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Properties

Label 18T6
Degree $18$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3 \times C_6$

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Show commands: Magma

magma: G := TransitiveGroup(18, 6);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3 \times C_6$
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:  (1,15,8,4,13,9)(2,16,7,3,14,10)(5,17,12)(6,18,11), (1,3,18,2,4,17)(5,8,10,6,7,9)(11,14,15,12,13,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 6: $C_6$, $D_{6}$

Degree 9: $S_3\times C_3$

Low degree siblings

12T18, 18T6, 36T6

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  36.12
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 3B 3C1 3C-1 6A1 6A-1 6B 6C1 6C-1 6D1 6D-1 6E1 6E-1
Size 1 1 3 3 1 1 2 2 2 1 1 2 2 2 3 3 3 3
2 P 1A 1A 1A 1A 3A-1 3A1 3B 3C-1 3C1 3A-1 3A1 3C-1 3B 3C1 3A1 3A-1 3A1 3A-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2B 2B 2C 2C
Type
36.12.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
36.12.1b R 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 1 1
36.12.1c R 1 −1 1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 −1 −1
36.12.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
36.12.1e1 C 1 1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1
36.12.1e2 C 1 1 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3
36.12.1f1 C 1 −1 −1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1
36.12.1f2 C 1 −1 −1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3
36.12.1g1 C 1 −1 1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1
36.12.1g2 C 1 −1 1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3
36.12.1h1 C 1 1 −1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
36.12.1h2 C 1 1 −1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3
36.12.2a R 2 2 0 0 2 2 −1 −1 −1 2 2 −1 −1 −1 0 0 0 0
36.12.2b R 2 −2 0 0 2 2 −1 −1 −1 −2 −2 1 1 1 0 0 0 0
36.12.2c1 C 2 2 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 0 0 0
36.12.2c2 C 2 2 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 0 0 0
36.12.2d1 C 2 −2 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 −2ζ3−1 −2ζ3 1 ζ3 ζ3−1 0 0 0 0
36.12.2d2 C 2 −2 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 −2ζ3 −2ζ3−1 1 ζ3−1 ζ3 0 0 0 0

magma: CharacterTable(G);