Properties

Label 41T5
Order \(328\)
n \(41\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{41}:C_{8}$

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Group action invariants

Degree $n$ :  $41$
Transitive number $t$ :  $5$
Group :  $C_{41}:C_{8}$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,27,32,3,40,14,9,38)(2,13,23,6,39,28,18,35)(4,26,5,12,37,15,36,29)(7,25,19,21,34,16,22,20)(8,11,10,24,33,30,31,17), (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:  $C_4$
8:  $C_8$

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

Group invariants

Order:  $328=2^{3} \cdot 41$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [328, 12]
Character table:   
      2  3   3  3   3   3  3   3  3   .   .   .   .   .
     41  1   .  .   .   .  .   .  .   1   1   1   1   1

        1a  8a 4a  8b  8c 4b  8d 2a 41a 41b 41c 41d 41e
     2P 1a  4a 2a  4b  4b 2a  4a 1a 41b 41c 41e 41a 41d
     3P 1a  8c 4b  8d  8a 4a  8b 2a 41a 41b 41c 41d 41e
     5P 1a  8d 4a  8c  8b 4b  8a 2a 41c 41e 41d 41b 41a
     7P 1a  8b 4b  8a  8d 4a  8c 2a 41d 41a 41b 41e 41c
    11P 1a  8c 4b  8d  8a 4a  8b 2a 41e 41d 41a 41c 41b
    13P 1a  8d 4a  8c  8b 4b  8a 2a 41b 41c 41e 41a 41d
    17P 1a  8a 4a  8b  8c 4b  8d 2a 41e 41d 41a 41c 41b
    19P 1a  8c 4b  8d  8a 4a  8b 2a 41d 41a 41b 41e 41c
    23P 1a  8b 4b  8a  8d 4a  8c 2a 41b 41c 41e 41a 41d
    29P 1a  8d 4a  8c  8b 4b  8a 2a 41c 41e 41d 41b 41a
    31P 1a  8b 4b  8a  8d 4a  8c 2a 41e 41d 41a 41c 41b
    37P 1a  8d 4a  8c  8b 4b  8a 2a 41c 41e 41d 41b 41a
    41P 1a  8a 4a  8b  8c 4b  8d 2a  1a  1a  1a  1a  1a

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

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