Properties

Label 36T50
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times A_4$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $50$
Group :  $S_3\times A_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10)(2,9)(3,12)(4,11)(13,18)(14,17)(15,20)(16,19)(21,24)(22,23)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35), (1,24,29,7,18,36)(2,23,30,8,17,35)(3,21,31,6,19,34)(4,22,32,5,20,33)(9,16,26,10,15,25)(11,13,28,12,14,27)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
12:  $A_4$
18:  $S_3\times C_3$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 4: None

Degree 6: $A_4$, $A_4\times C_2$

Degree 9: $S_3\times C_3$

Degree 12: $A_4\times C_2$

Degree 18: 18T31, 18T32

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 44]
Character table:   
      2  3  3  3  3  2  2   .   1  1   .   1  1
      3  2  .  1  1  1  2   2   1  2   2   1  2

        1a 2a 2b 2c 6a 3a  3b  6b 3c  3d  6c 3e
     2P 1a 1a 1a 1a 3a 3a  3d  3e 3e  3b  3c 3c
     3P 1a 2a 2b 2c 2b 1a  1a  2c 1a  1a  2c 1a
     5P 1a 2a 2b 2c 6a 3a  3d  6c 3e  3b  6b 3c

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

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