Properties

Label 36T11
Order \(36\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2:C_9$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $11$
Group :  $C_2^2:C_9$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,11,4,5,10)(2,7,12,3,6,9)(13,19,22,14,20,21)(15,18,23,16,17,24)(25,32,36,28,30,34)(26,31,35,27,29,33), (1,27,13,5,31,20,11,33,22)(2,28,14,6,32,19,12,34,21)(3,26,15,7,29,17,9,35,23)(4,25,16,8,30,18,10,36,24)
$|\Aut(F/K)|$:  $36$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
9:  $C_9$
12:  $A_4$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: $A_4$

Degree 6: $A_4$

Degree 9: $C_9$

Degree 12: $A_4$

Degree 18: $C_2^2 : C_9$

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

Group invariants

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

        1a 2a 3a  6a  6b 3b 9a 9b 9c 9d 9e 9f
     2P 1a 1a 3b  3b  3a 3a 9e 9d 9f 9a 9c 9b
     3P 1a 2a 1a  2a  2a 1a 3b 3b 3b 3a 3a 3a
     5P 1a 2a 3b  6b  6a 3a 9d 9f 9e 9b 9a 9c
     7P 1a 2a 3a  6a  6b 3b 9b 9c 9a 9f 9d 9e

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  C  D  E /E /D /C
X.5      1  1  A   A  /A /A  D  E  C /C /E /D
X.6      1  1  A   A  /A /A  E  C  D /D /C /E
X.7      1  1 /A  /A   A  A /C /D /E  E  D  C
X.8      1  1 /A  /A   A  A /E /C /D  D  C  E
X.9      1  1 /A  /A   A  A /D /E /C  C  E  D
X.10     3 -1  3  -1  -1  3  .  .  .  .  .  .
X.11     3 -1  B -/A  -A /B  .  .  .  .  .  .
X.12     3 -1 /B  -A -/A  B  .  .  .  .  .  .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)
  = (-3+3*Sqrt(-3))/2 = 3b3
C = -E(9)^4-E(9)^7
D = E(9)^7
E = E(9)^4