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

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  1  1  3  3  3  3  4  5  2  2  2  2   2   2  3   .   .  .
      3  4   2  3  2  4  4  2  2  1  1  3  1  2  2   1   1  .   2   2  .
      5  1   .  .  .  .  .  .  .  .  .  .  .  .  .   .   .  .   .   .  1

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

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  2  .  A /A  F /F  1  1 -1  1  I -I   J  /J -1  -J -/J  .
X.3      5  -3  2  . /A  A /F  F  1  1 -1  1 -I  I  /J   J -1 -/J  -J  .
X.4      6  -2  . -2 -3 -3  1  1  2  2  3 -1  1  1  -1  -1  .   .   .  1
X.5     10   2  1 -1  B /B  A /A  2 -2  1  1 -1 -1 -/J  -J  .   J  /J  .
X.6     10   2  1 -1 /B  B /A  A  2 -2  1  1 -1 -1  -J -/J  .  /J   J  .
X.7     15   7  3  1 -3 -3  1  1 -1  3  .  . -2 -2  -1  -1  1   .   .  .
X.8     15  -1  .  2  6  6  2  2  3 -1  3 -1 -1 -1   .   . -1   .   .  .
X.9     20   4 -1  1  2  2 -2 -2  .  4  5  1  1  1   .   .  .  -1  -1  .
X.10    24   8  3 -1  6  6  2  2  .  .  .  .  2  2   .   .  .   .   . -1
X.11    30 -10  3 -1  3  3 -1 -1 -2  2  3 -1 -1 -1   1   1  .   .   .  .
X.12    30   6  .  .  C /C  F /F  2  2 -3 -1  I -I  -J -/J  .   .   .  .
X.13    30   6  .  . /C  C /F  F  2  2 -3 -1 -I  I -/J  -J  .   .   .  .
X.14    40  -8  1  1  D /D  G /G  .  . -2  .  G /G   .   .  .   J  /J  .
X.15    40  -8  1  1 /D  D /G  G  .  . -2  . /G  G   .   .  .  /J   J  .
X.16    45  -3  .  .  E /E  H /H  1 -3  .  .  .  .   J  /J  1   .   .  .
X.17    45  -3  .  . /E  E /H  H  1 -3  .  .  .  .  /J   J  1   .   .  .
X.18    60  -4 -3 -1  6  6  2  2  .  4 -3  1 -1 -1   .   .  .   .   .  .
X.19    64   . -2  . -8 -8  .  .  .  .  4  .  .  .   .   .  .   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)-E(3)^2
  = Sqrt(-3) = i3
J = E(3)
  = (-1+Sqrt(-3))/2 = b3