Properties

Label 6T9
Order \(36\)
n \(6\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3^2$

Related objects

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Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $9$
Group :  $S_3^2$
CHM label :  $F_{18}(6):2 = [1/2.S(3)^{2}]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,5)(3,6), (2,4,6), (1,5)(2,4)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
12:  $D_{6}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Low degree siblings

9T8, 12T16, 18T9, 18T11 x 2, 36T13

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 $ $1$ $1$ $()$
$ 2, 2, 1, 1 $ $9$ $2$ $(3,5)(4,6)$
$ 3, 1, 1, 1 $ $4$ $3$ $(2,4,6)$
$ 2, 2, 2 $ $3$ $2$ $(1,2)(3,4)(5,6)$
$ 2, 2, 2 $ $3$ $2$ $(1,2)(3,6)(4,5)$
$ 6 $ $6$ $6$ $(1,2,3,4,5,6)$
$ 6 $ $6$ $6$ $(1,2,3,6,5,4)$
$ 3, 3 $ $2$ $3$ $(1,3,5)(2,4,6)$
$ 3, 3 $ $2$ $3$ $(1,3,5)(2,6,4)$

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 10]
Character table:   
     2  2  2  .  2  2  1  1  1  1
     3  2  .  2  1  1  1  1  2  2

       1a 2a 3a 2b 2c 6a 6b 3b 3c
    2P 1a 1a 3a 1a 1a 3b 3c 3b 3c
    3P 1a 2a 1a 2b 2c 2b 2c 1a 1a
    5P 1a 2a 3a 2b 2c 6a 6b 3b 3c

X.1     1  1  1  1  1  1  1  1  1
X.2     1 -1  1 -1  1 -1  1  1  1
X.3     1 -1  1  1 -1  1 -1  1  1
X.4     1  1  1 -1 -1 -1 -1  1  1
X.5     2  . -1  . -2  .  1  2 -1
X.6     2  . -1  .  2  . -1  2 -1
X.7     2  . -1 -2  .  1  . -1  2
X.8     2  . -1  2  . -1  . -1  2
X.9     4  .  1  .  .  .  . -2 -2