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

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

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

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