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

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

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