 0 0 0 0 −1 0 0
72.44.3b R 3 −1 −3 1 3 0 0 0 0 −1 0 0
72.44.6a R 6 −2 0 0 −3 0 0 0 0 1 0 0

magma: CharacterTable(G);