Properties

Label 36T21
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 $t$ :  $21$
Group :  $S_3\times A_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,27,3,16,25)(2,14,28,4,15,26)(5,9,31,33,19,21)(6,10,32,34,20,22)(7,12,29,36,17,23)(8,11,30,35,18,24), (1,34,7)(2,33,8)(3,36,6)(4,35,5)(9,17,15,10,18,16)(11,20,14,12,19,13)(21,29,28,22,30,27)(23,31,25,24,32,26)
$|\Aut(F/K)|$:  $12$

Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: None

Degree 6: $C_6$, $S_3$, $A_4$, $S_3\times C_3$, $A_4\times C_2$

Degree 9: $S_3\times C_3$

Degree 12: $A_4 \times C_2$

Degree 18: $S_3 \times C_3$, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30) (31,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 3)( 2, 4)( 5,33)( 6,34)( 7,36)( 8,35)( 9,19)(10,20)(11,18)(12,17)(13,16) (14,15)(21,31)(22,32)(23,29)(24,30)(25,27)(26,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 3)( 2, 4)( 5,33)( 6,34)( 7,36)( 8,35)( 9,20)(10,19)(11,17)(12,18)(13,15) (14,16)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 7,34)( 2, 8,33)( 3, 6,36)( 4, 5,35)( 9,15,18)(10,16,17)(11,14,19) (12,13,20)(21,28,30)(22,27,29)(23,25,32)(24,26,31)$
$ 6, 6, 6, 6, 3, 3, 3, 3 $ $6$ $6$ $( 1, 7,34)( 2, 8,33)( 3, 6,36)( 4, 5,35)( 9,16,18,10,15,17)(11,13,19,12,14,20) (21,27,30,22,28,29)(23,26,32,24,25,31)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1, 9,29)( 2,10,30)( 3,11,32)( 4,12,31)( 5,13,24)( 6,14,23)( 7,15,22) ( 8,16,21)(17,28,33)(18,27,34)(19,25,36)(20,26,35)$
$ 6, 6, 6, 6, 6, 6 $ $12$ $6$ $( 1,11,27,36,15,23)( 2,12,28,35,16,24)( 3, 9,25,34,14,22)( 4,10,26,33,13,21) ( 5,17,31, 8,20,30)( 6,18,32, 7,19,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,15,28)( 2,16,27)( 3,14,26)( 4,13,25)( 5,20,32)( 6,19,31)( 7,18,30) ( 8,17,29)( 9,21,34)(10,22,33)(11,24,36)(12,23,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1,21,18)( 2,22,17)( 3,24,19)( 4,23,20)( 5,25,12)( 6,26,11)( 7,28, 9) ( 8,27,10)(13,35,32)(14,36,31)(15,34,30)(16,33,29)$
$ 6, 6, 6, 6, 6, 6 $ $12$ $6$ $( 1,23,15,36,27,11)( 2,24,16,35,28,12)( 3,22,14,34,25, 9)( 4,21,13,33,26,10) ( 5,30,20, 8,31,17)( 6,29,19, 7,32,18)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,27,16)( 2,28,15)( 3,25,13)( 4,26,14)( 5,31,19)( 6,32,20)( 7,29,17) ( 8,30,18)( 9,33,21)(10,34,22)(11,35,24)(12,36,23)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
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  .  2  1   2   1  2   2   1  2

        1a 2a 2b 2c 3a 6a  3b  6b 3c  3d  6c 3e
     2P 1a 1a 1a 1a 3a 3a  3d  3e 3e  3b  3c 3c
     3P 1a 2a 2b 2c 1a 2a  1a  2b 1a  1a  2b 1a
     5P 1a 2a 2b 2c 3a 6a  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 -3  1  3 -1   .   .  .   .   .  .
X.11     3 -1  3 -1  3 -1   .   .  .   .   .  .
X.12     6 -2  .  . -3  1   .   .  .   .   .  .

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