Properties

Label 12T84
Order \(144\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $PSU(3,2):C_2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $84$
Group :  $PSU(3,2):C_2$
CHM label :  $[(1/3.3^{3}):2]D(4)_{4}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5)(2,10)(4,8)(7,11), (3,4)(6,10)(7,12)(8,11), (1,4,2,11,5,8,10,7)(3,9,12,6), (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:  $D_{4}$
16:  $QD_{16}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: None

Low degree siblings

9T19, 18T68, 18T71, 18T73, 24T278, 24T279, 24T280, 36T171, 36T172, 36T175

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 $ $12$ $2$ $( 3, 4)( 6,10)( 7,12)( 8,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 3,11)( 4, 8)( 5, 9)( 6,10)$
$ 3, 3, 3, 1, 1, 1 $ $8$ $3$ $( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 6, 3, 2, 1 $ $24$ $6$ $( 2, 6,10)( 3,12,11, 8, 7, 4)( 5, 9)$
$ 4, 4, 2, 2 $ $18$ $4$ $( 1, 2)( 3, 4,11, 8)( 5, 6, 9,10)( 7,12)$
$ 4, 4, 2, 2 $ $36$ $4$ $( 1, 3)( 2, 4, 6,12)( 5,11, 9, 7)( 8,10)$
$ 8, 4 $ $18$ $8$ $( 1, 3, 6, 8, 9, 7, 2, 4)( 5,11,10,12)$
$ 8, 4 $ $18$ $8$ $( 1, 3,10, 8)( 2, 4, 9,11, 6,12, 5, 7)$

Group invariants

Order:  $144=2^{4} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [144, 182]
Character table:   
     2  4  2  4  1  1  3  2  3  3
     3  2  1  .  2  1  .  .  .  .

       1a 2a 2b 3a 6a 4a 4b 8a 8b
    2P 1a 1a 1a 3a 3a 2b 2b 4a 4a
    3P 1a 2a 2b 1a 2a 4a 4b 8a 8b
    5P 1a 2a 2b 3a 6a 4a 4b 8b 8a
    7P 1a 2a 2b 3a 6a 4a 4b 8b 8a

X.1     1  1  1  1  1  1  1  1  1
X.2     1 -1  1  1 -1  1 -1  1  1
X.3     1 -1  1  1 -1  1  1 -1 -1
X.4     1  1  1  1  1  1 -1 -1 -1
X.5     2  .  2  2  . -2  .  .  .
X.6     2  . -2  2  .  .  .  A -A
X.7     2  . -2  2  .  .  . -A  A
X.8     8 -2  . -1  1  .  .  .  .
X.9     8  2  . -1 -1  .  .  .  .

A = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2