Properties

Label 27T993
Order \(25920\)
n \(27\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSp(4,3)$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $993$
Group :  $\PSp(4,3)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,20,25)(3,17,15,22)(4,8)(5,18,24,7)(6,19,26,9)(10,11,27,21)(13,14)(16,23), (1,16,24,4,5,8,7,21,25)(2,9,17,14,22,19,11,26,6)(3,20,10,12,27,18,15,13,23)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 9: None

Low degree siblings

36T12781, 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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 5, 1, 1 $ $5184$ $5$ $( 1,11,10,16,14)( 2,13,24, 9,19)( 3, 6,17,20,27)( 4,22,15,23,21) ( 5,12,25, 7,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $45$ $2$ $( 2, 3)( 4, 7)( 5,15)( 6,11)( 8,12)( 9,14)(16,19)(17,24)(18,23)(20,27)(21,26) (22,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1,13,10)( 2,11,14)( 3, 6, 9)( 4,20,18)( 5,16,21)( 7,27,23)( 8,24,25) (12,17,22)(15,19,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1,10,13)( 2,14,11)( 3, 9, 6)( 4,18,20)( 5,21,16)( 7,23,27)( 8,25,24) (12,22,17)(15,26,19)$
$ 4, 4, 4, 4, 4, 4, 1, 1, 1 $ $540$ $4$ $( 2,22, 3,25)( 4,19, 7,16)( 5,18,15,23)( 6, 8,11,12)( 9,24,14,17)(20,26,27,21)$
$ 6, 6, 6, 6, 3 $ $360$ $6$ $( 1,10,13)( 2, 9,11, 3,14, 6)( 4,23,20, 7,18,27)( 5,26,16,15,21,19) ( 8,22,24,12,25,17)$
$ 6, 6, 6, 6, 3 $ $360$ $6$ $( 1,13,10)( 2, 6,14, 3,11, 9)( 4,27,18, 7,20,23)( 5,19,21,15,16,26) ( 8,17,25,12,24,22)$
$ 12, 12, 3 $ $2160$ $12$ $( 1,13,10)( 2, 8, 9,22,11,24, 3,12,14,25, 6,17)( 4,21,23,19,20, 5, 7,26,18,16, 27,15)$
$ 12, 12, 3 $ $2160$ $12$ $( 1,10,13)( 2,24, 6,22,14, 8, 3,17,11,25, 9,12)( 4, 5,27,19,18,21, 7,15,20,16, 23,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $480$ $3$ $( 1,10,13)( 2,24,21)( 3,17,26)( 4,18,20)( 5,11,25)( 6,22,15)( 7,23,27) ( 8,16,14)( 9,12,19)$
$ 6, 6, 6, 6, 3 $ $1440$ $6$ $( 1,13,10)( 2,26,24, 3,21,17)( 4,27,18, 7,20,23)( 5,22,11,15,25, 6) ( 8, 9,16,12,14,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $270$ $2$ $( 1, 5)( 3, 8)( 6, 7)(10,16)(11,17)(12,18)(13,21)(14,20)(15,19)(25,27)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $3240$ $4$ $( 1,14, 5,20)( 2, 4)( 3,12, 8,18)( 6,17, 7,11)( 9,22)(10,27,16,25) (13,19,21,15)(23,24)$
$ 9, 9, 9 $ $2880$ $9$ $( 1,19,15, 3, 6, 5, 2,17,10)( 4, 8,20,24,13,16,26,12, 9)( 7,18,25,27,11,21,23, 22,14)$
$ 9, 9, 9 $ $2880$ $9$ $( 1,15, 6, 2,10,19, 3, 5,17)( 4,20,13,26, 9, 8,24,16,12)( 7,25,11,23,14,18,27, 21,22)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $240$ $3$ $( 2,16,25)( 3,19,22)( 5,24,14)( 6,26,12)( 8,11,21)( 9,15,17)$
$ 6, 6, 6, 2, 2, 2, 1, 1, 1 $ $720$ $6$ $( 1,13)( 2, 6,16,26,25,12)( 3,11,19,21,22, 8)( 4,20)( 5, 9,24,15,14,17)( 7,27)$
$ 6, 6, 6, 2, 2, 2, 1, 1, 1 $ $720$ $6$ $( 1,13)( 2,12,25,26,16, 6)( 3, 8,22,21,19,11)( 4,20)( 5,17,14,15,24, 9)( 7,27)$
$ 6, 6, 3, 3, 2, 2, 2, 2, 1 $ $2160$ $6$ $( 1,27)( 2, 5,16,24,25,14)( 3,22,19)( 4,20)( 6, 9,26,15,12,17)( 7,13) ( 8,21,11)(10,23)$

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

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

A = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3
B = -2*E(3)
  = 1-Sqrt(-3) = 1-i3
C = -2*E(3)+E(3)^2
  = (1-3*Sqrt(-3))/2 = -1-3b3
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 = 2*E(3)+8*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)
  = (-3+3*Sqrt(-3))/2 = 3b3
J = E(3)
  = (-1+Sqrt(-3))/2 = b3