# Properties

 Label 9T25 Order $$324$$ n $$9$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$

# Related objects

## Group action invariants

 Degree $n$ : $9$ Transitive number $t$ : $25$ Group : $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ CHM label : $[1/2.S(3)^{3}]3$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,9), (4,5)(7,8), (1,4,7)(2,5,8)(3,6,9) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
12:  $A_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

## Low degree siblings

12T132 x 2, 12T133, 18T141 x 2, 18T142, 18T143, 27T130, 27T131, 36T511, 36T512, 36T524, 36T546 x 2, 36T547

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 $1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 1, 1, 1, 1, 1, 1$ $6$ $3$ $(6,7,8)$ $2, 2, 1, 1, 1, 1, 1$ $27$ $2$ $(4,5)(7,8)$ $3, 3, 1, 1, 1$ $12$ $3$ $(3,4,5)(6,7,8)$ $3, 2, 2, 1, 1$ $54$ $6$ $(2,9)(4,5)(6,7,8)$ $3, 3, 3$ $4$ $3$ $(1,2,9)(3,4,5)(6,7,8)$ $3, 3, 3$ $4$ $3$ $(1,2,9)(3,4,5)(6,8,7)$ $3, 3, 3$ $36$ $3$ $(1,3,6)(2,4,7)(5,8,9)$ $9$ $36$ $9$ $(1,3,6,2,4,7,9,5,8)$ $9$ $36$ $9$ $(1,3,6,9,5,8,2,4,7)$ $3, 3, 3$ $36$ $3$ $(1,6,3)(2,7,4)(5,9,8)$ $9$ $36$ $9$ $(1,6,4,2,7,5,9,8,3)$ $9$ $36$ $9$ $(1,6,5,9,8,4,2,7,3)$

## Group invariants

 Order: $324=2^{2} \cdot 3^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [324, 160]
 Character table:  2 2 1 2 . 1 . . . . . . . . 3 4 3 1 3 1 4 4 2 2 2 2 2 2 1a 3a 2a 3b 6a 3c 3d 3e 9a 9b 3f 9c 9d 2P 1a 3a 1a 3b 3a 3d 3c 3f 9d 9c 3e 9b 9a 3P 1a 1a 2a 1a 2a 1a 1a 1a 3c 3d 1a 3c 3d 5P 1a 3a 2a 3b 6a 3d 3c 3f 9d 9c 3e 9b 9a 7P 1a 3a 2a 3b 6a 3c 3d 3e 9a 9b 3f 9c 9d X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 1 B B B /B /B /B X.3 1 1 1 1 1 1 1 /B /B /B B B B X.4 3 3 -1 3 -1 3 3 . . . . . . X.5 4 -2 . 1 . A /A B 1 /B /B B 1 X.6 4 -2 . 1 . /A A /B 1 B B /B 1 X.7 4 -2 . 1 . A /A /B B 1 B 1 /B X.8 4 -2 . 1 . /A A B /B 1 /B 1 B X.9 4 -2 . 1 . A /A 1 /B B 1 /B B X.10 4 -2 . 1 . /A A 1 B /B 1 B /B X.11 6 3 -2 . 1 -3 -3 . . . . . . X.12 6 3 2 . -1 -3 -3 . . . . . . X.13 12 . . -3 . 3 3 . . . . . . A = -E(3)+2*E(3)^2 = (-1-3*Sqrt(-3))/2 = -2-3b3 B = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3