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Properties

Label 14T39
Order \(2184\)
n \(14\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PGL(2,13)$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $39$
Group :  $\PGL(2,13)$
CHM label :  $L(14):2=PGL(2,13)$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,6)(3,4)(7,11)(9,10)(13,14), (1,2,4,8,3,6,12,11,9,5,10,7), (1,2,3,4,5,6,7,8,9,10,11,12,14)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: None

Low degree siblings

28T201, 42T284

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2 $ $78$ $2$ $( 1, 5)( 2,14)( 3,10)( 4, 8)( 6,12)( 7, 9)(11,13)$
$ 7, 7 $ $156$ $7$ $( 1, 6, 3, 8, 9,13,14)( 2, 5,12,10, 4, 7,11)$
$ 7, 7 $ $156$ $7$ $( 1, 9, 6,13, 3,14, 8)( 2, 4, 5, 7,12,11,10)$
$ 7, 7 $ $156$ $7$ $( 1, 3, 9,14, 6, 8,13)( 2,12, 4,11, 5,10, 7)$
$ 14 $ $156$ $14$ $( 1, 7, 6,11, 3, 2, 8, 5, 9,12,13,10,14, 4)$
$ 14 $ $156$ $14$ $( 1,11, 8,12,14, 7, 3, 5,13, 4, 6, 2, 9,10)$
$ 14 $ $156$ $14$ $( 1,12, 3, 4, 9,11,14, 5, 6,10, 8, 7,13, 2)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $91$ $2$ $( 1, 9)( 2, 8)( 3, 7)( 4, 6)(10,14)(11,12)$
$ 4, 4, 4, 1, 1 $ $182$ $4$ $( 1,12, 9,11)( 2, 7, 8, 3)( 4,10, 6,14)$
$ 3, 3, 3, 3, 1, 1 $ $182$ $3$ $( 1, 6, 8)( 2, 9, 4)( 3,12,14)( 7,11,10)$
$ 6, 6, 1, 1 $ $182$ $6$ $( 1, 2, 6, 9, 8, 4)( 3,10,12, 7,14,11)$
$ 12, 1, 1 $ $182$ $12$ $( 1,10, 2,12, 6, 7, 9,14, 8,11, 4, 3)$
$ 12, 1, 1 $ $182$ $12$ $( 1,14, 2,11, 6, 3, 9,10, 8,12, 4, 7)$
$ 13, 1 $ $168$ $13$ $( 1, 2, 5,13, 6, 9,10, 4, 3,11,14, 8, 7)$

Group invariants

Order:  $2184=2^{3} \cdot 3 \cdot 7 \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:    
      2  3   .  2  3  2  2  2   2   2  1  1  1   1   1   1
      3  1   .  .  1  1  1  1   1   1  .  .  .   .   .   .
      7  1   .  1  .  .  .  .   .   .  1  1  1   1   1   1
     13  1   1  .  .  .  .  .   .   .  .  .  .   .   .   .

        1a 13a 2a 2b 3a 4a 6a 12a 12b 7a 7b 7c 14a 14b 14c
     2P 1a 13a 1a 1a 3a 2b 3a  6a  6a 7c 7a 7b  7b  7c  7a
     3P 1a 13a 2a 2b 1a 4a 2b  4a  4a 7b 7c 7a 14b 14c 14a
     5P 1a 13a 2a 2b 3a 4a 6a 12b 12a 7c 7a 7b 14c 14a 14b
     7P 1a 13a 2a 2b 3a 4a 6a 12b 12a 1a 1a 1a  2a  2a  2a
    11P 1a 13a 2a 2b 3a 4a 6a 12a 12b 7b 7c 7a 14b 14c 14a
    13P 1a  1a 2a 2b 3a 4a 6a 12a 12b 7a 7b 7c 14a 14b 14c

X.1      1   1  1  1  1  1  1   1   1  1  1  1   1   1   1
X.2      1   1 -1  1  1 -1  1  -1  -1  1  1  1  -1  -1  -1
X.3     12  -1 -2  .  .  .  .   .   .  B  C  D   D   B   C
X.4     12  -1 -2  .  .  .  .   .   .  C  D  B   B   C   D
X.5     12  -1 -2  .  .  .  .   .   .  D  B  C   C   D   B
X.6     12  -1  2  .  .  .  .   .   .  B  C  D  -D  -B  -C
X.7     12  -1  2  .  .  .  .   .   .  C  D  B  -B  -C  -D
X.8     12  -1  2  .  .  .  .   .   .  D  B  C  -C  -D  -B
X.9     13   .  1  1  1 -1  1  -1  -1 -1 -1 -1   1   1   1
X.10    13   . -1  1  1  1  1   1   1 -1 -1 -1  -1  -1  -1
X.11    14   1  .  2 -1 -2 -1   1   1  .  .  .   .   .   .
X.12    14   1  .  2 -1  2 -1  -1  -1  .  .  .   .   .   .
X.13    14   1  . -2  2  . -2   .   .  .  .  .   .   .   .
X.14    14   1  . -2 -1  .  1   A  -A  .  .  .   .   .   .
X.15    14   1  . -2 -1  .  1  -A   A  .  .  .   .   .   .

A = -E(12)^7+E(12)^11
  = Sqrt(3) = r3
B = -E(7)-E(7)^6
C = -E(7)^3-E(7)^4
D = -E(7)^2-E(7)^5