# Properties

 Label 29T5 Degree $29$ Order $406$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{29}:C_{14}$

## Group action invariants

 Degree $n$: $29$ Transitive number $t$: $5$ Group: $C_{29}:C_{14}$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ 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)

## 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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $14, 14, 1$ $29$ $14$ $( 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28)( 3, 5, 7, 9,11,13,15,17,19,21,23, 25,27,29)$ $7, 7, 7, 7, 1$ $29$ $7$ $( 2, 6,10,14,18,22,26)( 3, 7,11,15,19,23,27)( 4, 8,12,16,20,24,28) ( 5, 9,13,17,21,25,29)$ $14, 14, 1$ $29$ $14$ $( 2, 8,14,20,26, 4,10,16,22,28, 6,12,18,24)( 3, 9,15,21,27, 5,11,17,23,29, 7, 13,19,25)$ $7, 7, 7, 7, 1$ $29$ $7$ $( 2,10,18,26, 6,14,22)( 3,11,19,27, 7,15,23)( 4,12,20,28, 8,16,24) ( 5,13,21,29, 9,17,25)$ $14, 14, 1$ $29$ $14$ $( 2,12,22, 4,14,24, 6,16,26, 8,18,28,10,20)( 3,13,23, 5,15,25, 7,17,27, 9,19, 29,11,21)$ $7, 7, 7, 7, 1$ $29$ $7$ $( 2,14,26,10,22, 6,18)( 3,15,27,11,23, 7,19)( 4,16,28,12,24, 8,20) ( 5,17,29,13,25, 9,21)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $29$ $2$ $( 2,16)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)( 8,22)( 9,23)(10,24)(11,25)(12,26) (13,27)(14,28)(15,29)$ $7, 7, 7, 7, 1$ $29$ $7$ $( 2,18, 6,22,10,26,14)( 3,19, 7,23,11,27,15)( 4,20, 8,24,12,28,16) ( 5,21, 9,25,13,29,17)$ $14, 14, 1$ $29$ $14$ $( 2,20,10,28,18, 8,26,16, 6,24,14, 4,22,12)( 3,21,11,29,19, 9,27,17, 7,25,15, 5,23,13)$ $7, 7, 7, 7, 1$ $29$ $7$ $( 2,22,14, 6,26,18,10)( 3,23,15, 7,27,19,11)( 4,24,16, 8,28,20,12) ( 5,25,17, 9,29,21,13)$ $14, 14, 1$ $29$ $14$ $( 2,24,18,12, 6,28,22,16,10, 4,26,20,14, 8)( 3,25,19,13, 7,29,23,17,11, 5,27, 21,15, 9)$ $7, 7, 7, 7, 1$ $29$ $7$ $( 2,26,22,18,14,10, 6)( 3,27,23,19,15,11, 7)( 4,28,24,20,16,12, 8) ( 5,29,25,21,17,13, 9)$ $14, 14, 1$ $29$ $14$ $( 2,28,26,24,22,20,18,16,14,12,10, 8, 6, 4)( 3,29,27,25,23,21,19,17,15,13,11, 9, 7, 5)$ $29$ $14$ $29$ $( 1, 2, 3, 7, 4,24, 8,14, 5,12,25,27, 9,20,15,29, 6,23,13,11,26,19,28,22,10, 18,21,17,16)$ $29$ $14$ $29$ $( 1, 3, 4, 8, 5,25, 9,15, 6,13,26,28,10,21,16, 2, 7,24,14,12,27,20,29,23,11, 19,22,18,17)$

## 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