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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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Show commands: Magma

magma: G := TransitiveGroup(27, 993);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $993$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSp(4,3)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $25920=2^{6} \cdot 3^{4} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  25920.a
magma: IdentifyGroup(G);
 
Character table:    
      2  6  5  3  .  3  3   .   .   6  4  3  3   2   2  2  2  2  1  2  1
      3  4  1  .  .  4  4   2   2   2  1  2  2   1   1  3  2  2  3  1  2
      5  1  .  .  1  .  .   .   .   .  .  .  .   .   .  .  .  .  .  .  .

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

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

magma: CharacterTable(G);