Properties

Label 36T14
Order \(36\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:S_3.C_2$

Learn more about

Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $14$
Group :  $C_3:S_3.C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,6,30)(2,15,5,29)(3,13,8,32)(4,14,7,31)(9,27,21,19)(10,28,22,20)(11,26,24,18)(12,25,23,17)(33,35,34,36), (1,8,28,21)(2,7,27,22)(3,5,26,23)(4,6,25,24)(9,31,18,33)(10,32,17,34)(11,29,20,36)(12,30,19,35)(13,16,14,15)
$|\Aut(F/K)|$:  $36$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: $C_3^2:C_4$ x 2

Degree 9: $C_3^2:C_4$

Degree 12: $(C_3\times C_3):C_4$ x 2

Degree 18: $C_3^2 : C_4$

Low degree siblings

6T10 x 2, 9T9

Siblings are shown with degree $\leq 10$

Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 2)( 3, 4)( 5,34)( 6,33)( 7,36)( 8,35)( 9,29)(10,30)(11,32)(12,31)(13,27) (14,28)(15,25)(16,26)(17,21)(18,22)(19,24)(20,23)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $9$ $4$ $( 1, 3, 2, 4)( 5,18,34,22)( 6,17,33,21)( 7,19,36,24)( 8,20,35,23)( 9,14,29,28) (10,13,30,27)(11,15,32,25)(12,16,31,26)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $9$ $4$ $( 1, 4, 2, 3)( 5,22,34,18)( 6,21,33,17)( 7,24,36,19)( 8,23,35,20)( 9,28,29,14) (10,27,30,13)(11,25,32,15)(12,26,31,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 5,33)( 2, 6,34)( 3, 7,35)( 4, 8,36)( 9,15,17)(10,16,18)(11,13,19) (12,14,20)(21,25,29)(22,26,30)(23,28,31)(24,27,32)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,12,32)( 2,11,31)( 3,10,29)( 4, 9,30)( 5,14,24)( 6,13,23)( 7,16,21) ( 8,15,22)(17,26,36)(18,25,35)(19,28,34)(20,27,33)$

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

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

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