Properties

Label 9T26
Order \(432\)
n \(9\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $((C_3^2:Q_8):C_3):C_2$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $26$
Group :  $((C_3^2:Q_8):C_3):C_2$
CHM label :  $E(9):2S_{4}$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,4,5,2,3,8,7), (1,2,9)(3,4,5)(6,7,8), (3,4,5)(6,8,7), (1,4,7)(2,5,8)(3,6,9)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
24:  $S_4$
48:  $\textrm{GL(2,3)}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

12T157, 18T157, 24T1325, 24T1326, 24T1327, 24T1334, 27T139, 36T689, 36T709

Siblings are shown with degree $\leq 47$

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

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1 $ $24$ $3$ $(3,4,5)(6,8,7)$
$ 2, 2, 2, 1, 1, 1 $ $36$ $2$ $(3,6)(4,7)(5,8)$
$ 8, 1 $ $54$ $8$ $(2,3,4,6,9,8,7,5)$
$ 6, 2, 1 $ $72$ $6$ $(2,3,5,9,8,6)(4,7)$
$ 8, 1 $ $54$ $8$ $(2,3,6,7,9,8,5,4)$
$ 4, 4, 1 $ $54$ $4$ $(2,3,9,8)(4,5,7,6)$
$ 2, 2, 2, 2, 1 $ $9$ $2$ $(2,9)(3,8)(4,7)(5,6)$
$ 3, 3, 3 $ $48$ $3$ $(1,2,3)(4,5,6)(7,8,9)$
$ 6, 3 $ $72$ $6$ $(1,2,3,4,8,6)(5,9,7)$
$ 3, 3, 3 $ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$

Group invariants

Order:  $432=2^{4} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [432, 734]
Character table:   
      2  4  1  2  3  1  3  3  4  .  1  1
      3  3  2  1  .  1  .  .  1  2  1  3

        1a 3a 2a 8a 6a 8b 4a 2b 3b 6b 3c
     2P 1a 3a 1a 4a 3a 4a 2b 1a 3b 3c 3c
     3P 1a 1a 2a 8a 2b 8b 4a 2b 1a 2a 1a
     5P 1a 3a 2a 8b 6a 8a 4a 2b 3b 6b 3c
     7P 1a 3a 2a 8b 6a 8a 4a 2b 3b 6b 3c

X.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
X.3      2 -1  .  . -1  .  2  2 -1  .  2
X.4      2 -1  .  A  1 -A  . -2 -1  .  2
X.5      2 -1  . -A  1  A  . -2 -1  .  2
X.6      3  . -1  1  .  1 -1  3  . -1  3
X.7      3  .  1 -1  . -1 -1  3  .  1  3
X.8      4  1  .  . -1  .  . -4  1  .  4
X.9      8  2 -2  .  .  .  .  . -1  1 -1
X.10     8  2  2  .  .  .  .  . -1 -1 -1
X.11    16 -2  .  .  .  .  .  .  1  . -2

A = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2