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Properties

Label 21T33
Degree $21$
Order $2520$
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)
$|\Aut(F/K)|$:  $1$
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)

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

Group invariants

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

       1a 7a 7b 2a 4a 3a 6a 3b 5a
    2P 1a 7a 7b 1a 2a 3a 3a 3b 5a
    3P 1a 7b 7a 2a 4a 1a 2a 1a 5a
    5P 1a 7b 7a 2a 4a 3a 6a 3b 1a
    7P 1a 1a 1a 2a 4a 3a 6a 3b 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  .  2  2 -1 -1
X.6    14  .  .  2  . -1 -1  2 -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