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Properties

Label 18T33
Degree $18$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times S_4$

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Show commands: Magma

magma: G := TransitiveGroup(18, 33);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times S_4$
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:  (1,8,6,12,3,10,2,7,5,11,4,9)(13,15,18,14,16,17), (1,9,15)(2,10,16)(3,12,17)(4,11,18)(5,7,14)(6,8,13)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: $S_4$

Degree 9: $S_3\times C_3$

Low degree siblings

12T45, 18T30, 24T80, 24T84, 36T20, 36T52

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  72.42
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A1 3A-1 3B 3C1 3C-1 4A 6A1 6A-1 6B1 6B-1 12A1 12A-1
Size 1 3 6 1 1 8 8 8 6 3 3 6 6 6 6
2 P 1A 1A 1A 3A-1 3A1 3C1 3B 3C-1 2A 3A1 3A-1 3A1 3A-1 6A1 6A-1
3 P 1A 2A 2B 1A 1A 1A 1A 1A 4A 2A 2A 2B 2B 4A 4A
Type
72.42.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.42.1b R 1 1 −1 1 1 1 1 1 −1 1 1 −1 −1 −1 −1
72.42.1c1 C 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
72.42.1c2 C 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
72.42.1d1 C 1 1 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
72.42.1d2 C 1 1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1
72.42.2a R 2 2 0 2 2 −1 −1 −1 0 2 2 0 0 0 0
72.42.2b1 C 2 2 0 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 2ζ3 2ζ3−1 0 0 0 0
72.42.2b2 C 2 2 0 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 2ζ3−1 2ζ3 0 0 0 0
72.42.3a R 3 −1 1 3 3 0 0 0 −1 −1 −1 1 1 −1 −1
72.42.3b R 3 −1 −1 3 3 0 0 0 1 −1 −1 −1 −1 1 1
72.42.3c1 C 3 −1 1 3ζ3−1 3ζ3 0 0 0 −1 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3
72.42.3c2 C 3 −1 1 3ζ3 3ζ3−1 0 0 0 −1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1
72.42.3d1 C 3 −1 −1 3ζ3−1 3ζ3 0 0 0 1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3
72.42.3d2 C 3 −1 −1 3ζ3 3ζ3−1 0 0 0 1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1

magma: CharacterTable(G);