↑

Properties

Label 18T30
Degree $18$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times S_4$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(18, 30);
 

Group invariants

Abstract group:  $C_3\times S_4$
magma: IdentifyGroup(G);
 
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
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $30$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,3,5)(2,4,6)(7,18,11,15,9,13,8,17,12,16,10,14)$, $(1,8,18)(2,7,17)(3,10,13)(4,9,14)(5,11,16)(6,12,15)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$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, 18T33, 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 TypeSizeOrderIndexRepresentative
1A $1^{18}$ $1$ $1$ $0$ $()$
2A $2^{6},1^{6}$ $3$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)(13,14)(15,16)(17,18)$
2B $2^{9}$ $6$ $2$ $9$ $( 1,12)( 2,11)( 3, 7)( 4, 8)( 5, 9)( 6,10)(13,14)(15,16)(17,18)$
3A1 $3^{6}$ $1$ $3$ $12$ $( 1, 5, 3)( 2, 6, 4)( 7,12, 9)( 8,11,10)(13,18,16)(14,17,15)$
3A-1 $3^{6}$ $1$ $3$ $12$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,16,18)(14,15,17)$
3B $3^{6}$ $8$ $3$ $12$ $( 1,13,11)( 2,14,12)( 3,16, 8)( 4,15, 7)( 5,18,10)( 6,17, 9)$
3C1 $3^{6}$ $8$ $3$ $12$ $( 1,16,10)( 2,15, 9)( 3,18,11)( 4,17,12)( 5,13, 8)( 6,14, 7)$
3C-1 $3^{6}$ $8$ $3$ $12$ $( 1,18, 8)( 2,17, 7)( 3,13,10)( 4,14, 9)( 5,16,11)( 6,15,12)$
4A $4^{3},1^{6}$ $6$ $4$ $9$ $( 1,13, 2,14)( 3,16, 4,15)( 5,18, 6,17)$
6A1 $6^{2},3^{2}$ $3$ $6$ $14$ $( 1, 6, 3, 2, 5, 4)( 7,12, 9)( 8,11,10)(13,17,16,14,18,15)$
6A-1 $6^{2},3^{2}$ $3$ $6$ $14$ $( 1, 4, 5, 2, 3, 6)( 7, 9,12)( 8,10,11)(13,15,18,14,16,17)$
6B1 $6^{3}$ $6$ $6$ $15$ $( 1, 7, 5,12, 3, 9)( 2, 8, 6,11, 4,10)(13,15,18,14,16,17)$
6B-1 $6^{3}$ $6$ $6$ $15$ $( 1, 9, 3,12, 5, 7)( 2,10, 4,11, 6, 8)(13,17,16,14,18,15)$
12A1 $12,3^{2}$ $6$ $12$ $15$ $( 1,15, 6,13, 3,17, 2,16, 5,14, 4,18)( 7, 9,12)( 8,10,11)$
12A-1 $12,3^{2}$ $6$ $12$ $15$ $( 1,17, 4,13, 5,15, 2,18, 3,14, 6,16)( 7,12, 9)( 8,11,10)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(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 3B 3C-1 3C1 2A 3A-1 3A1 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 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
72.42.1c2 C 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
72.42.1d1 C 1 1 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1
72.42.1d2 C 1 1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
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 2ζ3−1 2ζ3 0 0 0 0
72.42.2b2 C 2 2 0 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 2ζ3 2ζ3−1 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 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1
72.42.3c2 C 3 −1 1 3ζ3 3ζ3−1 0 0 0 −1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3
72.42.3d1 C 3 −1 −1 3ζ3−1 3ζ3 0 0 0 1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1
72.42.3d2 C 3 −1 −1 3ζ3 3ζ3−1 0 0 0 1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3

magma: CharacterTable(G);
 

Regular extensions

Data not computed