Properties

Label 36T6
Order \(36\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_6\times S_3$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $6$
Group :  $C_6\times S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,29,10,2,30,9)(3,32,12,4,31,11)(5,23,13,6,24,14)(7,21,15,8,22,16)(17,34,27,18,33,28)(19,35,26,20,36,25), (1,25,15,3,27,14)(2,26,16,4,28,13)(5,21,19,34,32,9)(6,22,20,33,31,10)(7,23,17,35,30,12)(8,24,18,36,29,11)
$|\Aut(F/K)|$:  $36$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
12:  $D_{6}$
18:  $S_3\times C_3$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$, $S_3$

Degree 4: $C_2^2$

Degree 6: $C_6$ x 3, $S_3$, $D_{6}$ x 2, $S_3\times C_3$

Degree 9: $S_3\times C_3$

Degree 12: $C_6\times C_2$, $D_6$, $C_6\times S_3$

Degree 18: $S_3 \times C_3$, $S_3 \times C_6$ x 2

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 12]
Character table:   
      2  2  2  2  2  1  1   1   1   2   2   2   2   1   1   2   2   2   2
      3  2  2  1  1  2  2   2   2   1   1   2   2   2   2   1   1   2   2

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

X.1      1  1  1  1  1  1   1   1   1   1   1   1   1   1   1   1   1   1
X.2      1 -1 -1  1  1 -1  -1   1   1  -1   1  -1  -1   1  -1   1   1  -1
X.3      1 -1  1 -1  1 -1  -1   1  -1   1   1  -1  -1   1   1  -1   1  -1
X.4      1  1 -1 -1  1  1   1   1  -1  -1   1   1   1   1  -1  -1   1   1
X.5      1 -1 -1  1  1 -1   A  -A  -A   A  -A   A  /A -/A  /A -/A -/A  /A
X.6      1 -1 -1  1  1 -1  /A -/A -/A  /A -/A  /A   A  -A   A  -A  -A   A
X.7      1 -1  1 -1  1 -1   A  -A   A  -A  -A   A  /A -/A -/A  /A -/A  /A
X.8      1 -1  1 -1  1 -1  /A -/A  /A -/A -/A  /A   A  -A  -A   A  -A   A
X.9      1  1 -1 -1  1  1 -/A -/A  /A  /A -/A -/A  -A  -A   A   A  -A  -A
X.10     1  1 -1 -1  1  1  -A  -A   A   A  -A  -A -/A -/A  /A  /A -/A -/A
X.11     1  1  1  1  1  1 -/A -/A -/A -/A -/A -/A  -A  -A  -A  -A  -A  -A
X.12     1  1  1  1  1  1  -A  -A  -A  -A  -A  -A -/A -/A -/A -/A -/A -/A
X.13     2 -2  .  . -1  1   1  -1   .   .   2  -2   1  -1   .   .   2  -2
X.14     2  2  .  . -1 -1  -1  -1   .   .   2   2  -1  -1   .   .   2   2
X.15     2 -2  .  . -1  1 -/A  /A   .   .   B  -B  -A   A   .   .  /B -/B
X.16     2 -2  .  . -1  1  -A   A   .   .  /B -/B -/A  /A   .   .   B  -B
X.17     2  2  .  . -1 -1  /A  /A   .   .   B   B   A   A   .   .  /B  /B
X.18     2  2  .  . -1 -1   A   A   .   .  /B  /B  /A  /A   .   .   B   B

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3