Properties

Label 29T5
Order \(406\)
n \(29\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{29}:C_{14}$

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Group action invariants

Degree $n$ :  $29$
Transitive number $t$ :  $5$
Group :  $C_{29}:C_{14}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10,12,14,16,18,20,22,24,26,28)(3,5,7,9,11,13,15,17,19,21,23,25,27,29), (1,2,3,7,4,24,8,14,5,12,25,27,9,20,15,29,6,23,13,11,26,19,28,22,10,18,21,17,16)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
7:  $C_7$
14:  $C_{14}$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings 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, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 14, 14, 1 $ $29$ $14$ $( 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28)( 3, 5, 7, 9,11,13,15,17,19,21,23, 25,27,29)$
$ 7, 7, 7, 7, 1 $ $29$ $7$ $( 2, 6,10,14,18,22,26)( 3, 7,11,15,19,23,27)( 4, 8,12,16,20,24,28) ( 5, 9,13,17,21,25,29)$
$ 14, 14, 1 $ $29$ $14$ $( 2, 8,14,20,26, 4,10,16,22,28, 6,12,18,24)( 3, 9,15,21,27, 5,11,17,23,29, 7, 13,19,25)$
$ 7, 7, 7, 7, 1 $ $29$ $7$ $( 2,10,18,26, 6,14,22)( 3,11,19,27, 7,15,23)( 4,12,20,28, 8,16,24) ( 5,13,21,29, 9,17,25)$
$ 14, 14, 1 $ $29$ $14$ $( 2,12,22, 4,14,24, 6,16,26, 8,18,28,10,20)( 3,13,23, 5,15,25, 7,17,27, 9,19, 29,11,21)$
$ 7, 7, 7, 7, 1 $ $29$ $7$ $( 2,14,26,10,22, 6,18)( 3,15,27,11,23, 7,19)( 4,16,28,12,24, 8,20) ( 5,17,29,13,25, 9,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $29$ $2$ $( 2,16)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)( 8,22)( 9,23)(10,24)(11,25)(12,26) (13,27)(14,28)(15,29)$
$ 7, 7, 7, 7, 1 $ $29$ $7$ $( 2,18, 6,22,10,26,14)( 3,19, 7,23,11,27,15)( 4,20, 8,24,12,28,16) ( 5,21, 9,25,13,29,17)$
$ 14, 14, 1 $ $29$ $14$ $( 2,20,10,28,18, 8,26,16, 6,24,14, 4,22,12)( 3,21,11,29,19, 9,27,17, 7,25,15, 5,23,13)$
$ 7, 7, 7, 7, 1 $ $29$ $7$ $( 2,22,14, 6,26,18,10)( 3,23,15, 7,27,19,11)( 4,24,16, 8,28,20,12) ( 5,25,17, 9,29,21,13)$
$ 14, 14, 1 $ $29$ $14$ $( 2,24,18,12, 6,28,22,16,10, 4,26,20,14, 8)( 3,25,19,13, 7,29,23,17,11, 5,27, 21,15, 9)$
$ 7, 7, 7, 7, 1 $ $29$ $7$ $( 2,26,22,18,14,10, 6)( 3,27,23,19,15,11, 7)( 4,28,24,20,16,12, 8) ( 5,29,25,21,17,13, 9)$
$ 14, 14, 1 $ $29$ $14$ $( 2,28,26,24,22,20,18,16,14,12,10, 8, 6, 4)( 3,29,27,25,23,21,19,17,15,13,11, 9, 7, 5)$
$ 29 $ $14$ $29$ $( 1, 2, 3, 7, 4,24, 8,14, 5,12,25,27, 9,20,15,29, 6,23,13,11,26,19,28,22,10, 18,21,17,16)$
$ 29 $ $14$ $29$ $( 1, 3, 4, 8, 5,25, 9,15, 6,13,26,28,10,21,16, 2, 7,24,14,12,27,20,29,23,11, 19,22,18,17)$

Group invariants

Order:  $406=2 \cdot 7 \cdot 29$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [406, 1]
Character table:   
      2  1   1   1   1   1   1   1  1   1   1   1   1   1   1   .   .
      7  1   1   1   1   1   1   1  1   1   1   1   1   1   1   .   .
     29  1   .   .   .   .   .   .  .   .   .   .   .   .   .   1   1

        1a 14a  7a 14b  7b 14c  7c 2a  7d 14d  7e 14e  7f 14f 29a 29b
     2P 1a  7a  7b  7c  7d  7e  7f 1a  7a  7b  7c  7d  7e  7f 29b 29a
     3P 1a 14b  7c 14d  7f 14a  7b 2a  7e 14f  7a 14c  7d 14e 29b 29a
     5P 1a 14c  7e 14a  7c 14e  7a 2a  7f 14b  7d 14f  7b 14d 29a 29b
     7P 1a  2a  1a  2a  1a  2a  1a 2a  1a  2a  1a  2a  1a  2a 29a 29b
    11P 1a 14e  7d 14c  7a 14f  7e 2a  7b 14a  7f 14d  7c 14b 29b 29a
    13P 1a 14f  7f 14e  7e 14d  7d 2a  7c 14c  7b 14b  7a 14a 29a 29b
    17P 1a 14b  7c 14d  7f 14a  7b 2a  7e 14f  7a 14c  7d 14e 29b 29a
    19P 1a 14c  7e 14a  7c 14e  7a 2a  7f 14b  7d 14f  7b 14d 29b 29a
    23P 1a 14d  7b 14f  7d 14b  7f 2a  7a 14e  7c 14a  7e 14c 29a 29b
    29P 1a 14a  7a 14b  7b 14c  7c 2a  7d 14d  7e 14e  7f 14f  1a  1a

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

A = -E(7)
B = -E(7)^2
C = -E(7)^3
D = E(29)^2+E(29)^3+E(29)^8+E(29)^10+E(29)^11+E(29)^12+E(29)^14+E(29)^15+E(29)^17+E(29)^18+E(29)^19+E(29)^21+E(29)^26+E(29)^27
  = (-1-Sqrt(29))/2 = -1-b29