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Properties

Label 27T993
Degree $27$
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 $n$:  $27$
Transitive number $t$:  $993$
Group:  $\PSp(4,3)$
Parity:  $1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$\card{\Aut(F/K)}$:  $1$
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)

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

Group invariants

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

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