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Properties

Label 28T323
Order \(6048\)
n \(28\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSU(3,3)$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $323$
Group :  $\PSU(3,3)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,7,17,19,25,8,22,14,12,9,4)(2,20,27)(3,11,26,28,21,23,18,5,16,24,15,13), (1,19,5,20,6,21,8)(2,17,10,24,16,13,7)(3,22,11,4,23,27,26)(9,15,18,28,12,14,25)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: None

Degree 14: None

Low degree siblings

36T6815

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $63$ $2$ $( 1,24)( 2,15)( 3, 9)( 4, 7)( 5,10)( 6, 8)(11,27)(12,26)(13,28)(14,25)(20,21) (22,23)$
$ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1 $ $63$ $4$ $( 1,15,24, 2)( 3,12, 9,26)( 4,14, 7,25)( 5,13,10,28)( 6,11, 8,27)(20,23,21,22)$
$ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1 $ $63$ $4$ $( 1, 2,24,15)( 3,26, 9,12)( 4,25, 7,14)( 5,28,10,13)( 6,27, 8,11)(20,22,21,23)$
$ 8, 8, 8, 2, 1, 1 $ $756$ $8$ $( 1, 8,15,27,24, 6, 2,11)( 3, 7,12,25, 9, 4,26,14)( 5,22,13,20,10,23,28,21) (16,17)$
$ 8, 8, 8, 2, 1, 1 $ $756$ $8$ $( 1,11, 2, 6,24,27,15, 8)( 3,14,26, 4, 9,25,12, 7)( 5,21,28,23,10,20,13,22) (16,17)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $56$ $3$ $( 2,15,24)( 3,25,20)( 4,26,21)( 5,27,22)( 6,28,23)( 7,11,16)( 8,12,17) ( 9,13,18)(10,14,19)$
$ 6, 6, 6, 6, 3, 1 $ $504$ $6$ $( 2,24,15)( 3,21,25, 4,20,26)( 5,23,27, 6,22,28)( 7,18,11, 9,16,13) ( 8,19,12,10,17,14)$
$ 12, 12, 3, 1 $ $504$ $12$ $( 2,15,24)( 3,28,21, 5,25,23, 4,27,20, 6,26,22)( 7,14,18, 8,11,19, 9,12,16,10, 13,17)$
$ 12, 12, 3, 1 $ $504$ $12$ $( 2,15,24)( 3,27,21, 6,25,22, 4,28,20, 5,26,23)( 7,12,18,10,11,17, 9,14,16, 8, 13,19)$
$ 4, 4, 4, 4, 4, 4, 2, 2 $ $378$ $4$ $( 1,11,22,10)( 2, 6,21,13)( 3, 9)( 4,14, 7,25)( 5,24,27,23)( 8,20,28,15) (12,26)(16,18,17,19)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $672$ $3$ $( 1,13,28)( 2,12,25)( 3,19,16)( 4,22,20)( 5, 6, 9)( 8,17,21)(10,23,18) (11,27,15)(14,26,24)$
$ 7, 7, 7, 7 $ $864$ $7$ $( 1, 8,15,21,19,18,22)( 2,28, 7, 6, 9,13,24)( 3,26,23,10,20,14, 4) ( 5,12,16,11,27,17,25)$
$ 7, 7, 7, 7 $ $864$ $7$ $( 1,22,18,19,21,15, 8)( 2,24,13, 9, 6, 7,28)( 3, 4,14,20,10,23,26) ( 5,25,17,27,11,16,12)$

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  .  2  5  5  5  2   2   2  .  .  3  3  4
      3  3  2  3  1  1  1  1   1   1  .  .  .  .  .
      7  1  .  .  .  .  .  .   .   .  1  1  .  .  .

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

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

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