Properties

Label 43T4
Order \(258\)
n \(43\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{43}:C_{6}$

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Group action invariants

Degree $n$ :  $43$
Transitive number $t$ :  $4$
Group :  $C_{43}:C_{6}$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,37,36,42,6,7)(2,31,29,41,12,14)(3,25,22,40,18,21)(4,19,15,39,24,28)(5,13,8,38,30,35)(9,32,23,34,11,20)(10,26,16,33,17,27), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $43$ $3$ $( 2, 7,37)( 3,13,30)( 4,19,23)( 5,25,16)( 6,31, 9)( 8,43,38)(10,12,24) (11,18,17)(14,36,39)(15,42,32)(20,29,40)(21,35,33)(22,41,26)(27,28,34)$
$ 6, 6, 6, 6, 6, 6, 6, 1 $ $43$ $6$ $( 2, 8, 7,43,37,38)( 3,15,13,42,30,32)( 4,22,19,41,23,26)( 5,29,25,40,16,20) ( 6,36,31,39, 9,14)(10,21,12,35,24,33)(11,28,18,34,17,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $43$ $3$ $( 2,37, 7)( 3,30,13)( 4,23,19)( 5,16,25)( 6, 9,31)( 8,38,43)(10,24,12) (11,17,18)(14,39,36)(15,32,42)(20,40,29)(21,33,35)(22,26,41)(27,34,28)$
$ 6, 6, 6, 6, 6, 6, 6, 1 $ $43$ $6$ $( 2,38,37,43, 7, 8)( 3,32,30,42,13,15)( 4,26,23,41,19,22)( 5,20,16,40,25,29) ( 6,14, 9,39,31,36)(10,33,24,35,12,21)(11,27,17,34,18,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $43$ $2$ $( 2,43)( 3,42)( 4,41)( 5,40)( 6,39)( 7,38)( 8,37)( 9,36)(10,35)(11,34)(12,33) (13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)$
$ 43 $ $6$ $43$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43)$
$ 43 $ $6$ $43$ $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43, 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42)$
$ 43 $ $6$ $43$ $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40,43, 3, 6, 9,12,15,18,21,24,27,30, 33,36,39,42, 2, 5, 8,11,14,17,20,23,26,29,32,35,38,41)$
$ 43 $ $6$ $43$ $( 1, 5, 9,13,17,21,25,29,33,37,41, 2, 6,10,14,18,22,26,30,34,38,42, 3, 7,11, 15,19,23,27,31,35,39,43, 4, 8,12,16,20,24,28,32,36,40)$
$ 43 $ $6$ $43$ $( 1, 6,11,16,21,26,31,36,41, 3, 8,13,18,23,28,33,38,43, 5,10,15,20,25,30,35, 40, 2, 7,12,17,22,27,32,37,42, 4, 9,14,19,24,29,34,39)$
$ 43 $ $6$ $43$ $( 1,10,19,28,37, 3,12,21,30,39, 5,14,23,32,41, 7,16,25,34,43, 9,18,27,36, 2, 11,20,29,38, 4,13,22,31,40, 6,15,24,33,42, 8,17,26,35)$
$ 43 $ $6$ $43$ $( 1,11,21,31,41, 8,18,28,38, 5,15,25,35, 2,12,22,32,42, 9,19,29,39, 6,16,26, 36, 3,13,23,33,43,10,20,30,40, 7,17,27,37, 4,14,24,34)$

Group invariants

Order:  $258=2 \cdot 3 \cdot 43$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [258, 1]
Character table:   
      2  1  1   1  1   1  1   .   .   .   .   .   .   .
      3  1  1   1  1   1  1   .   .   .   .   .   .   .
     43  1  .   .  .   .  .   1   1   1   1   1   1   1

        1a 3a  6a 3b  6b 2a 43a 43b 43c 43d 43e 43f 43g
     2P 1a 3b  3a 3a  3b 1a 43b 43d 43a 43e 43g 43c 43f
     3P 1a 1a  2a 1a  2a 2a 43c 43a 43f 43b 43d 43g 43e
     5P 1a 3b  6b 3a  6a 2a 43e 43g 43d 43f 43c 43b 43a
     7P 1a 3a  6a 3b  6b 2a 43a 43b 43c 43d 43e 43f 43g
    11P 1a 3b  6b 3a  6a 2a 43f 43c 43g 43a 43b 43e 43d
    13P 1a 3a  6a 3b  6b 2a 43e 43g 43d 43f 43c 43b 43a
    17P 1a 3b  6b 3a  6a 2a 43g 43f 43e 43c 43a 43d 43b
    19P 1a 3a  6a 3b  6b 2a 43d 43e 43b 43g 43f 43a 43c
    23P 1a 3b  6b 3a  6a 2a 43f 43c 43g 43a 43b 43e 43d
    29P 1a 3b  6b 3a  6a 2a 43b 43d 43a 43e 43g 43c 43f
    31P 1a 3a  6a 3b  6b 2a 43b 43d 43a 43e 43g 43c 43f
    37P 1a 3a  6a 3b  6b 2a 43a 43b 43c 43d 43e 43f 43g
    41P 1a 3b  6b 3a  6a 2a 43b 43d 43a 43e 43g 43c 43f
    43P 1a 3a  6a 3b  6b 2a  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  1  -1  1  -1 -1   1   1   1   1   1   1   1
X.3      1  A -/A /A  -A -1   1   1   1   1   1   1   1
X.4      1 /A  -A  A -/A -1   1   1   1   1   1   1   1
X.5      1  A  /A /A   A  1   1   1   1   1   1   1   1
X.6      1 /A   A  A  /A  1   1   1   1   1   1   1   1
X.7      6  .   .  .   .  .   B   F   H   E   G   C   D
X.8      6  .   .  .   .  .   C   H   D   B   F   G   E
X.9      6  .   .  .   .  .   D   C   G   H   B   E   F
X.10     6  .   .  .   .  .   E   G   F   D   C   B   H
X.11     6  .   .  .   .  .   F   E   B   G   D   H   C
X.12     6  .   .  .   .  .   G   D   E   C   H   F   B
X.13     6  .   .  .   .  .   H   B   C   F   E   D   G

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(43)^9+E(43)^11+E(43)^20+E(43)^23+E(43)^32+E(43)^34
C = E(43)^5+E(43)^8+E(43)^13+E(43)^30+E(43)^35+E(43)^38
D = E(43)^4+E(43)^15+E(43)^19+E(43)^24+E(43)^28+E(43)^39
E = E(43)+E(43)^6+E(43)^7+E(43)^36+E(43)^37+E(43)^42
F = E(43)^3+E(43)^18+E(43)^21+E(43)^22+E(43)^25+E(43)^40
G = E(43)^2+E(43)^12+E(43)^14+E(43)^29+E(43)^31+E(43)^41
H = E(43)^10+E(43)^16+E(43)^17+E(43)^26+E(43)^27+E(43)^33