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Properties

Label 45T666
Order \(25920\)
n \(45\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSp(4,3)$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $666$
Group :  $\PSp(4,3)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,6,12,19)(2,5,10,18,11)(4,8,15,22,29)(7,14,20,28,40)(9,16,24,33,13)(17,25,34,36,31)(21,30,41,45,32)(23,35,42,37,39)(26,27,38,44,43), (1,2,4,7,13)(3,5,9,14,8)(6,11,15,21,29)(10,17,24,31,19)(12,18,26,37,43)(16,23,34,35,33)(20,27,38,44,40)(22,32,41,45,30)(25,36,42,28,39)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: None

Degree 5: None

Degree 9: None

Degree 15: None

Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

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, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 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 $ $45$ $2$ $( 1,33)( 2,22)( 3,40)( 5,28)( 6,29)( 7,36)( 8,35)(10,24)(11,12)(13,44)(14,45) (16,38)(20,34)(21,43)(23,42)(27,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $3$ $( 2,34, 3)( 4,39,19)( 6, 7,27)( 8,10,42)( 9,18,41)(11,44,14)(12,13,45) (15,32,25)(17,31,37)(20,40,22)(23,35,24)(29,36,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $3$ $( 2, 3,34)( 4,19,39)( 6,27, 7)( 8,42,10)( 9,41,18)(11,14,44)(12,45,13) (15,25,32)(17,37,31)(20,22,40)(23,24,35)(29,30,36)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1 $ $540$ $4$ $( 1,38,33,16)( 2,29,22, 6)( 3,30,40,27)( 4,15)( 5,43,28,21)( 7,34,36,20) ( 8,12,35,11)( 9,17)(10,13,24,44)(14,42,45,23)(18,31)(19,25)(32,39)(37,41)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 2, 2, 1 $ $360$ $6$ $( 1,33)( 2,40,34,22, 3,20)( 4,19,39)( 5,28)( 6,30, 7,29,27,36)( 8,23,10,35,42, 24)( 9,41,18)(11,45,44,12,14,13)(15,25,32)(16,38)(17,37,31)(21,43)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 2, 2, 1 $ $360$ $6$ $( 1,33)( 2,20, 3,22,34,40)( 4,39,19)( 5,28)( 6,36,27,29, 7,30)( 8,24,42,35,10, 23)( 9,18,41)(11,13,14,12,44,45)(15,32,25)(16,38)(17,31,37)(21,43)$
$ 12, 12, 6, 6, 4, 4, 1 $ $2160$ $12$ $( 1,16,33,38)( 2, 7,40,29,34,27,22,36, 3, 6,20,30)( 4,32,19,15,39,25) ( 5,21,28,43)( 8,44,23,12,10,14,35,13,42,11,24,45)( 9,31,41,17,18,37)$
$ 12, 12, 6, 6, 4, 4, 1 $ $2160$ $12$ $( 1,16,33,38)( 2,27,20,29, 3, 7,22,30,34, 6,40,36)( 4,25,39,15,19,32) ( 5,21,28,43)( 8,14,24,12,42,44,35,45,10,11,23,13)( 9,37,18,17,41,31)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $480$ $3$ $( 1,33,26)( 2,22,17)( 3,40,37)( 5,16,43)( 6,29,41)( 7,36, 9)( 8,42,10) (11,25,12)(13,44,15)(14,32,45)(18,27,30)(20,31,34)(21,38,28)(23,35,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $240$ $3$ $( 2,13, 7)( 3,12, 6)( 5,43,16)( 8,42,10)( 9,17,15)(11,29,40)(14,30,20) (18,31,32)(21,38,28)(22,44,36)(23,24,35)(25,41,37)(27,34,45)$
$ 6, 6, 6, 6, 6, 3, 3, 3, 2, 1, 1, 1, 1 $ $720$ $6$ $( 2,45,13,27, 7,34)( 3, 6,12)( 4,39)( 5,35,43,23,16,24)( 8,21,42,38,10,28) ( 9,31,17,32,15,18)(11,40,29)(14,44,30,36,20,22)(25,37,41)$
$ 6, 6, 6, 6, 6, 3, 3, 3, 2, 1, 1, 1, 1 $ $720$ $6$ $( 2,34, 7,27,13,45)( 3,12, 6)( 4,39)( 5,24,16,23,43,35)( 8,28,10,38,42,21) ( 9,18,15,32,17,31)(11,29,40)(14,22,20,36,30,44)(25,41,37)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $270$ $2$ $( 1,33)( 2,30)( 3,40)( 4,39)( 5,42)( 6,29)( 7,14)( 8,16)( 9,32)(10,43)(11,12) (13,20)(15,31)(17,18)(21,24)(22,27)(23,28)(34,44)(35,38)(36,45)$
$ 6, 6, 6, 6, 6, 6, 3, 2, 2, 1, 1 $ $2160$ $6$ $( 1,33)( 2,14,13,30, 7,20)( 3,29,12,40, 6,11)( 4,39)( 5, 8,43,42,16,10) ( 9,31,17,32,15,18)(21,23,38,24,28,35)(22,45,44,27,36,34)(25,37,41)$
$ 6, 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ $1440$ $6$ $( 1,33,26)( 2,30,17,27,22,18)( 3,40,37)( 4,39)( 5,35,43,23,16,24)( 6,29,41) ( 7,14, 9,45,36,32)( 8,28,10,38,42,21)(11,25,12)(13,20,15,34,44,31)$
$ 9, 9, 9, 9, 3, 3, 3 $ $2880$ $9$ $( 1,40, 5)( 2, 7,42,34,45, 9, 3,19,11)( 4,32,41,27,35,14,12,29,10) ( 6,33,18,13,16,44,39,28, 8)(15,36,26,23,25,21,30,24,43)(17,38,37)(20,22,31)$
$ 9, 9, 9, 9, 3, 3, 3 $ $2880$ $9$ $( 1, 5,40)( 2,42,45, 3,11, 7,34, 9,19)( 4,41,35,12,10,32,27,14,29) ( 6,18,16,39, 8,33,13,44,28)(15,26,25,30,43,36,23,21,24)(17,37,38)(20,31,22)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 1 $ $3240$ $4$ $( 2, 4)( 3,39, 6,27)( 5,33,10,22)( 7,13)( 8,14,20,16)( 9,30,15,25) (11,36,18,44)(12,34,19,45)(17,38,26,35)(21,23,32,31)(24,41,40,43)(28,42,29,37)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $5184$ $5$ $( 1,37, 3,45,14)( 2,25,38,35,30)( 4,26,32,40,33)( 5,27,24, 6,43) ( 7,19,36, 9,39)( 8,34,21,12,10)(11,22,15,31,42)(13,41,23,28,20) (16,29,44,17,18)$

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

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

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