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Properties

Label 14T22
Order \(588\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $22$
CHM label :  $[1/6_-.F_{42}(7)^{2}]2_{2}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10,12,14), (1,6,13,8)(2,9,12,5)(3,4,11,10)(7,14), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8), (1,11,9)(2,4,8)(3,5,13)(6,12,10)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
12:  $C_3 : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

28T75, 42T119, 42T125

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 $ $1$ $1$ $()$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $12$ $7$ $( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $12$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $12$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $12$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $49$ $2$ $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 3, 3, 3, 3, 1, 1 $ $98$ $3$ $( 3, 5, 9)( 4,10, 6)( 7,13,11)( 8,12,14)$
$ 6, 6, 1, 1 $ $98$ $6$ $( 3,11, 9,13, 5, 7)( 4, 8, 6,14,10,12)$
$ 4, 4, 4, 2 $ $147$ $4$ $( 1, 6,13, 8)( 2, 9,12, 5)( 3, 4,11,10)( 7,14)$
$ 4, 4, 4, 2 $ $147$ $4$ $( 1,12,11, 8)( 2, 7, 4, 5)( 3,14, 9, 6)(10,13)$

Group invariants

Order:  $588=2^{2} \cdot 3 \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [588, 33]
Character table:    
      2  2  .  .  .  .  2  1  1  2  2
      3  1  .  .  .  .  1  1  1  .  .
      7  2  2  2  2  2  .  .  .  .  .

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

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  1  1  1  1 -1  1 -1  D -D
X.4      1  1  1  1  1 -1  1 -1 -D  D
X.5      2  2  2  2  2 -2 -1  1  .  .
X.6      2  2  2  2  2  2 -1 -1  .  .
X.7     12  5 -2 -2 -2  .  .  .  .  .
X.8     12 -2  A  C  B  .  .  .  .  .
X.9     12 -2  B  A  C  .  .  .  .  .
X.10    12 -2  C  B  A  .  .  .  .  .

A = E(7)^2-2*E(7)^3-2*E(7)^4+E(7)^5
B = E(7)-2*E(7)^2-2*E(7)^5+E(7)^6
C = -2*E(7)+E(7)^3+E(7)^4-2*E(7)^6
D = -E(4)
  = -Sqrt(-1) = -i