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Properties

Label 21T2
Order \(21\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7:C_3$

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $2$
Group :  $C_7:C_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,10)(2,6,11)(3,4,12)(7,16,13)(8,17,14)(9,18,15)(19,20,21), (1,4,7,11,14,18,19)(2,5,8,12,15,16,20)(3,6,9,10,13,17,21)
$|\Aut(F/K)|$:  $21$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: $C_7:C_3$

Low degree siblings

7T3

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1, 2, 3)( 4, 8,13)( 5, 9,14)( 6, 7,15)(10,19,16)(11,20,17)(12,21,18)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1, 3, 2)( 4,13, 8)( 5,14, 9)( 6,15, 7)(10,16,19)(11,17,20)(12,18,21)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 4, 7,11,14,18,19)( 2, 5, 8,12,15,16,20)( 3, 6, 9,10,13,17,21)$
$ 7, 7, 7 $ $3$ $7$ $( 1,11,19, 7,18, 4,14)( 2,12,20, 8,16, 5,15)( 3,10,21, 9,17, 6,13)$

Group invariants

Order:  $21=3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [21, 1]
Character table:    
     3  1  1  1  .  .
     7  1  .  .  1  1

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

X.1     1  1  1  1  1
X.2     1  A /A  1  1
X.3     1 /A  A  1  1
X.4     3  .  .  B /B
X.5     3  .  . /B  B

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(7)+E(7)^2+E(7)^4
  = (-1+Sqrt(-7))/2 = b7