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Properties

Label 14T9
Order \(112\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times F_8$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $9$
Group :  $C_2\times F_8$
CHM label :  $[2^{4}]7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8)(2,9)(4,11), (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:  $C_2$
7:  $C_7$
14:  $C_{14}$
56:  $C_2^3:C_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7$

Low degree siblings

16T196, 28T19 x 3, 28T20

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, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 4,11)( 5,12)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 5,12)( 6,13)( 7,14)$
$ 14 $ $8$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
$ 7, 7 $ $8$ $7$ $( 1, 2, 3, 4,12, 6, 7)( 5,13,14, 8, 9,10,11)$
$ 7, 7 $ $8$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 14 $ $8$ $14$ $( 1, 3, 5,14, 9,11, 6, 8,10,12, 7, 2, 4,13)$
$ 14 $ $8$ $14$ $( 1, 4, 7,10,13, 2, 5, 8,11,14, 3, 6, 9,12)$
$ 7, 7 $ $8$ $7$ $( 1, 4,14,10,13, 2, 5)( 3, 6, 9,12, 8,11, 7)$
$ 14 $ $8$ $14$ $( 1, 5, 2, 6,10,14,11, 8,12, 9,13, 3, 7, 4)$
$ 7, 7 $ $8$ $7$ $( 1, 5, 9,13, 3, 7,11)( 2, 6,10,14, 4, 8,12)$
$ 7, 7 $ $8$ $7$ $( 1, 6, 4, 2, 7,12, 3)( 5,10, 8,13,11, 9,14)$
$ 14 $ $8$ $14$ $( 1, 6, 4, 9,14,12,10, 8,13,11, 2, 7, 5, 3)$
$ 7, 7 $ $8$ $7$ $( 1, 7, 6,12, 4, 3, 2)( 5,11,10, 9, 8,14,13)$
$ 14 $ $8$ $14$ $( 1, 7,13, 5, 4,10, 9, 8,14, 6,12,11, 3, 2)$
$ 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:  $112=2^{4} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [112, 41]
Character table:    
      2  4  4  4   1   1   1   1   1   1   1   1   1   1   1   1  4
      7  1  .  .   1   1   1   1   1   1   1   1   1   1   1   1  1

        1a 2a 2b 14a  7a  7b 14b 14c  7c 14d  7d  7e 14e  7f 14f 2c
     2P 1a 1a 1a  7b  7b  7d  7d  7f  7f  7a  7a  7c  7c  7e  7e 1a
     3P 1a 2a 2b 14c  7c  7f 14f 14b  7b 14e  7e  7a 14a  7d 14d 2c
     5P 1a 2a 2b 14e  7e  7c 14c 14a  7a 14f  7f  7d 14d  7b 14b 2c
     7P 1a 2a 2b  2c  1a  1a  2c  2c  1a  2c  1a  1a  2c  1a  2c 2c
    11P 1a 2a 2b 14d  7d  7a 14a 14e  7e 14b  7b  7f 14f  7c 14c 2c
    13P 1a 2a 2b 14f  7f  7e 14e 14d  7d 14c  7c  7b 14b  7a 14a 2c

X.1      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 -1
X.3      1 -1  1   A  -A  -B   B   C  -C  /C -/C -/B  /B -/A  /A -1
X.4      1 -1  1   B  -B -/C  /C  /A -/A   A  -A  -C   C -/B  /B -1
X.5      1 -1  1   C  -C -/A  /A   B  -B  /B -/B  -A   A -/C  /C -1
X.6      1 -1  1  /C -/C  -A   A  /B -/B   B  -B -/A  /A  -C   C -1
X.7      1 -1  1  /B -/B  -C   C   A  -A  /A -/A -/C  /C  -B   B -1
X.8      1 -1  1  /A -/A -/B  /B  /C -/C   C  -C  -B   B  -A   A -1
X.9      1  1  1 -/A -/A -/B -/B -/C -/C  -C  -C  -B  -B  -A  -A  1
X.10     1  1  1 -/B -/B  -C  -C  -A  -A -/A -/A -/C -/C  -B  -B  1
X.11     1  1  1 -/C -/C  -A  -A -/B -/B  -B  -B -/A -/A  -C  -C  1
X.12     1  1  1  -C  -C -/A -/A  -B  -B -/B -/B  -A  -A -/C -/C  1
X.13     1  1  1  -B  -B -/C -/C -/A -/A  -A  -A  -C  -C -/B -/B  1
X.14     1  1  1  -A  -A  -B  -B  -C  -C -/C -/C -/B -/B -/A -/A  1
X.15     7  1 -1   .   .   .   .   .   .   .   .   .   .   .   . -7
X.16     7 -1 -1   .   .   .   .   .   .   .   .   .   .   .   .  7

A = -E(7)
B = -E(7)^2
C = -E(7)^3