Properties

Label 21T33
Order \(2520\)
n \(21\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $A_7$

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $33$
Group :  $A_7$
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), (4,6,5)(9,11,10)(13,15,14)(16,18,17)(19,20,21)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 7: None

Low degree siblings

7T6, 15T47 x 2, 35T28, 42T294, 42T299

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 $ $1$ $1$ $()$
$ 7, 7, 7 $ $360$ $7$ $( 1,17,13, 6, 8,19,15)( 2, 3,16,20,11,10,14)( 4,18, 9,21, 7, 5,12)$
$ 7, 7, 7 $ $360$ $7$ $( 1,15,19, 8, 6,13,17)( 2,14,10,11,20,16, 3)( 4,12, 5, 7,21, 9,18)$
$ 5, 5, 5, 5, 1 $ $504$ $5$ $( 1,12, 4, 8,13)( 2, 3,16, 9, 7)( 5,17,19,10,14)( 6,18,20,11,15)$
$ 3, 3, 3, 3, 3, 3, 3 $ $280$ $3$ $( 1,17,20)( 2,12,13)( 3,16, 4)( 5,18, 9)( 6, 8,19)( 7,14,15)(10,21,11)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $70$ $3$ $( 1, 6, 5)( 7,15,14)( 8,18,17)( 9,20,19)(10,11,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $105$ $2$ $( 1, 7)( 3,13)( 4,12)( 5,14)( 6,15)( 8, 9)(17,19)(18,20)$
$ 6, 6, 3, 2, 2, 1, 1 $ $210$ $6$ $( 1,15, 5, 7, 6,14)( 3,13)( 4,12)( 8,20,17, 9,18,19)(10,11,21)$
$ 4, 4, 4, 4, 2, 2, 1 $ $630$ $4$ $( 1, 9, 7, 8)( 2,16)( 3, 4,13,12)( 5,20,14,18)( 6,19,15,17)(10,11)$

Group invariants

Order:  $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:   
     2  3  3  2  2  .  .  .  2  .
     3  2  1  2  1  2  .  .  .  .
     5  1  .  .  .  .  .  .  .  1
     7  1  .  .  .  .  1  1  .  .

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

X.1     1  1  1  1  1  1  1  1  1
X.2     6  2  3 -1  . -1 -1  .  1
X.3    10 -2  1  1  1  A /A  .  .
X.4    10 -2  1  1  1 /A  A  .  .
X.5    14  2  2  2 -1  .  .  . -1
X.6    14  2 -1 -1  2  .  .  . -1
X.7    15 -1  3 -1  .  1  1 -1  .
X.8    21  1 -3  1  .  .  . -1  1
X.9    35 -1 -1 -1 -1  .  .  1  .

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