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Properties

Label 9T28
Degree $9$
Order $648$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3 \wr C_3 $

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Show commands: Magma

magma: G := TransitiveGroup(9, 28);
 

Group action invariants

Degree $n$:  $9$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $28$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3 \wr C_3 $
CHM label:   $[S(3)^{3}]3=S(3)wr3$
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), (1,2), (1,4,7)(2,5,8)(3,6,9)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Low degree siblings

12T176, 18T197 x 2, 18T198 x 2, 18T202, 18T204, 18T206, 18T207, 24T1528, 24T1539, 27T210, 27T213, 36T1094 x 2, 36T1095, 36T1096 x 2, 36T1097, 36T1099, 36T1101, 36T1102, 36T1103, 36T1137, 36T1238

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{9}$ $1$ $1$ $0$ $()$
2A $2,1^{7}$ $9$ $2$ $1$ $(1,2)$
2B $2^{3},1^{3}$ $27$ $2$ $3$ $(1,2)(4,5)(7,8)$
2C $2^{2},1^{5}$ $27$ $2$ $2$ $(4,5)(7,8)$
3A $3,1^{6}$ $6$ $3$ $2$ $(1,9,2)$
3B $3^{3}$ $8$ $3$ $6$ $(1,9,2)(3,5,4)(6,8,7)$
3C $3^{2},1^{3}$ $12$ $3$ $4$ $(1,9,2)(6,8,7)$
3D1 $3^{3}$ $36$ $3$ $6$ $(1,7,4)(2,8,5)(3,9,6)$
3D-1 $3^{3}$ $36$ $3$ $6$ $(1,4,7)(2,5,8)(3,6,9)$
6A $3,2,1^{4}$ $18$ $6$ $3$ $(1,2)(3,5,4)$
6B $3,2,1^{4}$ $18$ $6$ $3$ $(1,2)(6,8,7)$
6C $3^{2},2,1$ $36$ $6$ $5$ $(1,2)(3,5,4)(6,8,7)$
6D $3,2^{2},1^{2}$ $54$ $6$ $4$ $(1,9,2)(4,5)(7,8)$
6E1 $6,3$ $108$ $6$ $7$ $(1,5,7,2,4,8)(3,6,9)$
6E-1 $6,3$ $108$ $6$ $7$ $(1,8,4,2,7,5)(3,9,6)$
9A1 $9$ $72$ $9$ $8$ $(1,3,6,9,5,8,2,4,7)$
9A-1 $9$ $72$ $9$ $8$ $(1,6,5,2,7,3,9,8,4)$

Malle's constant $a(G)$:     $1$

magma: ConjugacyClasses(G);
 

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:  648.705
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 3D1 3D-1 6A 6B 6C 6D 6E1 6E-1 9A1 9A-1
Size 1 9 27 27 6 8 12 36 36 18 18 36 54 108 108 72 72
2 P 1A 1A 1A 1A 3A 3B 3C 3D-1 3D1 3A 3A 3C 3A 3D1 3D-1 9A-1 9A1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2A 2A 2A 2C 2B 2B 3B 3B
Type
648.705.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
648.705.1b R 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 1 −1 −1 1 1
648.705.1c1 C 1 1 1 1 1 1 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
648.705.1c2 C 1 1 1 1 1 1 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
648.705.1d1 C 1 −1 −1 1 1 1 1 ζ3−1 ζ3 −1 −1 −1 1 −ζ3 −ζ3−1 ζ3−1 ζ3
648.705.1d2 C 1 −1 −1 1 1 1 1 ζ3 ζ3−1 −1 −1 −1 1 −ζ3−1 −ζ3 ζ3 ζ3−1
648.705.3a R 3 −1 3 −1 3 3 3 0 0 −1 −1 −1 −1 0 0 0 0
648.705.3b R 3 1 −3 −1 3 3 3 0 0 1 1 1 −1 0 0 0 0
648.705.6a R 6 4 0 2 3 −3 0 0 0 1 1 −2 −1 0 0 0 0
648.705.6b R 6 0 0 −2 3 −3 0 0 0 −3 3 0 1 0 0 0 0
648.705.6c R 6 0 0 −2 3 −3 0 0 0 3 −3 0 1 0 0 0 0
648.705.6d R 6 −4 0 2 3 −3 0 0 0 −1 −1 2 −1 0 0 0 0
648.705.8a R 8 0 0 0 −4 −1 2 2 2 0 0 0 0 0 0 −1 −1
648.705.8b1 C 8 0 0 0 −4 −1 2 2ζ3−1 2ζ3 0 0 0 0 0 0 −ζ3−1 −ζ3
648.705.8b2 C 8 0 0 0 −4 −1 2 2ζ3 2ζ3−1 0 0 0 0 0 0 −ζ3 −ζ3−1
648.705.12a R 12 4 0 0 0 3 −3 0 0 −2 −2 1 0 0 0 0 0
648.705.12b R 12 −4 0 0 0 3 −3 0 0 2 2 −1 0 0 0 0 0

magma: CharacterTable(G);