Properties

Label 6T10
Order \(36\)
n \(6\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2:C_4$

Related objects

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Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $10$
Group :  $C_3^2:C_4$
CHM label :  $F_{36}(6) = 1/2[S(3)^{2}]2$
Parity:  $1$
Primitive:  No
Generators:  (1,4,5,2)(3,6), (2,4,6)
$|\Aut(F/K)|$:  $1$
Low degree resolvents:  
2: $C_2$
4: $C_4$

Subfields

Degree 2: $C_2$

Degree 3: None

Low degree siblings

6T10, 9T9, 12T17 x 2, 18T10, 36T14
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 $ $4$ $3$ $(2,4,6)$
$ 4, 2 $ $9$ $4$ $(1,2)(3,4,5,6)$
$ 4, 2 $ $9$ $4$ $(1,2)(3,6,5,4)$
$ 3, 3 $ $4$ $3$ $(1,3,5)(2,4,6)$

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 9]
Character table:  
     2  2  2  .  2  2  .
     3  2  .  2  .  .  2

       1a 2a 3a 4a 4b 3b
    2P 1a 1a 3a 2a 2a 3b
    3P 1a 2a 1a 4b 4a 1a

X.1     1  1  1  1  1  1
X.2     1  1  1 -1 -1  1
X.3     1 -1  1  A -A  1
X.4     1 -1  1 -A  A  1
X.5     4  .  1  .  . -2
X.6     4  . -2  .  .  1

A = -E(4)
  = -Sqrt(-1) = -i