Properties

Label 36T6815
Order \(6048\)
n \(36\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSU(3,3)$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $6815$
Group :  $\PSU(3,3)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (4,5,6)(7,8,9)(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,27,29)(26,28,30)(31,35,34)(32,36,33), (5,6)(8,9)(10,11)(13,14)(17,18)(19,20)(23,24)(27,29)(28,30)(31,33)(32,34)(35,36), (3,5,4,6)(7,12)(8,10,11,9)(13,14)(16,21)(17,19,20,18)(23,24)(25,35,26,36)(27,32,33,30)(28,31,34,29), (1,2,22,15)(3,17,26,11,4,20,25,8)(5,18,36,10,6,19,35,9)(7,24,12,23)(13,16,14,21)(27,31,32,28,33,29,30,34)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Degree 9: None

Degree 12: None

Degree 18: 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $63$ $2$ $( 1,33)( 2,32)( 3,36)( 5,18)( 6, 8)( 9,17)(11,25)(12,23)(13,21)(15,30)(19,26) (22,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $56$ $3$ $( 1, 9,15)( 2,19,23)( 3,25, 8)( 4,34,16)( 5,27,21)( 6,36,11)( 7,31,35) (10,14,24)(12,32,26)(13,18,22)(17,30,33)(20,29,28)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2 $ $63$ $4$ $( 1, 8,33, 6)( 2, 5,32,18)( 3,17,36, 9)( 4,35)( 7,34)(10,20)(11,15,25,30) (12,13,23,21)(14,29)(16,31)(19,27,26,22)(24,28)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2 $ $63$ $4$ $( 1, 6,33, 8)( 2,18,32, 5)( 3, 9,36,17)( 4,35)( 7,34)(10,20)(11,30,25,15) (12,21,23,13)(14,29)(16,31)(19,22,26,27)(24,28)$
$ 6, 6, 6, 6, 3, 3, 3, 3 $ $504$ $6$ $( 1,30, 9,33,15,17)( 2,12,19,32,23,26)( 3, 6,25,36, 8,11)( 4,16,34) ( 5,13,27,18,21,22)( 7,35,31)(10,24,14)(20,28,29)$
$ 12, 12, 6, 6 $ $504$ $12$ $( 1,36,30, 8, 9,11,33, 3,15, 6,17,25)( 2,22,12, 5,19,13,32,27,23,18,26,21) ( 4, 7,16,35,34,31)(10,29,24,20,14,28)$
$ 12, 12, 6, 6 $ $504$ $12$ $( 1, 3,30, 6, 9,25,33,36,15, 8,17,11)( 2,27,12,18,19,21,32,22,23, 5,26,13) ( 4, 7,16,35,34,31)(10,29,24,20,14,28)$
$ 8, 8, 8, 4, 4, 4 $ $756$ $8$ $( 1, 7,36, 5)( 2,14,16,17,12, 3,22,28)( 4,33,32, 8,13,29,19,10)( 6,27,15,31) ( 9,35,11,21)(18,20,23,24,34,30,26,25)$
$ 8, 8, 8, 4, 4, 4 $ $756$ $8$ $( 1, 5,36, 7)( 2,28,22, 3,12,17,16,14)( 4,10,19,29,13, 8,32,33)( 6,31,15,27) ( 9,21,11,35)(18,25,26,30,34,24,23,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $672$ $3$ $( 1,29,18)( 2, 7,26)( 4,21,22)( 5,36,32)( 6,13,10)( 8,28, 9)(11,24,35) (12,19,33)(14,34,23)(15,20,31)(16,27,17)$
$ 7, 7, 7, 7, 7, 1 $ $864$ $7$ $( 1,33, 3,14, 8, 7,17)( 2,10,25,27,34,13,36)( 4,30, 9,23,29,22, 5) ( 6,18,28,20,35,31,11)(12,32,16,24,15,21,26)$
$ 7, 7, 7, 7, 7, 1 $ $864$ $7$ $( 1,17, 7, 8,14, 3,33)( 2,36,13,34,27,25,10)( 4, 5,22,29,23, 9,30) ( 6,11,31,35,20,28,18)(12,26,21,15,24,16,32)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $378$ $4$ $( 1,12,25, 4)( 2,13,28,33)( 3,32)( 5,30,20, 7)( 6,22,31,10)( 8,16)( 9,26) (11,14,21,19)(15,34)(17,24,29,36)$

Group invariants

Order:  $6048=2^{5} \cdot 3^{3} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:   
      2  5  5  2  5  5  2   2   2  3  3  .  .  .  4
      3  3  1  3  1  1  1   1   1  .  .  .  .  2  .
      7  1  .  .  .  .  .   .   .  .  .  1  1  .  .

        1a 2a 3a 4a 4b 6a 12a 12b 8a 8b 7a 7b 3b 4c
     2P 1a 1a 3a 2a 2a 3a  6a  6a 4b 4a 7a 7b 3b 2a
     3P 1a 2a 1a 4b 4a 2a  4a  4b 8b 8a 7b 7a 1a 4c
     5P 1a 2a 3a 4a 4b 6a 12a 12b 8a 8b 7b 7a 3b 4c
     7P 1a 2a 3a 4b 4a 6a 12b 12a 8b 8a 1a 1a 3b 4c
    11P 1a 2a 3a 4b 4a 6a 12b 12a 8b 8a 7a 7b 3b 4c

X.1      1  1  1  1  1  1   1   1  1  1  1  1  1  1
X.2      6 -2 -3 -2 -2  1   1   1  .  . -1 -1  .  2
X.3      7 -1 -2  3  3  2   .   . -1 -1  .  .  1 -1
X.4      7  3 -2  A /A  .   D  /D  E -E  .  .  1  1
X.5      7  3 -2 /A  A  .  /D   D -E  E  .  .  1  1
X.6     14 -2  5  2  2  1  -1  -1  .  .  .  . -1  2
X.7     21  5  3  1  1 -1   1   1 -1 -1  .  .  .  1
X.8     21  1  3  B /B  1   E  -E -E  E  .  .  . -1
X.9     21  1  3 /B  B  1  -E   E  E -E  .  .  . -1
X.10    27  3  .  3  3  .   .   .  1  1 -1 -1  . -1
X.11    28 -4  1  C -C -1  -E   E  .  .  .  .  1  .
X.12    28 -4  1 -C  C -1   E  -E  .  .  .  .  1  .
X.13    32  . -4  .  .  .   .   .  .  .  F /F -1  .
X.14    32  . -4  .  .  .   .   .  .  . /F  F -1  .

A = -1-2*E(4)
  = -1-2*Sqrt(-1) = -1-2i
B = -3-2*E(4)
  = -3-2*Sqrt(-1) = -3-2i
C = -4*E(4)
  = -4*Sqrt(-1) = -4i
D = -1-E(4)
  = -1-Sqrt(-1) = -1-i
E = -E(4)
  = -Sqrt(-1) = -i
F = -E(7)-E(7)^2-E(7)^4
  = (1-Sqrt(-7))/2 = -b7