Properties

Label 36T39
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_6:S_3$

Learn more about

Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $39$
Group :  $D_6:S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,34,4,6,35)(2,7,33,3,5,36)(9,26,18,22,14,29)(10,25,17,21,13,30)(11,27,20,23,15,31)(12,28,19,24,16,32), (1,24,2,23)(3,21,4,22)(5,31,6,32)(7,30,8,29)(9,15,10,16)(11,13,12,14)(17,19,18,20)(25,35,26,36)(27,34,28,33)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
8:  $D_{4}$
12:  $D_{6}$ x 2
36:  $S_3^2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 2

Degree 4: $D_{4}$

Degree 6: $S_3$, $D_{6}$

Degree 9: $S_3^2$

Degree 12: $(C_6\times C_2):C_2$, $(C_6\times C_2):C_2$

Degree 18: $S_3^2$

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 22]
Character table:   
      2  3  2  3  2  2  2  2  2  2  2  1  1  2  2  2
      3  2  1  2  1  .  2  2  1  1  1  2  2  1  2  2

        1a 2a 2b 2c 4a 6a 3a 6b 6c 6d 3b 6e 6f 3c 6g
     2P 1a 1a 1a 1a 2b 3a 3a 3a 3a 3c 3b 3b 3c 3c 3c
     3P 1a 2a 2b 2c 4a 2b 1a 2c 2c 2a 1a 2b 2a 1a 2b
     5P 1a 2a 2b 2c 4a 6a 3a 6c 6b 6f 3b 6e 6d 3c 6g

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

A = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3