# Properties

 Label 29T3 Degree $29$ Order $116$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{29}:C_{4}$

## Group action invariants

 Degree $n$: $29$ Transitive number $t$: $3$ Group: $C_{29}:C_{4}$ Parity: $-1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (2,9,16,23)(3,10,17,24)(4,11,18,25)(5,12,19,26)(6,13,20,27)(7,14,21,28)(8,15,22,29), (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)
$2$:  $C_2$
$4$:  $C_4$

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

## Group invariants

 Order: $116=2^{2} \cdot 29$ Cyclic: no Abelian: no Solvable: yes GAP id: [116, 3]
 Character table:  2 2 2 2 2 . . . . . . . 29 1 . . . 1 1 1 1 1 1 1 1a 4a 2a 4b 29a 29b 29c 29d 29e 29f 29g 2P 1a 2a 1a 2a 29b 29c 29d 29e 29f 29g 29a 3P 1a 4b 2a 4a 29f 29g 29a 29b 29c 29d 29e 5P 1a 4a 2a 4b 29b 29c 29d 29e 29f 29g 29a 7P 1a 4b 2a 4a 29f 29g 29a 29b 29c 29d 29e 11P 1a 4b 2a 4a 29e 29f 29g 29a 29b 29c 29d 13P 1a 4a 2a 4b 29e 29f 29g 29a 29b 29c 29d 17P 1a 4a 2a 4b 29a 29b 29c 29d 29e 29f 29g 19P 1a 4b 2a 4a 29c 29d 29e 29f 29g 29a 29b 23P 1a 4b 2a 4a 29g 29a 29b 29c 29d 29e 29f 29P 1a 4a 2a 4b 1a 1a 1a 1a 1a 1a 1a X.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 X.3 1 A -1 -A 1 1 1 1 1 1 1 X.4 1 -A -1 A 1 1 1 1 1 1 1 X.5 4 . . . B G H D F C E X.6 4 . . . C E B G H D F X.7 4 . . . D F C E B G H X.8 4 . . . E B G H D F C X.9 4 . . . F C E B G H D X.10 4 . . . G H D F C E B X.11 4 . . . H D F C E B G A = -E(4) = -Sqrt(-1) = -i B = E(29)^11+E(29)^13+E(29)^16+E(29)^18 C = E(29)^4+E(29)^10+E(29)^19+E(29)^25 D = E(29)+E(29)^12+E(29)^17+E(29)^28 E = E(29)^8+E(29)^9+E(29)^20+E(29)^21 F = E(29)^2+E(29)^5+E(29)^24+E(29)^27 G = E(29)^3+E(29)^7+E(29)^22+E(29)^26 H = E(29)^6+E(29)^14+E(29)^15+E(29)^23