Properties

Label 14T5
Order \(42\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_7:C_3) \times C_2$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $5$
Group :  $(C_7:C_3) \times C_2$
CHM label :  $F_{21}(7)[x]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
21:  $C_7:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $C_7:C_3$

Low degree siblings

42T2

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 1, 1 $ $7$ $3$ $( 2,10,12)( 3, 5, 9)( 4,14, 6)( 7,13,11)$
$ 3, 3, 3, 3, 1, 1 $ $7$ $3$ $( 2,12,10)( 3, 9, 5)( 4, 6,14)( 7,11,13)$
$ 14 $ $3$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
$ 6, 6, 2 $ $7$ $6$ $( 1, 2,11, 8, 9, 4)( 3, 6, 5,10,13,12)( 7,14)$
$ 6, 6, 2 $ $7$ $6$ $( 1, 2,13, 8, 9, 6)( 3,10)( 4, 7,12,11,14, 5)$
$ 7, 7 $ $3$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 14 $ $3$ $14$ $( 1, 4, 7,10,13, 2, 5, 8,11,14, 3, 6, 9,12)$
$ 7, 7 $ $3$ $7$ $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$
$ 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$

Group invariants

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

        1a 3a 3b 14a  6a  6b  7a 14b  7b 2a
     2P 1a 3b 3a  7a  3b  3a  7a  7b  7b 1a
     3P 1a 1a 1a 14b  2a  2a  7b 14a  7a 2a
     5P 1a 3b 3a 14b  6b  6a  7b 14a  7a 2a
     7P 1a 3a 3b  2a  6a  6b  1a  2a  1a 2a
    11P 1a 3b 3a 14a  6b  6a  7a 14b  7b 2a
    13P 1a 3a 3b 14b  6a  6b  7b 14a  7a 2a

X.1      1  1  1   1   1   1   1   1   1  1
X.2      1  1  1  -1  -1  -1   1  -1   1 -1
X.3      1  A /A  -1  -A -/A   1  -1   1 -1
X.4      1 /A  A  -1 -/A  -A   1  -1   1 -1
X.5      1  A /A   1   A  /A   1   1   1  1
X.6      1 /A  A   1  /A   A   1   1   1  1
X.7      3  .  .   B   .   .  -B  /B -/B -3
X.8      3  .  .  /B   .   . -/B   B  -B -3
X.9      3  .  . -/B   .   . -/B  -B  -B  3
X.10     3  .  .  -B   .   .  -B -/B -/B  3

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