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

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

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