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Properties

Label 21T67
Order \(20160\)
n \(21\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSL(3,4)$

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $67$
Group :  $\PSL(3,4)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,12,16,19,21,6)(2,8,13,17,20,5,11)(3,9,14,18,4,10,15), (2,14,18,20,8)(3,7,12,13,19)(4,21,17,15,10)(5,11,16,6,9)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 7: None

Low degree siblings

21T67

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 1 $ $4032$ $5$ $( 1,13,14, 4, 9)( 2,15,12,17,11)( 3,18,21,20,16)( 5, 8,19,10, 6)$
$ 5, 5, 5, 5, 1 $ $4032$ $5$ $( 1, 4,13, 9,14)( 2,17,15,11,12)( 3,20,18,16,21)( 5,10, 8, 6,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $315$ $2$ $( 1,14)( 3, 6)( 4,13)( 8,18)(10,15)(11,19)(12,21)(16,17)$
$ 4, 4, 4, 4, 2, 2, 1 $ $1260$ $4$ $( 1,17,14,16)( 2, 5)( 3,10, 6,15)( 4,12,13,21)( 7,20)( 8,11,18,19)$
$ 7, 7, 7 $ $2880$ $7$ $( 1,21, 8,19,13, 4, 7)( 2,12, 9, 6,16,18,20)( 3,14,11,15, 5,10,17)$
$ 7, 7, 7 $ $2880$ $7$ $( 1, 7, 4,13,19, 8,21)( 2,20,18,16, 6, 9,12)( 3,17,10, 5,15,11,14)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $2240$ $3$ $( 1,21, 5)( 2, 4, 8)( 3,16,15)( 6,11,12)( 7,14,10)( 9,13,20)$
$ 4, 4, 4, 4, 2, 2, 1 $ $1260$ $4$ $( 1,13)( 2,11, 3,12)( 4, 9)( 5,21, 8,15)( 6,18,20, 7)(10,16,17,19)$
$ 4, 4, 4, 4, 2, 2, 1 $ $1260$ $4$ $( 1,15, 4,11)( 2, 5)( 3,16,18,21)( 6,17, 8,12)( 7,20)(10,13,19,14)$

Group invariants

Order:  $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:    
      2  6  6  4  4  4  .  .  .  .  .
      3  2  .  .  .  .  .  .  .  .  2
      5  1  .  .  .  .  1  1  .  .  .
      7  1  .  .  .  .  .  .  1  1  .

        1a 2a 4a 4b 4c 5a 5b 7a 7b 3a
     2P 1a 1a 2a 2a 2a 5b 5a 7a 7b 3a
     3P 1a 2a 4a 4b 4c 5b 5a 7b 7a 1a
     5P 1a 2a 4a 4b 4c 1a 1a 7b 7a 3a
     7P 1a 2a 4a 4b 4c 5b 5a 1a 1a 3a

X.1      1  1  1  1  1  1  1  1  1  1
X.2     20  4  .  .  .  .  . -1 -1  2
X.3     35  3  3 -1 -1  .  .  .  . -1
X.4     35  3 -1 -1  3  .  .  .  . -1
X.5     35  3 -1  3 -1  .  .  .  . -1
X.6     45 -3  1  1  1  .  .  B /B  .
X.7     45 -3  1  1  1  .  . /B  B  .
X.8     63 -1 -1 -1 -1  A *A  .  .  .
X.9     63 -1 -1 -1 -1 *A  A  .  .  .
X.10    64  .  .  .  . -1 -1  1  1  1

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
B = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7