Properties

Label 40T14345
Degree $40$
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 $n$:  $40$
Transitive number $t$:  $14345$
Group:  $\PSp(4,3)$
Parity:  $1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$\card{\Aut(F/K)}$:  $1$
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)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: None

Degree 8: None

Degree 10: None

Degree 20: None

Low degree siblings

27T993, 36T12781, 40T14344, 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, 1, 1 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $5184$ $5$ $( 1, 5,20, 7,38)( 2,14,23,34, 6)( 3,22,12,11,26)( 4,33,24,15,30) ( 8,19,39,35,13)( 9,29,21,31,16)(10,25,27,37,40)(17,36,32,28,18)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $40$ $3$ $( 1,13,29)( 3,24, 6)( 4,30,25)( 7,38,39)( 8,20,21)( 9,15,22)(10,36,27) (11,28,37)(12,17,19)(14,40,18)(23,35,32)(31,34,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $40$ $3$ $( 1,29,13)( 3, 6,24)( 4,25,30)( 7,39,38)( 8,21,20)( 9,22,15)(10,27,36) (11,37,28)(12,19,17)(14,18,40)(23,32,35)(31,33,34)$
$ 9, 9, 9, 9, 3, 1 $ $2880$ $9$ $( 1,31,15,13,34,22,29,33, 9)( 2, 5,26)( 3,25, 7,24, 4,38, 6,30,39) ( 8,14,17,20,40,19,21,18,12)(10,32,28,36,23,37,27,35,11)$
$ 9, 9, 9, 9, 3, 1 $ $2880$ $9$ $( 1,15,34,29, 9,31,13,22,33)( 2,26, 5)( 3, 7, 4, 6,39,25,24,38,30) ( 8,17,40,21,12,14,20,19,18)(10,28,23,27,11,32,36,37,35)$
$ 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 $ $45$ $2$ $( 2,21)( 3,18)( 4,28)( 7,33)( 8,16)( 9,40)(17,38)(19,34)(20,26)(22,24)(23,37) (25,35)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2 $ $540$ $4$ $( 1,31)( 2, 3,21,18)( 4,22,28,24)( 5,10)( 6,11)( 7,35,33,25)( 8,37,16,23) ( 9,34,40,19)(12,14)(13,15)(17,20,38,26)(27,36)(29,30)(32,39)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $270$ $2$ $( 1,30)( 2,21)( 3,18)( 4,17)( 5,15)( 6,14)( 8, 9)(10,13)(11,12)(16,40)(19,23) (20,22)(24,26)(27,39)(28,38)(29,31)(32,36)(34,37)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 1 $ $3240$ $4$ $( 1,22,27,24)( 2,10,21,13)( 3,11,18,12)( 4,19,17,23)( 5,15)( 7,33,35,25) ( 8,32,16,31)( 9,36,40,29)(20,39,26,30)(28,37,38,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $240$ $3$ $( 1,13,29)( 2,26,16)( 3,22,40)( 4,37,35)( 6,15,14)( 7,17,34)( 8,21,20) ( 9,18,24)(10,36,27)(11,32,30)(12,31,39)(19,33,38)(23,25,28)$
$ 6, 6, 6, 6, 6, 6, 3, 1 $ $2160$ $6$ $( 1,10,13,36,29,27)( 2,20,26, 8,16,21)( 3,34,22, 7,40,17)( 4,11,37,32,35,30) ( 6,38,15,19,14,33)( 9,31,18,39,24,12)(23,28,25)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 3, 1 $ $720$ $6$ $( 1,10,13,36,29,27)( 2,16,26)( 3,34,22, 7,40,17)( 4,35,37)( 6,31,15,39,14,12) ( 8,20,21)( 9,38,18,19,24,33)(11,30,32)(23,28,25)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 3, 1 $ $720$ $6$ $( 1,27,29,36,13,10)( 2,26,16)( 3,17,40, 7,22,34)( 4,37,35)( 6,12,14,39,15,31) ( 8,21,20)( 9,33,24,19,18,38)(11,32,30)(23,25,28)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1 $ $360$ $6$ $( 1,36,29,10,13,27)( 3, 4, 6,25,24,30)( 7,38,39)( 8,20,21)( 9,11,22,37,15,28) (12,17,19)(14,23,18,32,40,35)(31,34,33)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1 $ $360$ $6$ $( 1,27,13,10,29,36)( 3,30,24,25, 6, 4)( 7,39,38)( 8,21,20)( 9,28,15,37,22,11) (12,19,17)(14,35,40,32,18,23)(31,33,34)$
$ 12, 12, 6, 6, 2, 2 $ $2160$ $12$ $( 1,18,36,32,29,40,10,35,13,14,27,23)( 2,16)( 3, 9, 4,11, 6,22,25,37,24,15,30, 28)( 5,26)( 7,17,38,19,39,12)( 8,34,20,33,21,31)$
$ 12, 12, 6, 6, 2, 2 $ $2160$ $12$ $( 1,40,27,32,13,18,10,23,29,14,36,35)( 2,16)( 3,22,30,11,24, 9,25,28, 6,15, 4, 37)( 5,26)( 7,12,39,19,38,17)( 8,31,21,33,20,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ $480$ $3$ $( 1, 6,31)( 3, 7,35)( 5,12,15)( 8,34,40)( 9,16,19)(10,29,32)(13,27,30) (14,39,36)(17,22,26)(18,33,25)(20,38,24)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 2, 2, 2, 1 $ $1440$ $6$ $( 1,31, 6)( 2,23)( 3, 5, 7,12,35,15)( 4,37)( 8,40,34)( 9,10,16,29,19,32) (13,20,27,38,30,24)(14,36,39)(17,26,22)(18,25,33)(21,28)$

Group invariants

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

        1a 2a 3a 6a  2b 4a 3b 3c 6b 6c 6d 6e 3d 6f 12a 12b 4b  9a  9b 5a
     2P 1a 1a 3a 3a  1a 2b 3c 3b 3a 3a 3c 3b 3d 3d  6d  6e 2a  9b  9a 5a
     3P 1a 2a 1a 2a  2b 4a 1a 1a 2b 2b 2b 2b 1a 2b  4a  4a 4b  3b  3c 5a
     5P 1a 2a 3a 6a  2b 4a 3c 3b 6c 6b 6e 6d 3d 6f 12b 12a 4b  9b  9a 1a
     7P 1a 2a 3a 6a  2b 4a 3b 3c 6b 6c 6d 6e 3d 6f 12a 12b 4b  9a  9b 5a
    11P 1a 2a 3a 6a  2b 4a 3c 3b 6c 6b 6e 6d 3d 6f 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  1 -1  1  -3  1  A /A  F -F  H /H  2  .   J  /J -1 -/J  -J  .
X.3      5  1 -1  1  -3  1 /A  A -F  F /H  H  2  .  /J   J -1  -J -/J  .
X.4      6  2  3 -1  -2  2 -3 -3  1  1  1  1  . -2  -1  -1  .   .   .  1
X.5     10 -2  1  1   2  2  B /B -1 -1 /A  A  1 -1 -/J  -J  .  /J   J  .
X.6     10 -2  1  1   2  2 /B  B -1 -1  A /A  1 -1  -J -/J  .   J  /J  .
X.7     15  3  .  .   7 -1 -3 -3 -2 -2  1  1  3  1  -1  -1  1   .   .  .
X.8     15 -1  3 -1  -1  3  6  6 -1 -1  2  2  .  2   .   . -1   .   .  .
X.9     20  4  5  1   4  .  2  2  1  1 -2 -2 -1  1   .   .  .  -1  -1  .
X.10    24  .  .  .   8  .  6  6  2  2  2  2  3 -1   .   .  .   .   . -1
X.11    30  2  3 -1 -10 -2  3  3 -1 -1 -1 -1  3 -1   1   1  .   .   .  .
X.12    30  2 -3 -1   6  2  C /C  F -F  H /H  .  .  -J -/J  .   .   .  .
X.13    30  2 -3 -1   6  2 /C  C -F  F /H  H  .  . -/J  -J  .   .   .  .
X.14    40  . -2  .  -8  .  D /D  G /G  G /G  1  1   .   .  .  /J   J  .
X.15    40  . -2  .  -8  . /D  D /G  G /G  G  1  1   .   .  .   J  /J  .
X.16    45 -3  .  .  -3  1  E /E  .  .  I /I  .  .   J  /J  1   .   .  .
X.17    45 -3  .  .  -3  1 /E  E  .  . /I  I  .  .  /J   J  1   .   .  .
X.18    60  4 -3  1  -4  .  6  6 -1 -1  2  2 -3 -1   .   .  .   .   .  .
X.19    64  .  4  .   .  . -8 -8  .  .  .  . -2  .   .   .  .   1   1 -1
X.20    81 -3  .  .   9 -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)+E(3)^2
  = -Sqrt(-3) = -i3
G = -2*E(3)
  = 1-Sqrt(-3) = 1-i3
H = 2*E(3)+E(3)^2
  = (-3+Sqrt(-3))/2 = -1+b3
I = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
J = E(3)
  = (-1+Sqrt(-3))/2 = b3