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Properties

Label 43T3
Degree $43$
Order $129$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_{43}:C_{3}$

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Show commands: Magma

magma: G := TransitiveGroup(43, 3);
 

Group action invariants

Degree $n$:  $43$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{43}:C_{3}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,36,6)(2,29,12)(3,22,18)(4,15,24)(5,8,30)(7,37,42)(9,23,11)(10,16,17)(13,38,35)(14,31,41)(19,39,28)(20,32,34)(21,25,40)(26,33,27), (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)
magma: Generators(G);
 

Low degree resolvents

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

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $129=3 \cdot 43$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  129.1
magma: IdentifyGroup(G);
 
Character table:    
      3  1  1  1   .   .   .   .   .   .   .   .   .   .   .   .   .   .
     43  1  .  .   1   1   1   1   1   1   1   1   1   1   1   1   1   1

        1a 3a 3b 43a 43b 43c 43d 43e 43f 43g 43h 43i 43j 43k 43l 43m 43n
     2P 1a 3b 3a 43b 43d 43a 43e 43h 43j 43c 43l 43n 43k 43i 43m 43f 43g
     3P 1a 1a 1a 43c 43a 43g 43b 43d 43m 43n 43e 43k 43f 43j 43h 43l 43i
     5P 1a 3b 3a 43e 43h 43d 43l 43m 43i 43b 43f 43c 43n 43g 43j 43k 43a
     7P 1a 3a 3b 43f 43j 43m 43k 43i 43a 43l 43n 43e 43b 43d 43g 43c 43h
    11P 1a 3b 3a 43g 43c 43n 43a 43b 43l 43i 43d 43j 43m 43f 43e 43h 43k
    13P 1a 3a 3b 43i 43n 43k 43g 43c 43e 43j 43a 43m 43h 43l 43b 43d 43f
    17P 1a 3b 3a 43h 43l 43e 43m 43f 43n 43d 43j 43a 43g 43c 43k 43i 43b
    19P 1a 3a 3b 43k 43i 43j 43n 43g 43d 43f 43c 43l 43e 43h 43a 43b 43m
    23P 1a 3b 3a 43g 43c 43n 43a 43b 43l 43i 43d 43j 43m 43f 43e 43h 43k
    29P 1a 3b 3a 43b 43d 43a 43e 43h 43j 43c 43l 43n 43k 43i 43m 43f 43g
    31P 1a 3a 3b 43j 43k 43f 43i 43n 43b 43m 43g 43h 43d 43e 43c 43a 43l
    37P 1a 3a 3b 43f 43j 43m 43k 43i 43a 43l 43n 43e 43b 43d 43g 43c 43h
    41P 1a 3b 3a 43j 43k 43f 43i 43n 43b 43m 43g 43h 43d 43e 43c 43a 43l
    43P 1a 3a 3b  1a  1a  1a  1a  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   1   1   1
X.2      1  A /A   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   1   1   1
X.4      3  .  .   B  /D   F  /E  /C  /B   G  /H   C   D   E  /G  /F   H
X.5      3  .  .   C   H   E   G   F  /C   D   B  /F  /H  /G  /D  /E  /B
X.6      3  .  .   D   E  /B   C   H  /D  /F   G  /H  /E  /C   F   B  /G
X.7      3  .  .   E   C   D   H   G  /E  /B   F  /G  /C  /H   B  /D  /F
X.8      3  .  .  /E  /C  /D  /H  /G   E   B  /F   G   C   H  /B   D   F
X.9      3  .  .   F   B   G  /D  /E  /F   H  /C   E  /B   D  /H  /G   C
X.10     3  .  .   G   F   H   B  /D  /G   C  /E   D  /F  /B  /C  /H   E
X.11     3  .  .  /G  /F  /H  /B   D   G  /C   E  /D   F   B   C   H  /E
X.12     3  .  .  /D  /E   B  /C  /H   D   F  /G   H   E   C  /F  /B   G
X.13     3  .  .   H   G   C   F   B  /H   E  /D  /B  /G  /F  /E  /C   D
X.14     3  .  .  /B   D  /F   E   C   B  /G   H  /C  /D  /E   G   F  /H
X.15     3  .  .  /H  /G  /C  /F  /B   H  /E   D   B   G   F   E   C  /D
X.16     3  .  .  /F  /B  /G   D   E   F  /H   C  /E   B  /D   H   G  /C
X.17     3  .  .  /C  /H  /E  /G  /F   C  /D  /B   F   H   G   D   E   B

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(43)^21+E(43)^25+E(43)^40
C = E(43)^4+E(43)^15+E(43)^24
D = E(43)+E(43)^6+E(43)^36
E = E(43)^2+E(43)^12+E(43)^29
F = E(43)^20+E(43)^32+E(43)^34
G = E(43)^10+E(43)^16+E(43)^17
H = E(43)^5+E(43)^8+E(43)^30

magma: CharacterTable(G);