Properties

Label 36T12
Order \(36\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times A_4$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $12$
Group :  $C_3\times A_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,33,15)(2,34,16)(3,35,13)(4,36,14)(5,28,19)(6,27,20)(7,26,18)(8,25,17)(9,29,21)(10,30,22)(11,32,23)(12,31,24), (1,28,23)(2,27,24)(3,26,21)(4,25,22)(5,29,14)(6,30,13)(7,32,16)(8,31,15)(9,34,17)(10,33,18)(11,36,20)(12,35,19)
$|\Aut(F/K)|$:  $36$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4
9:  $C_3^2$
12:  $A_4$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: $C_3$ x 4

Degree 4: $A_4$

Degree 6: $A_4$

Degree 9: $C_3^2$

Degree 12: $A_4$, $C_3\times A_4$ x 3

Degree 18: $A_4 \times C_3$

Low degree siblings

There are no siblings 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 $ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23) (22,24)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)$
$ 6, 6, 6, 6, 6, 6 $ $3$ $6$ $( 1, 5, 9, 2, 6,10)( 3, 7,11, 4, 8,12)(13,20,23,15,17,21)(14,19,24,16,18,22) (25,30,35,28,32,34)(26,29,36,27,31,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 6, 9)( 2, 5,10)( 3, 8,11)( 4, 7,12)(13,17,23)(14,18,24)(15,20,21) (16,19,22)(25,32,35)(26,31,36)(27,29,33)(28,30,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 9, 6)( 2,10, 5)( 3,11, 8)( 4,12, 7)(13,23,17)(14,24,18)(15,21,20) (16,22,19)(25,35,32)(26,36,31)(27,33,29)(28,34,30)$
$ 6, 6, 6, 6, 6, 6 $ $3$ $6$ $( 1,10, 6, 2, 9, 5)( 3,12, 8, 4,11, 7)(13,21,17,15,23,20)(14,22,18,16,24,19) (25,34,32,28,35,30)(26,33,31,27,36,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,13,34)( 2,14,33)( 3,15,36)( 4,16,35)( 5,18,27)( 6,17,28)( 7,19,25) ( 8,20,26)( 9,23,30)(10,24,29)(11,21,31)(12,22,32)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,17,30)( 2,18,29)( 3,20,31)( 4,19,32)( 5,24,33)( 6,23,34)( 7,22,35) ( 8,21,36)( 9,13,28)(10,14,27)(11,15,26)(12,16,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,21,27)( 2,22,28)( 3,23,25)( 4,24,26)( 5,16,30)( 6,15,29)( 7,14,31) ( 8,13,32)( 9,20,33)(10,19,34)(11,17,35)(12,18,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,25,24)( 2,26,23)( 3,27,22)( 4,28,21)( 5,31,13)( 6,32,14)( 7,30,15) ( 8,29,16)( 9,35,18)(10,36,17)(11,33,19)(12,34,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,29,20)( 2,30,19)( 3,32,17)( 4,31,18)( 5,34,22)( 6,33,21)( 7,36,24) ( 8,35,23)( 9,27,15)(10,28,16)(11,25,13)(12,26,14)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,33,15)( 2,34,16)( 3,35,13)( 4,36,14)( 5,28,19)( 6,27,20)( 7,26,18) ( 8,25,17)( 9,29,21)(10,30,22)(11,32,23)(12,31,24)$

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 11]
Character table:   
      2  2  2   2  2  2   2  .  .  .  .  .  .
      3  2  1   1  2  2   1  2  2  2  2  2  2

        1a 2a  6a 3a 3b  6b 3c 3d 3e 3f 3g 3h
     2P 1a 1a  3b 3b 3a  3a 3h 3g 3f 3e 3d 3c
     3P 1a 2a  2a 1a 1a  2a 1a 1a 1a 1a 1a 1a
     5P 1a 2a  6b 3b 3a  6a 3h 3g 3f 3e 3d 3c

X.1      1  1   1  1  1   1  1  1  1  1  1  1
X.2      1  1   1  1  1   1  A  A  A /A /A /A
X.3      1  1   1  1  1   1 /A /A /A  A  A  A
X.4      1  1   A  A /A  /A  1  A /A  A /A  1
X.5      1  1  /A /A  A   A  1 /A  A /A  A  1
X.6      1  1   A  A /A  /A  A /A  1  1  A /A
X.7      1  1  /A /A  A   A /A  A  1  1 /A  A
X.8      1  1   A  A /A  /A /A  1  A /A  1  A
X.9      1  1  /A /A  A   A  A  1 /A  A  1 /A
X.10     3 -1  -1  3  3  -1  .  .  .  .  .  .
X.11     3 -1 -/A  B /B  -A  .  .  .  .  .  .
X.12     3 -1  -A /B  B -/A  .  .  .  .  .  .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)
  = (-3+3*Sqrt(-3))/2 = 3b3