Properties

Label 40T14345
Order \(25920\)
n \(40\)
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)
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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $270$ $2$ $( 1,10)( 2, 7)( 3,35)( 5,31)( 6,36)( 8,34)( 9,23)(11,27)(12,39)(13,30)(14,32) (15,29)(16,19)(17,38)(18,25)(20,26)(21,33)(37,40)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $240$ $3$ $( 1,12,20)( 2,37,11)( 3,13,34)( 4,22,24)( 5,38,14)( 6,19,25)( 7,40,27) ( 8,35,30)( 9,29,21)(10,39,26)(15,33,23)(16,18,36)(17,32,31)$
$ 6, 6, 6, 6, 6, 6, 3, 1 $ $2160$ $6$ $( 1,26,12,10,20,39)( 2,27,37, 7,11,40)( 3, 8,13,35,34,30)( 4,24,22) ( 5,32,38,31,14,17)( 6,18,19,36,25,16)( 9,33,29,23,21,15)$
$ 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 $ $45$ $2$ $( 1,14)( 3,35)( 5,12)( 6,36)( 8,34)(10,32)(13,30)(16,19)(17,26)(18,25)(20,38) (31,39)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 3, 1 $ $720$ $6$ $( 1,38,12,14,20, 5)( 2,11,37)( 3, 8,13,35,34,30)( 4,24,22)( 6,18,19,36,25,16) ( 7,27,40)( 9,21,29)(10,17,39,32,26,31)(15,23,33)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 3, 1 $ $720$ $6$ $( 1, 5,20,14,12,38)( 2,37,11)( 3,30,34,35,13, 8)( 4,22,24)( 6,16,25,36,19,18) ( 7,40,27)( 9,29,21)(10,31,26,32,39,17)(15,33,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ $480$ $3$ $( 1,32,19)( 2,11,37)( 4,29,15)( 5,39,18)( 6,20,17)( 7,40,27)( 9,23,24) (10,16,14)(12,31,25)(21,33,22)(26,36,38)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 2, 2, 2, 1 $ $1440$ $6$ $( 1,16,32,14,19,10)( 2,37,11)( 3,35)( 4,15,29)( 5,25,39,12,18,31) ( 6,26,20,36,17,38)( 7,27,40)( 8,34)( 9,24,23)(13,30)(21,22,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $40$ $3$ $( 1, 6,31)( 3,34,13)( 4,23,21)( 5,16,26)( 7,40,27)( 8,30,35)( 9,22,15) (10,38,18)(12,19,17)(14,36,39)(20,25,32)(24,33,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $40$ $3$ $( 1,31, 6)( 3,13,34)( 4,21,23)( 5,26,16)( 7,27,40)( 8,35,30)( 9,15,22) (10,18,38)(12,17,19)(14,39,36)(20,32,25)(24,29,33)$
$ 9, 9, 9, 9, 3, 1 $ $2880$ $9$ $( 1,18, 5, 6,10,16,31,38,26)( 2,11,37)( 3,30, 7,34,35,40,13, 8,27) ( 4,20,12,23,25,19,21,32,17)( 9,14,29,22,36,24,15,39,33)$
$ 9, 9, 9, 9, 3, 1 $ $2880$ $9$ $( 1, 5,10,31,26,18, 6,16,38)( 2,37,11)( 3, 7,35,13,27,30,34,40, 8) ( 4,12,25,21,17,20,23,19,32)( 9,29,36,15,33,14,22,24,39)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2 $ $540$ $4$ $( 1,15, 8,37)( 2, 5)( 3,23)( 4,34, 7,14)( 6,32)( 9,21,28,29)(10,17) (11,13,22,20)(12,27)(16,26)(18,25,31,39)(19,36)(24,35)(30,33,38,40)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1 $ $360$ $6$ $( 1,21,13, 8,29,20)( 3, 6,24)( 4,39,30, 7,25,38)( 9,11,15,28,22,37)(10,27,36) (12,19,17)(14,31,40,34,18,33)(23,32,35)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1 $ $360$ $6$ $( 1,20,29, 8,13,21)( 3,24, 6)( 4,38,25, 7,30,39)( 9,37,22,28,15,11)(10,36,27) (12,17,19)(14,33,18,34,40,31)(23,35,32)$
$ 12, 12, 6, 6, 2, 2 $ $2160$ $12$ $( 1,11,21,15,13,28, 8,22,29,37,20, 9)( 2, 5)( 3,35, 6,23,24,32) ( 4,40,39,34,30,18, 7,33,25,14,38,31)(10,19,27,17,36,12)(16,26)$
$ 12, 12, 6, 6, 2, 2 $ $2160$ $12$ $( 1,28,20,15,29,11, 8, 9,13,37,21,22)( 2, 5)( 3,32,24,23, 6,35) ( 4,18,38,34,25,40, 7,31,30,14,39,33)(10,12,36,17,27,19)(16,26)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 1 $ $3240$ $4$ $( 1,32,16, 8)( 2,28,40,15)( 3,18,31,12)( 4,33,29,23)( 5,35,25,39)( 7,21, 9,37) (10,19,34,14)(11,27,24,22)(13,26,30,17)(20,38)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $5184$ $5$ $( 1,29,12,32,35)( 2,20,36,40,33)( 3,10,18, 4,16)( 5,14,27,26, 9) ( 6,23,38,13,30)( 7,34,15, 8,11)(17,37,19,25,24)(21,39,28,31,22)$

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

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

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