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

|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

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 1, 1, 1, 1, 1, 1$ $6$ $3$ $(1,2,9)$ $3, 3, 1, 1, 1$ $12$ $3$ $(1,2,9)(3,4,5)$ $3, 3, 3$ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$ $2, 1, 1, 1, 1, 1, 1, 1$ $9$ $2$ $(2,9)$ $3, 2, 1, 1, 1, 1$ $18$ $6$ $(2,9)(3,4,5)$ $3, 2, 1, 1, 1, 1$ $18$ $6$ $(2,9)(6,7,8)$ $3, 3, 2, 1$ $36$ $6$ $(2,9)(3,4,5)(6,7,8)$ $2, 2, 1, 1, 1, 1, 1$ $27$ $2$ $(2,9)(4,5)$ $3, 2, 2, 1, 1$ $54$ $6$ $(2,9)(4,5)(6,7,8)$ $2, 2, 2, 1, 1, 1$ $27$ $2$ $(2,9)(4,5)(7,8)$ $3, 3, 3$ $36$ $3$ $(1,4,7)(2,5,8)(3,6,9)$ $9$ $72$ $9$ $(1,4,7,2,5,8,9,3,6)$ $6, 3$ $108$ $6$ $(1,4,7)(2,5,8,9,3,6)$ $3, 3, 3$ $36$ $3$ $(1,7,4)(2,8,5)(3,9,6)$ $9$ $72$ $9$ $(1,7,4,2,8,5,9,6,3)$ $6, 3$ $108$ $6$ $(1,7,4)(2,8,5,9,6,3)$

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:  2 3 2 1 . 3 2 2 1 3 2 3 1 . 1 1 . 1 3 4 3 3 4 2 2 2 2 1 1 1 2 2 1 2 2 1 1a 3a 3b 3c 2a 6a 6b 6c 2b 6d 2c 3d 9a 6e 3e 9b 6f 2P 1a 3a 3b 3c 1a 3a 3a 3b 1a 3a 1a 3e 9b 3e 3d 9a 3d 3P 1a 1a 1a 1a 2a 2a 2a 2a 2b 2b 2c 1a 3c 2c 1a 3c 2c 5P 1a 3a 3b 3c 2a 6a 6b 6c 2b 6d 2c 3e 9b 6f 3d 9a 6e 7P 1a 3a 3b 3c 2a 6a 6b 6c 2b 6d 2c 3d 9a 6e 3e 9b 6f X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 1 1 -1 X.3 1 1 1 1 -1 -1 -1 -1 1 1 -1 A A -A /A /A -/A X.4 1 1 1 1 -1 -1 -1 -1 1 1 -1 /A /A -/A A A -A X.5 1 1 1 1 1 1 1 1 1 1 1 A A A /A /A /A X.6 1 1 1 1 1 1 1 1 1 1 1 /A /A /A A A A X.7 3 3 3 3 -1 -1 -1 -1 -1 -1 3 . . . . . . X.8 3 3 3 3 1 1 1 1 -1 -1 -3 . . . . . . X.9 6 3 . -3 4 1 1 -2 2 -1 . . . . . . . X.10 6 3 . -3 -4 -1 -1 2 2 -1 . . . . . . . X.11 6 3 . -3 . -3 3 . -2 1 . . . . . . . X.12 6 3 . -3 . 3 -3 . -2 1 . . . . . . . X.13 8 -4 2 -1 . . . . . . . 2 -1 . 2 -1 . X.14 8 -4 2 -1 . . . . . . . B -A . /B -/A . X.15 8 -4 2 -1 . . . . . . . /B -/A . B -A . X.16 12 . -3 3 -4 2 2 -1 . . . . . . . . . X.17 12 . -3 3 4 -2 -2 1 . . . . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 

magma: CharacterTable(G);