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

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

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

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