Properties

Label 41T3
Order \(164\)
n \(41\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{41}:C_{4}$

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Group action invariants

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

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

Group invariants

Order:  $164=2^{2} \cdot 41$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [164, 3]
Character table:   
      2  2  2  2  2   .   .   .   .   .   .   .   .   .   .
     41  1  .  .  .   1   1   1   1   1   1   1   1   1   1

        1a 4a 4b 2a 41a 41b 41c 41d 41e 41f 41g 41h 41i 41j
     2P 1a 2a 2a 1a 41b 41d 41e 41g 41i 41c 41j 41f 41h 41a
     3P 1a 4b 4a 2a 41c 41e 41a 41i 41b 41j 41h 41g 41d 41f
     5P 1a 4a 4b 2a 41d 41g 41i 41j 41h 41e 41a 41c 41f 41b
     7P 1a 4b 4a 2a 41f 41c 41j 41e 41a 41g 41i 41d 41b 41h
    11P 1a 4b 4a 2a 41h 41f 41g 41c 41j 41d 41e 41b 41a 41i
    13P 1a 4a 4b 2a 41e 41i 41b 41h 41d 41a 41f 41j 41g 41c
    17P 1a 4a 4b 2a 41h 41f 41g 41c 41j 41d 41e 41b 41a 41i
    19P 1a 4b 4a 2a 41f 41c 41j 41e 41a 41g 41i 41d 41b 41h
    23P 1a 4b 4a 2a 41b 41d 41e 41g 41i 41c 41j 41f 41h 41a
    29P 1a 4a 4b 2a 41i 41h 41d 41f 41g 41b 41c 41a 41j 41e
    31P 1a 4b 4a 2a 41g 41j 41h 41a 41f 41i 41b 41e 41c 41d
    37P 1a 4a 4b 2a 41d 41g 41i 41j 41h 41e 41a 41c 41f 41b
    41P 1a 4a 4b 2a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a

X.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
X.3      1  A -A -1   1   1   1   1   1   1   1   1   1   1
X.4      1 -A  A -1   1   1   1   1   1   1   1   1   1   1
X.5      4  .  .  .   B   E   G   D   C   J   H   F   K   I
X.6      4  .  .  .   C   K   E   F   D   B   J   I   H   G
X.7      4  .  .  .   D   H   K   I   F   C   B   G   J   E
X.8      4  .  .  .   E   D   C   H   K   G   I   J   F   B
X.9      4  .  .  .   F   J   H   G   I   D   C   E   B   K
X.10     4  .  .  .   G   C   B   K   E   I   F   H   D   J
X.11     4  .  .  .   H   I   F   B   J   K   E   C   G   D
X.12     4  .  .  .   I   B   J   E   G   F   D   K   C   H
X.13     4  .  .  .   J   G   I   C   B   H   K   D   E   F
X.14     4  .  .  .   K   F   D   J   H   E   G   B   I   C

A = -E(4)
  = -Sqrt(-1) = -i
B = E(41)^2+E(41)^18+E(41)^23+E(41)^39
C = E(41)^12+E(41)^15+E(41)^26+E(41)^29
D = E(41)^8+E(41)^10+E(41)^31+E(41)^33
E = E(41)^4+E(41)^5+E(41)^36+E(41)^37
F = E(41)^7+E(41)^19+E(41)^22+E(41)^34
G = E(41)^6+E(41)^13+E(41)^28+E(41)^35
H = E(41)^16+E(41)^20+E(41)^21+E(41)^25
I = E(41)+E(41)^9+E(41)^32+E(41)^40
J = E(41)^3+E(41)^14+E(41)^27+E(41)^38
K = E(41)^11+E(41)^17+E(41)^24+E(41)^30