↑

Properties

Label 40T14345
Order \(25920\)
n \(40\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSp(4,3)$

Learn more about

Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $14345$
Group :  $\PSp(4,3)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,4,7,10)(3,6,12,21,20)(5,9,16,27,38)(8,14,24,32,34)(11,19,23,31,17)(13,22,30,40,37)(15,25,28,18,29)(26,33,39,36,35), (1,3,4,7,12)(2,5,10,18,22)(6,9,16,27,38)(8,15,26,36,37)(11,20,30,40,34)(13,23,19,29,17)(14,25,28,21,31)(24,33,39,32,35)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 4: None

Degree 5: None

Degree 8: None

Degree 10: None

Degree 20: None

Low degree siblings

There are no siblings with degree $\leq 10$
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, 1, 1 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $5184$ $5$ $( 1,28,16,36,12)( 2, 5,27, 6,11)( 3,39,19,29,40)( 4,22,21, 9, 8) ( 7,24,14,17,13)(10,32,31,37,26)(15,20,33,18,23)(25,34,38,35,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $270$ $2$ $( 1,21)( 2,18)( 3,30)( 4,17)( 5,38)( 6,15)( 7,36)( 8, 9)(10,16)(11,40)(12,13) (14,28)(20,35)(23,25)(24,31)(26,37)(27,39)(29,34)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 1 $ $3240$ $4$ $( 1,40,21,11)( 2,29,18,34)( 3,16,30,10)( 4,31,17,24)( 5,14,38,28)( 6,25,15,23) ( 7, 8,36, 9)(12,37,13,26)(19,33)(20,39,35,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $40$ $3$ $( 1,11,31)( 2,15, 9)( 3,16,24)( 4,19,13)( 5,21,20)( 7,29,23)(10,38,37) (12,32,27)(14,26,40)(17,39,33)(25,36,34)(28,30,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $40$ $3$ $( 1,31,11)( 2, 9,15)( 3,24,16)( 4,13,19)( 5,20,21)( 7,23,29)(10,37,38) (12,27,32)(14,40,26)(17,33,39)(25,34,36)(28,35,30)$
$ 9, 9, 9, 9, 3, 1 $ $2880$ $9$ $( 1,20,15,11, 5, 9,31,21, 2)( 3,30,23,16,35, 7,24,28,29)( 4,26,38,19,40,37,13, 14,10)( 6, 8,22)(12,17,25,32,39,36,27,33,34)$
$ 9, 9, 9, 9, 3, 1 $ $2880$ $9$ $( 1,15, 5,31, 2,20,11, 9,21)( 3,23,35,24,29,30,16, 7,28)( 4,38,40,13,10,26,19, 37,14)( 6,22, 8)(12,25,39,27,34,17,32,36,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $45$ $2$ $( 1,32)( 2, 7)( 5,39)( 9,23)(10,14)(11,27)(12,31)(15,29)(17,20)(21,33)(26,38) (37,40)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2 $ $540$ $4$ $( 1,10,32,14)( 2,33, 7,21)( 3, 6)( 4,19)( 5,37,39,40)( 8,28)( 9,31,23,12) (11,17,27,20)(13,18)(15,26,29,38)(16,24)(22,34)(25,36)(30,35)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1 $ $360$ $6$ $( 1,37,17,32,40,20)( 2,23,15, 7, 9,29)( 3,13,28)( 4,30,16)( 5,11,10,39,27,14) ( 6,18, 8)(12,26,21,31,38,33)(19,35,24)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1 $ $360$ $6$ $( 1,20,40,32,17,37)( 2,29, 9, 7,15,23)( 3,28,13)( 4,16,30)( 5,14,27,39,10,11) ( 6, 8,18)(12,33,38,31,21,26)(19,24,35)$
$ 12, 12, 6, 6, 2, 2 $ $2160$ $12$ $( 1,11,37,10,17,39,32,27,40,14,20, 5)( 2,38,23,33,15,12, 7,26, 9,21,29,31) ( 3, 8,13, 6,28,18)( 4,24,30,19,16,35)(22,34)(25,36)$
$ 12, 12, 6, 6, 2, 2 $ $2160$ $12$ $( 1,39,20,10,40,11,32, 5,17,14,37,27)( 2,12,29,33, 9,38, 7,31,15,21,23,26) ( 3,18,28, 6,13, 8)( 4,35,16,19,30,24)(22,34)(25,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $240$ $3$ $( 1,34, 2)( 3,16, 6)( 4,32,14)( 5,31,28)( 7, 8,10)( 9,25,12)(11,13,26) (15,40,21)(17,38,22)(18,23,39)(19,35,36)(20,30,27)(29,33,37)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 3, 1 $ $720$ $6$ $( 1, 2,34)( 3,29,16,33, 6,37)( 4,27,32,20,14,30)( 5,26,31,11,28,13) ( 7,39, 8,18,10,23)( 9,12,25)(15,21,40)(17,22,38)(19,36,35)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 3, 1 $ $720$ $6$ $( 1,34, 2)( 3,37, 6,33,16,29)( 4,30,14,20,32,27)( 5,13,28,11,31,26) ( 7,23,10,18, 8,39)( 9,25,12)(15,40,21)(17,38,22)(19,35,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ $480$ $3$ $( 1,28,26)( 2,31,13)( 3,16, 6)( 4,12,20)( 5,11,34)( 7,18,17)( 8,23,38) ( 9,30,32)(10,39,22)(14,25,27)(29,37,33)$
$ 6, 6, 6, 6, 6, 6, 3, 1 $ $2160$ $6$ $( 1,17,39,16,30,35)( 2,14,24,31,23,13)( 3,15,26,29,19,11)( 4,28, 9,40,33, 7) ( 5,12,36, 6,10,32)( 8,38,25)(18,37,20,21,34,22)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 2, 2, 2, 1 $ $1440$ $6$ $( 1,33, 2,32,21, 7)( 3,18,30)( 4,28, 6)( 5,39)( 8,16,13)( 9,37,12,23,40,31) (10,14)(11,27)(15,20,38,29,17,26)(19,35,24)(22,25,34)$

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

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

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  1  1  1  2  .   J  /J -1 -/J  -J
X.3      5  .  -3 -1 -A  A /C  C /H  H  1  1  1  2  .  /J   J -1  -J -/J
X.4      6  1  -2  3  1  1 -3 -3  1  1  2  2 -1  . -2  -1  -1  .   .   .
X.5     10  .   2  1 -1 -1  D /D /C  C  2 -2  1  1 -1  -J -/J  .   J  /J
X.6     10  .   2  1 -1 -1 /D  D  C /C  2 -2  1  1 -1 -/J  -J  .  /J   J
X.7     15  .   7  . -2 -2 -3 -3  1  1 -1  3  .  3  1  -1  -1  1   .   .
X.8     15  .  -1  3 -1 -1  6  6  2  2  3 -1 -1  .  2   .   . -1   .   .
X.9     20  .   4  5  1  1  2  2 -2 -2  .  4  1 -1  1   .   .  .  -1  -1
X.10    24 -1   8  .  2  2  6  6  2  2  .  .  .  3 -1   .   .  .   .   .
X.11    30  . -10  3 -1 -1  3  3 -1 -1 -2  2 -1  3 -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  .  .  .  /J   J  1   .   .
X.17    45  .  -3  .  .  . /G  G /I  I  1 -3  .  .  .   J  /J  1   .   .
X.18    60  .  -4 -3 -1 -1  6  6  2  2  .  4  1 -3 -1   .   .  .   .   .
X.19    64 -1   .  4  .  . -8 -8  .  .  .  .  . -2  .   .   .  .   1   1
X.20    81  1   9  .  .  .  .  .  .  . -3 -3  .  .  .   .   . -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 = 2*E(3)+E(3)^2
  = (-3+Sqrt(-3))/2 = -1+b3
I = 3*E(3)
  = (-3+3*Sqrt(-3))/2 = 3b3
J = E(3)
  = (-1+Sqrt(-3))/2 = b3