Properties

Label 36T12781
Order \(25920\)
n \(36\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSp(4,3)$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $12781$
Group :  $\PSp(4,3)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,4,8,16)(3,6,7,14,25)(5,11,21,32,35)(9,17,18,28,29)(10,19,30,22,33)(12,13,20,24,31)(15,27,34,36,23), (2,3,5,10,18)(4,8,15,26,30)(6,12,17,25,22)(7,13,23,24,29)(9,14,19,27,31)(11,20,16,28,34)(21,32,35,33,36)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Degree 9: None

Degree 12: None

Degree 18: None

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 $ $45$ $2$ $( 1,32)( 2, 8)( 4,18)( 6,22)( 7,14)(11,17)(12,36)(16,21)(20,34)(24,31)(25,33) (28,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 7,25)( 2,36, 6)( 3,26,13)( 4,28,17)( 5,27,10)( 8,12,22)( 9,19,30) (11,18,35)(14,33,32)(15,29,23)(16,20,31)(21,34,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1,25, 7)( 2, 6,36)( 3,13,26)( 4,17,28)( 5,10,27)( 8,22,12)( 9,30,19) (11,35,18)(14,32,33)(15,23,29)(16,31,20)(21,24,34)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2 $ $540$ $4$ $( 1,20,32,34)( 2, 4, 8,18)( 3,10)( 5,26)( 6,17,22,11)( 7,31,14,24)( 9,29) (12,35,36,28)(13,27)(15,30)(16,33,21,25)(19,23)$
$ 6, 6, 6, 6, 3, 3, 3, 3 $ $360$ $6$ $( 1,33, 7,32,25,14)( 2,22,36, 8, 6,12)( 3,13,26)( 4,11,28,18,17,35)( 5,10,27) ( 9,30,19)(15,23,29)(16,24,20,21,31,34)$
$ 6, 6, 6, 6, 3, 3, 3, 3 $ $360$ $6$ $( 1,14,25,32, 7,33)( 2,12, 6, 8,36,22)( 3,26,13)( 4,35,17,18,28,11)( 5,27,10) ( 9,19,30)(15,29,23)(16,34,31,21,20,24)$
$ 12, 12, 6, 6 $ $2160$ $12$ $( 1,24,33,20, 7,21,32,31,25,34,14,16)( 2,35,22, 4,36,11, 8,28, 6,18,12,17) ( 3, 5,13,10,26,27)( 9,23,30,29,19,15)$
$ 12, 12, 6, 6 $ $2160$ $12$ $( 1,21,14,20,25,24,32,16, 7,34,33,31)( 2,11,12, 4, 6,35, 8,17,36,18,22,28) ( 3,27,26,10,13, 5)( 9,15,19,29,30,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $480$ $3$ $( 1,25, 7)( 2,34,35)( 3,10,15)( 4,22,16)( 5,29,26)( 6,21,18)( 8,20,28) (11,36,24)(12,31,17)(13,27,23)(14,32,33)$
$ 6, 6, 6, 6, 3, 3, 3, 1, 1, 1 $ $1440$ $6$ $( 1,14,25,32, 7,33)( 2,28,34, 8,35,20)( 3,15,10)( 4,21,22,18,16, 6)( 5,26,29) (11,31,36,17,24,12)(13,23,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $270$ $2$ $( 1,11)( 3,15)( 6,20)( 7,14)(10,30)(12,25)(13,19)(16,35)(17,32)(21,28)(22,34) (23,27)(24,31)(33,36)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1 $ $3240$ $4$ $( 1,21,16,32)( 2, 9, 8,29)( 3,13)( 4,18)( 6,34,25,36)( 7,30,14,10) (11,28,35,17)(12,33,20,22)(15,19)(23,24,27,31)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $240$ $3$ $( 2,18,24)( 3,23, 5)( 4,31, 8)( 6,11,34)( 9,30,19)(10,13,29)(12,28,16) (15,27,26)(17,20,22)(21,36,35)$
$ 6, 6, 6, 3, 3, 3, 3, 2, 2, 2 $ $720$ $6$ $( 1,32)( 2,31,18, 8,24, 4)( 3, 5,23)( 6,20,11,22,34,17)( 7,14)( 9,19,30) (10,29,13)(12,21,28,36,16,35)(15,26,27)(25,33)$
$ 6, 6, 6, 3, 3, 3, 3, 2, 2, 2 $ $720$ $6$ $( 1,32)( 2, 4,24, 8,18,31)( 3,23, 5)( 6,17,34,22,11,20)( 7,14)( 9,30,19) (10,13,29)(12,35,16,36,28,21)(15,27,26)(25,33)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1, 1 $ $2160$ $6$ $( 1,25)( 2, 4,24, 8,18,31)( 3,27, 5,15,23,26)( 6,12,34,16,11,28)( 9,30,19) (10,13,29)(17,35,22,36,20,21)(32,33)$
$ 5, 5, 5, 5, 5, 5, 5, 1 $ $5184$ $5$ $( 1,12, 8, 9, 2)( 4,18,28,24,22)( 5,26,32,27,36)( 6,11,25,35,15) ( 7,23,14,34,21)(10,33,17,30,31)(13,20,29,16,19)$
$ 9, 9, 9, 9 $ $2880$ $9$ $( 1,10, 5,23,32,14, 6,34,24)( 2,25,19, 7,29,31,15,36, 8)( 3, 4,11,26,17,18,13, 28,35)( 9,21,12,20,27,30,22,33,16)$
$ 9, 9, 9, 9 $ $2880$ $9$ $( 1, 5,32, 6,24,10,23,14,34)( 2,19,29,15, 8,25, 7,31,36)( 3,11,17,13,35, 4,26, 18,28)( 9,12,27,22,16,21,20,30,33)$

Group invariants

Order:  $25920=2^{6} \cdot 3^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:   
      2  6   6  3  3  4  3  3   2   2  2  2  2  5  2  3  1   .   .  .  1
      3  4   2  4  4  1  2  2   1   1  3  2  2  1  1  .  3   2   2  .  2
      5  1   .  .  .  .  .  .   .   .  .  .  .  .  .  .  .   .   .  1  .

        1a  2a 3a 3b 4a 6a 6b 12a 12b 3c 6c 6d 2b 6e 4b 3d  9a  9b 5a 6f
     2P 1a  1a 3b 3a 2a 3a 3b  6a  6b 3c 3c 3c 1a 3c 2b 3d  9b  9a 5a 3d
     3P 1a  2a 1a 1a 4a 2a 2a  4a  4a 1a 2a 2a 2b 2b 4b 1a  3b  3a 5a 2a
     5P 1a  2a 3b 3a 4a 6b 6a 12b 12a 3c 6d 6c 2b 6e 4b 3d  9b  9a 1a 6f
     7P 1a  2a 3a 3b 4a 6a 6b 12a 12b 3c 6c 6d 2b 6e 4b 3d  9a  9b 5a 6f
    11P 1a  2a 3b 3a 4a 6b 6a 12b 12a 3c 6d 6c 2b 6e 4b 3d  9b  9a 5a 6f

X.1      1   1  1  1  1  1  1   1   1  1  1  1  1  1  1  1   1   1  1  1
X.2      5  -3  A /A  1  F /F   I  /I -1  J -J  1  1 -1  2 -/I  -I  .  .
X.3      5  -3 /A  A  1 /F  F  /I   I -1 -J  J  1  1 -1  2  -I -/I  .  .
X.4      6  -2 -3 -3  2  1  1  -1  -1  3  1  1  2 -1  .  .   .   .  1 -2
X.5     10   2  B /B  2  A /A -/I  -I  1 -1 -1 -2  1  .  1  /I   I  . -1
X.6     10   2 /B  B  2 /A  A  -I -/I  1 -1 -1 -2  1  .  1   I  /I  . -1
X.7     15   7 -3 -3 -1  1  1  -1  -1  . -2 -2  3  .  1  3   .   .  .  1
X.8     15  -1  6  6  3  2  2   .   .  3 -1 -1 -1 -1 -1  .   .   .  .  2
X.9     20   4  2  2  . -2 -2   .   .  5  1  1  4  1  . -1  -1  -1  .  1
X.10    24   8  6  6  .  2  2   .   .  .  2  2  .  .  .  3   .   . -1 -1
X.11    30 -10  3  3 -2 -1 -1   1   1  3 -1 -1  2 -1  .  3   .   .  . -1
X.12    30   6  C /C  2  F /F  -I -/I -3  J -J  2 -1  .  .   .   .  .  .
X.13    30   6 /C  C  2 /F  F -/I  -I -3 -J  J  2 -1  .  .   .   .  .  .
X.14    40  -8  D /D  .  G /G   .   . -2  G /G  .  .  .  1  /I   I  .  1
X.15    40  -8 /D  D  . /G  G   .   . -2 /G  G  .  .  .  1   I  /I  .  1
X.16    45  -3  E /E  1  H /H   I  /I  .  .  . -3  .  1  .   .   .  .  .
X.17    45  -3 /E  E  1 /H  H  /I   I  .  .  . -3  .  1  .   .   .  .  .
X.18    60  -4  6  6  .  2  2   .   . -3 -1 -1  4  1  . -3   .   .  . -1
X.19    64   . -8 -8  .  .  .   .   .  4  .  .  .  .  . -2   1   1 -1  .
X.20    81   9  .  . -3  .  .   .   .  .  .  . -3  . -1  .   .   .  1  .

A = -2*E(3)+E(3)^2
  = (1-3*Sqrt(-3))/2 = -1-3b3
B = 5*E(3)+2*E(3)^2
  = (-7+3*Sqrt(-3))/2 = -2+3b3
C = 6*E(3)-3*E(3)^2
  = (-3+9*Sqrt(-3))/2 = 3+9b3
D = 2*E(3)+8*E(3)^2
  = -5-3*Sqrt(-3) = -5-3i3
E = -9*E(3)^2
  = (9+9*Sqrt(-3))/2 = 9+9b3
F = E(3)+2*E(3)^2
  = (-3-Sqrt(-3))/2 = -2-b3
G = -2*E(3)^2
  = 1+Sqrt(-3) = 1+i3
H = 3*E(3)
  = (-3+3*Sqrt(-3))/2 = 3b3
I = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
J = E(3)-E(3)^2
  = Sqrt(-3) = i3