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Properties

Label 9T30
Degree $9$
Order $648$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$

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Show commands: Magma

magma: G := TransitiveGroup(9, 30);
 

Group action invariants

Degree $n$:  $9$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $30$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
CHM label:   $1/2[S(3)^{3}]S(3)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,9), (4,5)(7,8), (1,4,7)(2,5,8)(3,6,9), (1,2)(3,6)(4,7)(5,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Low degree siblings

12T177 x 2, 12T178, 18T217, 18T218, 18T221, 18T222, 24T1529 x 2, 24T1530, 27T211, 27T216, 36T1121, 36T1122, 36T1123, 36T1124, 36T1125, 36T1130, 36T1140, 36T1239 x 2, 36T1240

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrder IndexRepresentative
1A $1^{9}$ $1$ $1$ $0$ $()$
2A $2^{2},1^{5}$ $27$ $2$ $2$ $(1,2)(6,8)$
2B $2^{4},1$ $54$ $2$ $4$ $(1,8)(2,6)(4,5)(7,9)$
3A $3,1^{6}$ $6$ $3$ $2$ $(3,4,5)$
3B $3^{3}$ $8$ $3$ $6$ $(1,2,9)(3,4,5)(6,8,7)$
3C $3^{2},1^{3}$ $12$ $3$ $4$ $(1,2,9)(3,5,4)$
3D $3^{3}$ $72$ $3$ $6$ $(1,6,4)(2,8,5)(3,9,7)$
4A $4,2,1^{3}$ $54$ $4$ $4$ $(1,6,2,8)(7,9)$
6A $3,2^{2},1^{2}$ $54$ $6$ $4$ $(1,2)(3,5,4)(6,8)$
6B $6,2,1$ $108$ $6$ $6$ $(1,5,2,4,9,3)(6,8)$
9A $9$ $72$ $9$ $8$ $(1,6,4,9,7,3,2,8,5)$
9B $9$ $72$ $9$ $8$ $(1,6,4,2,8,5,9,7,3)$
12A1 $4,3,2$ $54$ $12$ $6$ $(1,6,2,8)(3,5,4)(7,9)$
12A5 $4,3,2$ $54$ $12$ $6$ $(1,6,2,8)(3,4,5)(7,9)$

magma: ConjugacyClasses(G);
 

Malle's constant $a(G)$:     $1/2$

Group invariants

Order:  $648=2^{3} \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  648.703
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 3B 3C 3D 4A 6A 6B 9A 9B 12A1 12A5
Size 1 27 54 6 8 12 72 54 54 108 72 72 54 54
2 P 1A 1A 1A 3A 3B 3C 3D 2A 3A 3C 9A 9B 6A 6A
3 P 1A 2A 2B 1A 1A 1A 1A 4A 2A 2B 3B 3B 4A 4A
Type
648.703.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
648.703.1b R 1 1 −1 1 1 1 1 −1 1 −1 1 1 −1 −1
648.703.2a R 2 2 0 2 2 2 −1 0 2 0 −1 −1 0 0
648.703.3a R 3 −1 1 3 3 3 0 −1 −1 1 0 0 −1 −1
648.703.3b R 3 −1 −1 3 3 3 0 1 −1 −1 0 0 1 1
648.703.6a R 6 2 0 3 −3 0 0 2 −1 0 0 0 −1 −1
648.703.6b R 6 2 0 3 −3 0 0 −2 −1 0 0 0 1 1
648.703.6c1 R 6 −2 0 3 −3 0 0 0 1 0 0 0 −ζ12−1−ζ12 ζ12−1+ζ12
648.703.6c2 R 6 −2 0 3 −3 0 0 0 1 0 0 0 ζ12−1+ζ12 −ζ12−1−ζ12
648.703.8a R 8 0 0 −4 −1 2 −1 0 0 0 −1 2 0 0
648.703.8b R 8 0 0 −4 −1 2 −1 0 0 0 2 −1 0 0
648.703.8c R 8 0 0 −4 −1 2 2 0 0 0 −1 −1 0 0
648.703.12a R 12 0 2 0 3 −3 0 0 0 −1 0 0 0 0
648.703.12b R 12 0 −2 0 3 −3 0 0 0 1 0 0 0 0

magma: CharacterTable(G);