Properties

Label 40T14344
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 $t$:  $14344$
Group:  $\PSp(4,3)$
Parity:  $1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$\card{\Aut(F/K)}$:  $1$
Generators:  (1,3,8,16,7)(2,6,13,19,17)(4,9,5,10,20)(11,23,36,26,35)(12,21,31,25,18)(14,27,29,22,34)(15,28,30,38,32)(24,37,40,39,33), (1,2,5,10,19)(3,4,7,15,28)(6,12,21,13,25)(8,16,29,38,14)(9,18,31,20,27)(11,22,33,37,34)(17,30,23,35,32)(24,26,39,40,36)

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

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  3   6  1  1  .  3  3  4  3  3   2   2  2  2  2  2   .   .
      3  4  1  .   2  3  2  .  4  4  1  2  2   1   1  3  1  2  2   2   2
      5  1  .  .   .  .  .  1  .  .  .  .  .   .   .  .  .  .  .   .   .

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

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