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Properties

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

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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 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 $ $1$ $1$ $()$
$ 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