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Properties

Label 36T20
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times S_4$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $20$
Group :  $C_3\times S_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,28,36,15,23,2,11,27,35,16,24)(3,10,25,34,14,22,4,9,26,33,13,21)(5,17,31,7,20,30)(6,18,32,8,19,29), (1,4)(2,3)(5,34)(6,33)(7,36)(8,35)(9,20)(10,19)(11,17)(12,18)(13,15)(14,16)(21,31)(22,32)(23,29)(24,30)(25,27)(26,28)
$|\Aut(F/K)|$:  $12$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
24:  $S_4$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: None

Degree 6: $C_6$, $S_3$, $S_3\times C_3$, $S_4$, $S_4$

Degree 9: $S_3\times C_3$

Degree 12: $S_4$

Degree 18: $S_3 \times C_3$, 18T30, 18T33

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 42]
Character table:    
      2  3  3  2  2  .   .   2   2   3  3   .   2   2  3   3
      3  2  1  1  1  2   2   1   1   1  2   2   1   1  2   1

        1a 2a 4a 2b 3a  3b 12a  6a  6b 3c  3d  6c 12b 3e  6d
     2P 1a 1a 2a 1a 3a  3d  6d  3e  3e 3e  3b  3c  6b 3c  3c
     3P 1a 2a 4a 2b 1a  1a  4a  2b  2a 1a  1a  2b  4a 1a  2a
     5P 1a 2a 4a 2b 3a  3d 12b  6c  6d 3e  3b  6a 12a 3c  6b
     7P 1a 2a 4a 2b 3a  3b 12a  6a  6b 3c  3d  6c 12b 3e  6d
    11P 1a 2a 4a 2b 3a  3d 12b  6c  6d 3e  3b  6a 12a 3c  6b

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

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