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

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  2  1  .
     5  1  .  .  .  .  .  .  .  1
     7  1  1  1  .  .  .  .  .  .

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

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