Properties

Label 43T6
Order \(602\)
n \(43\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{43}:C_{14}$

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Group action invariants

Degree $n$ :  $43$
Transitive number $t$ :  $6$
Group :  $C_{43}:C_{14}$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,27,41,32,4,22,35,42,16,2,11,39,21,8)(3,38,37,10,12,23,19,40,5,6,33,31,20,24)(7,17,29,9,28,25,30,36,26,14,34,15,18,13), (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:  $C_2$
7:  $C_7$
14:  $C_{14}$

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

Group invariants

Order:  $602=2 \cdot 7 \cdot 43$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [602, 1]
Character table:   
      2  1   .   .   .  1  1  1  1  1  1   1   1  1   1   1   1   1
      7  1   .   .   .  1  1  1  1  1  1   1   1  1   1   1   1   1
     43  1   1   1   1  .  .  .  .  .  .   .   .  .   .   .   .   .

        1a 43a 43b 43c 7a 7b 7c 7d 7e 7f 14a 14b 2a 14c 14d 14e 14f
     2P 1a 43a 43b 43c 7c 7d 7e 7f 7a 7b  7a  7b 1a  7f  7d  7c  7e
     3P 1a 43b 43c 43a 7b 7c 7d 7e 7f 7a 14b 14e 2a 14a 14f 14d 14c
     5P 1a 43b 43c 43a 7f 7a 7b 7c 7d 7e 14c 14a 2a 14f 14e 14b 14d
     7P 1a 43c 43a 43b 1a 1a 1a 1a 1a 1a  2a  2a 2a  2a  2a  2a  2a
    11P 1a 43a 43b 43c 7e 7f 7a 7b 7c 7d 14f 14c 2a 14d 14b 14a 14e
    13P 1a 43c 43a 43b 7d 7e 7f 7a 7b 7c 14d 14f 2a 14e 14a 14c 14b
    17P 1a 43c 43a 43b 7b 7c 7d 7e 7f 7a 14b 14e 2a 14a 14f 14d 14c
    19P 1a 43b 43c 43a 7f 7a 7b 7c 7d 7e 14c 14a 2a 14f 14e 14b 14d
    23P 1a 43b 43c 43a 7c 7d 7e 7f 7a 7b 14e 14d 2a 14b 14c 14f 14a
    29P 1a 43c 43a 43b 7a 7b 7c 7d 7e 7f 14a 14b 2a 14c 14d 14e 14f
    31P 1a 43b 43c 43a 7b 7c 7d 7e 7f 7a 14b 14e 2a 14a 14f 14d 14c
    37P 1a 43b 43c 43a 7c 7d 7e 7f 7a 7b 14e 14d 2a 14b 14c 14f 14a
    41P 1a 43a 43b 43c 7d 7e 7f 7a 7b 7c 14d 14f 2a 14e 14a 14c 14b
    43P 1a  1a  1a  1a 7a 7b 7c 7d 7e 7f 14a 14b 2a 14c 14d 14e 14f

X.1      1   1   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  -1  -1  -1
X.3      1   1   1   1  D  F  E /D /F /E -/F -/E -1 -/D  -F  -D  -E
X.4      1   1   1   1  E /D /F /E  D  F  -D  -F -1 -/E -/D  -E -/F
X.5      1   1   1   1  F  E /D /F /E  D -/E  -D -1 -/F  -E  -F -/D
X.6      1   1   1   1 /F /E  D  F  E /D  -E -/D -1  -F -/E -/F  -D
X.7      1   1   1   1 /E  D  F  E /D /F -/D -/F -1  -E  -D -/E  -F
X.8      1   1   1   1 /D /F /E  D  F  E  -F  -E -1  -D -/F -/D -/E
X.9      1   1   1   1  D  F  E /D /F /E  /F  /E  1  /D   F   D   E
X.10     1   1   1   1  E /D /F /E  D  F   D   F  1  /E  /D   E  /F
X.11     1   1   1   1  F  E /D /F /E  D  /E   D  1  /F   E   F  /D
X.12     1   1   1   1 /F /E  D  F  E /D   E  /D  1   F  /E  /F   D
X.13     1   1   1   1 /E  D  F  E /D /F  /D  /F  1   E   D  /E   F
X.14     1   1   1   1 /D /F /E  D  F  E   F   E  1   D  /F  /D  /E
X.15    14   A   B   C  .  .  .  .  .  .   .   .  .   .   .   .   .
X.16    14   B   C   A  .  .  .  .  .  .   .   .  .   .   .   .   .
X.17    14   C   A   B  .  .  .  .  .  .   .   .  .   .   .   .   .

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