Properties

Label 12T47
Order \(72\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $PSU(3,2)$

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $47$
Group :  $PSU(3,2)$
CHM label :  $[(1/3.3^{3}):2]E(4)_{4}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,5,10)(3,12)(4,11,8,7)(6,9), (1,5)(2,10)(4,8)(7,11), (1,4,5,8)(2,7,10,11)(3,6)(9,12), (2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: None

Low degree siblings

9T14, 18T35 x 3, 24T82, 36T55

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 3,11)( 4, 8)( 5, 9)( 6,10)$
$ 3, 3, 3, 1, 1, 1 $ $8$ $3$ $( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 4, 4, 2, 2 $ $18$ $4$ $( 1, 2)( 3, 4,11, 8)( 5, 6, 9,10)( 7,12)$
$ 4, 4, 2, 2 $ $18$ $4$ $( 1, 3)( 2, 4, 6,12)( 5,11, 9, 7)( 8,10)$
$ 4, 4, 2, 2 $ $18$ $4$ $( 1, 4, 9, 8)( 2, 3, 6, 7)( 5,12)(10,11)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 41]
Character table:   
     2  3  3  .  2  2  2
     3  2  .  2  .  .  .

       1a 2a 3a 4a 4b 4c
    2P 1a 1a 3a 2a 2a 2a
    3P 1a 2a 1a 4a 4b 4c

X.1     1  1  1  1  1  1
X.2     1  1  1 -1 -1  1
X.3     1  1  1 -1  1 -1
X.4     1  1  1  1 -1 -1
X.5     2 -2  2  .  .  .
X.6     8  . -1  .  .  .