Properties

Label 36T37
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times C_3:C_4$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $37$
Group :  $S_3\times C_3:C_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,24,6,28,33,31,2,23,5,27,34,32)(3,22,7,25,35,30,4,21,8,26,36,29)(9,19,14,11,18,15,10,20,13,12,17,16), (1,17,2,18)(3,19,4,20)(5,14,6,13)(7,16,8,15)(9,33,10,34)(11,35,12,36)(21,23,22,24)(25,32,26,31)(27,29,28,30)
$|\Aut(F/K)|$:  $12$

Low degree resolvents

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

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

Degree 9: $S_3^2$

Degree 12: $C_3 : C_4$, $S_3 \times C_4$

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 20]
Character table:   
      2  3  3  3  3  3  3  3  3  2  2   2   2  1  2  1  2  2  2
      3  2  1  2  1  1  .  1  .  2  2   1   1  2  1  2  1  2  2

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

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  1 -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  1  1  1
X.5      1 -1 -1  1  A -A -A  A  1 -1  -A   A -1  1  1 -1  1 -1
X.6      1 -1 -1  1 -A  A  A -A  1 -1   A  -A -1  1  1 -1  1 -1
X.7      1  1 -1 -1  A  A -A -A  1 -1  -A   A -1 -1  1  1  1 -1
X.8      1  1 -1 -1 -A -A  A  A  1 -1   A  -A -1 -1  1  1  1 -1
X.9      2 -2 -2  2  .  .  .  .  2 -2   .   .  1 -1 -1  1 -1  1
X.10     2 -2  2 -2  .  .  .  .  2  2   .   . -1  1 -1  1 -1 -1
X.11     2  2 -2 -2  .  .  .  .  2 -2   .   .  1  1 -1 -1 -1  1
X.12     2  2  2  2  .  .  .  .  2  2   .   . -1 -1 -1 -1 -1 -1
X.13     2  .  2  . -2  . -2  . -1 -1   1   1 -1  . -1  .  2  2
X.14     2  .  2  .  2  .  2  . -1 -1  -1  -1 -1  . -1  .  2  2
X.15     2  . -2  .  B  . -B  . -1  1   A  -A  1  . -1  .  2 -2
X.16     2  . -2  . -B  .  B  . -1  1  -A   A  1  . -1  .  2 -2
X.17     4  .  4  .  .  .  .  . -2 -2   .   .  1  .  1  . -2 -2
X.18     4  . -4  .  .  .  .  . -2  2   .   . -1  .  1  . -2  2

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i