Properties

Label 29T4
Order \(203\)
n \(29\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{29}:C_{7}$

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Group action invariants

Degree $n$ :  $29$
Transitive number $t$ :  $4$
Group :  $C_{29}:C_{7}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (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), (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)
7:  $C_7$

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

Group invariants

Order:  $203=7 \cdot 29$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [203, 1]
Character table:   
      7  1  1  1  1  1  1  1   .   .   .   .
     29  1  .  .  .  .  .  .   1   1   1   1

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

X.1      1  1  1  1  1  1  1   1   1   1   1
X.2      1  A  B  C /C /B /A   1   1   1   1
X.3      1  B /C /A  A  C /B   1   1   1   1
X.4      1  C /A  B /B  A /C   1   1   1   1
X.5      1 /C  A /B  B /A  C   1   1   1   1
X.6      1 /B  C  A /A /C  B   1   1   1   1
X.7      1 /A /B /C  C  B  A   1   1   1   1
X.8      7  .  .  .  .  .  .   D   E  /D  /E
X.9      7  .  .  .  .  .  .   E  /D  /E   D
X.10     7  .  .  .  .  .  .  /E   D   E  /D
X.11     7  .  .  .  .  .  .  /D  /E   D   E

A = E(7)^6
B = E(7)^5
C = E(7)^4
D = E(29)^4+E(29)^5+E(29)^6+E(29)^9+E(29)^13+E(29)^22+E(29)^28
E = E(29)^8+E(29)^10+E(29)^12+E(29)^15+E(29)^18+E(29)^26+E(29)^27