Properties

Label 36T5
Order \(36\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times C_3:C_4$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $5$
Group :  $C_3\times C_3:C_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,27,6,16,31,2,20,28,5,15,32)(3,17,26,7,13,29,4,18,25,8,14,30)(9,24,33,12,22,35,10,23,34,11,21,36), (1,17,22)(2,18,21)(3,20,23)(4,19,24)(5,11,26)(6,12,25)(7,10,27)(8,9,28)(13,32,36)(14,31,35)(15,30,33)(16,29,34)
$|\Aut(F/K)|$:  $36$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: $C_4$

Degree 6: $C_6$, $S_3$, $S_3\times C_3$

Degree 9: $S_3\times C_3$

Degree 12: $C_{12}$, $C_3 : C_4$, $C_3\times (C_3 : C_4)$

Degree 18: $S_3 \times C_3$

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 6]
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 4a 4b 6a 3a  3b  6b 12a 12b  6c  3c  6d  3d 12c 12d  6e  3e
     2P 1a 1a 2a 2a 3a 3a  3d  3d  6e  6e  3e  3e  3b  3b  6c  6c  3c  3c
     3P 1a 2a 4b 4a 2a 1a  1a  2a  4a  4b  2a  1a  2a  1a  4b  4a  2a  1a
     5P 1a 2a 4a 4b 6a 3a  3d  6d 12d 12c  6e  3e  6b  3b 12b 12a  6c  3c
     7P 1a 2a 4b 4a 6a 3a  3b  6b 12b 12a  6c  3c  6d  3d 12d 12c  6e  3e
    11P 1a 2a 4b 4a 6a 3a  3d  6d 12c 12d  6e  3e  6b  3b 12a 12b  6c  3c

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  A -A -1  1   1  -1  -A   A  -1   1  -1   1   A  -A  -1   1
X.4      1 -1 -A  A -1  1   1  -1   A  -A  -1   1  -1   1  -A   A  -1   1
X.5      1 -1  A -A -1  1   B  -B   C  -C  -B   B -/B  /B  /C -/C -/B  /B
X.6      1 -1  A -A -1  1  /B -/B -/C  /C -/B  /B  -B   B  -C   C  -B   B
X.7      1 -1 -A  A -1  1   B  -B  -C   C  -B   B -/B  /B -/C  /C -/B  /B
X.8      1 -1 -A  A -1  1  /B -/B  /C -/C -/B  /B  -B   B   C  -C  -B   B
X.9      1  1 -1 -1  1  1   B   B  -B  -B   B   B  /B  /B -/B -/B  /B  /B
X.10     1  1 -1 -1  1  1  /B  /B -/B -/B  /B  /B   B   B  -B  -B   B   B
X.11     1  1  1  1  1  1   B   B   B   B   B   B  /B  /B  /B  /B  /B  /B
X.12     1  1  1  1  1  1  /B  /B  /B  /B  /B  /B   B   B   B   B   B   B
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  -B   B   .   .   D  -D  /B -/B   .   .  /D -/D
X.16     2 -2  .  .  1 -1 -/B  /B   .   .  /D -/D   B  -B   .   .   D  -D
X.17     2  2  .  . -1 -1  -B  -B   .   .  -D  -D -/B -/B   .   . -/D -/D
X.18     2  2  .  . -1 -1 -/B -/B   .   . -/D -/D  -B  -B   .   .  -D  -D

A = -E(4)
  = -Sqrt(-1) = -i
B = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
C = E(12)^11
D = -2*E(3)^2
  = 1+Sqrt(-3) = 1+i3