Properties

Label 43T5
Order \(301\)
n \(43\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{43}:C_{7}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
7:  $C_7$

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

Group invariants

Order:  $301=7 \cdot 43$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [301, 1]
Character table:   
      7  1  1  1  1  1  1  1   .   .   .   .   .   .
     43  1  .  .  .  .  .  .   1   1   1   1   1   1

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

A = E(7)^6
B = E(7)^5
C = E(7)^4
D = E(43)^2+E(43)^8+E(43)^22+E(43)^27+E(43)^32+E(43)^39+E(43)^42
E = E(43)^3+E(43)^5+E(43)^12+E(43)^19+E(43)^20+E(43)^33+E(43)^37
F = E(43)^7+E(43)^18+E(43)^26+E(43)^28+E(43)^29+E(43)^30+E(43)^34