Properties

Label 36T12781
Degree $36$
Order $25920$
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)
$\card{\Aut(F/K)}$:  $1$
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)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

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

27T993, 40T14344, 40T14345, 45T666

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

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

Group invariants

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

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

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