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Properties

Label 21T4
Degree $21$
Order $42$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_7$

Related objects

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Group action invariants

Degree $n$:  $21$
Transitive number $t$:  $4$
Group:  $F_7$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $3$
Generators:  (1,9,5,18,10,14)(2,8,6,16,12,13)(3,7,4,17,11,15)(19,21,20), (1,4,8,12,15,18,19)(2,5,7,11,14,16,20)(3,6,9,10,13,17,21)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: $F_7$

Low degree siblings

7T4, 14T4, 42T4

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

Group invariants

Order:  $42=2 \cdot 3 \cdot 7$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [42, 1]
Character table:    
     2  1  1  1   1   1  1  .
     3  1  1  1   1   1  1  .
     7  1  .  .   .   .  .  1

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

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3