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Properties

Label 41T4
Degree $41$
Order $205$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_{41}:C_{5}$

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Group action invariants

Degree $n$:  $41$
Transitive number $t$:  $4$
Group:  $C_{41}:C_{5}$
Parity:  $1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,10,18,16,37)(2,20,36,32,33)(3,30,13,7,29)(4,40,31,23,25)(5,9,8,39,21)(6,19,26,14,17)(11,28,34,12,38)(15,27,24,35,22), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$5$:  $C_5$

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ $41$ $5$ $( 2,11,19,17,38)( 3,21,37,33,34)( 4,31,14, 8,30)( 5,41,32,24,26) ( 6,10, 9,40,22)( 7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ $41$ $5$ $( 2,17,11,38,19)( 3,33,21,34,37)( 4, 8,31,30,14)( 5,24,41,26,32) ( 6,40,10,22, 9)( 7,15,20,18,27)(12,13,29,39,35)(16,36,28,23,25)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ $41$ $5$ $( 2,19,38,11,17)( 3,37,34,21,33)( 4,14,30,31, 8)( 5,32,26,41,24) ( 6, 9,22,10,40)( 7,27,18,20,15)(12,35,39,29,13)(16,25,23,28,36)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ $41$ $5$ $( 2,38,17,19,11)( 3,34,33,37,21)( 4,30, 8,14,31)( 5,26,24,32,41) ( 6,22,40, 9,10)( 7,18,15,27,20)(12,39,13,35,29)(16,23,36,25,28)$
$ 41 $ $5$ $41$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$
$ 41 $ $5$ $41$ $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41, 2, 4, 6, 8, 10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40)$
$ 41 $ $5$ $41$ $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40, 2, 5, 8,11,14,17,20,23,26,29,32, 35,38,41, 3, 6, 9,12,15,18,21,24,27,30,33,36,39)$
$ 41 $ $5$ $41$ $( 1, 5, 9,13,17,21,25,29,33,37,41, 4, 8,12,16,20,24,28,32,36,40, 3, 7,11,15, 19,23,27,31,35,39, 2, 6,10,14,18,22,26,30,34,38)$
$ 41 $ $5$ $41$ $( 1, 6,11,16,21,26,31,36,41, 5,10,15,20,25,30,35,40, 4, 9,14,19,24,29,34,39, 3, 8,13,18,23,28,33,38, 2, 7,12,17,22,27,32,37)$
$ 41 $ $5$ $41$ $( 1, 7,13,19,25,31,37, 2, 8,14,20,26,32,38, 3, 9,15,21,27,33,39, 4,10,16,22, 28,34,40, 5,11,17,23,29,35,41, 6,12,18,24,30,36)$
$ 41 $ $5$ $41$ $( 1,12,23,34, 4,15,26,37, 7,18,29,40,10,21,32, 2,13,24,35, 5,16,27,38, 8,19, 30,41,11,22,33, 3,14,25,36, 6,17,28,39, 9,20,31)$
$ 41 $ $5$ $41$ $( 1,16,31, 5,20,35, 9,24,39,13,28, 2,17,32, 6,21,36,10,25,40,14,29, 3,18,33, 7,22,37,11,26,41,15,30, 4,19,34, 8,23,38,12,27)$

Group invariants

Order:  $205=5 \cdot 41$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [205, 1]
Character table:    
      5  1  1  1  1  1   .   .   .   .   .   .   .   .
     41  1  .  .  .  .   1   1   1   1   1   1   1   1

        1a 5a 5b 5c 5d 41a 41b 41c 41d 41e 41f 41g 41h
     2P 1a 5c 5a 5d 5b 41b 41d 41f 41e 41a 41g 41h 41c
     3P 1a 5b 5d 5a 5c 41c 41f 41e 41g 41h 41a 41b 41d
     5P 1a 1a 1a 1a 1a 41e 41a 41h 41b 41d 41c 41f 41g
     7P 1a 5c 5a 5d 5b 41c 41f 41e 41g 41h 41a 41b 41d
    11P 1a 5a 5b 5c 5d 41g 41h 41b 41c 41f 41d 41e 41a
    13P 1a 5b 5d 5a 5c 41c 41f 41e 41g 41h 41a 41b 41d
    17P 1a 5c 5a 5d 5b 41f 41g 41a 41h 41c 41b 41d 41e
    19P 1a 5d 5c 5b 5a 41f 41g 41a 41h 41c 41b 41d 41e
    23P 1a 5b 5d 5a 5c 41d 41e 41g 41a 41b 41h 41c 41f
    29P 1a 5d 5c 5b 5a 41c 41f 41e 41g 41h 41a 41b 41d
    31P 1a 5a 5b 5c 5d 41d 41e 41g 41a 41b 41h 41c 41f
    37P 1a 5c 5a 5d 5b 41a 41b 41c 41d 41e 41f 41g 41h
    41P 1a 5a 5b 5c 5d  1a  1a  1a  1a  1a  1a  1a  1a

X.1      1  1  1  1  1   1   1   1   1   1   1   1   1
X.2      1  A /B  B /A   1   1   1   1   1   1   1   1
X.3      1  B  A /A /B   1   1   1   1   1   1   1   1
X.4      1 /B /A  A  B   1   1   1   1   1   1   1   1
X.5      1 /A  B /B  A   1   1   1   1   1   1   1   1
X.6      5  .  .  .  .   C  /F  /E  /C   F   D   E  /D
X.7      5  .  .  .  .  /C   F   E   C  /F  /D  /E   D
X.8      5  .  .  .  .   D   E   C  /D  /E  /F  /C   F
X.9      5  .  .  .  .   E  /D  /F  /E   D  /C   F   C
X.10     5  .  .  .  .   F   C  /D  /F  /C  /E   D   E
X.11     5  .  .  .  .  /E   D   F   E  /D   C  /F  /C
X.12     5  .  .  .  .  /D  /E  /C   D   E   F   C  /F
X.13     5  .  .  .  .  /F  /C   D   F   C   E  /D  /E

A = E(5)^4
B = E(5)^3
C = E(41)^3+E(41)^7+E(41)^13+E(41)^29+E(41)^30
D = E(41)+E(41)^10+E(41)^16+E(41)^18+E(41)^37
E = E(41)^2+E(41)^20+E(41)^32+E(41)^33+E(41)^36
F = E(41)^15+E(41)^22+E(41)^24+E(41)^27+E(41)^35