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Magma
magma: G := TransitiveGroup(43, 3);
Group action invariants
Degree $n$: | $43$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $3$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{43}:C_{3}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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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);
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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
Label | Cycle Type | Size | Order | Representative |
$1^{43}$ | $1$ | $1$ | $()$ | |
$3^{14},1$ | $43$ | $3$ | $( 2, 7,37)( 3,13,30)( 4,19,23)( 5,25,16)( 6,31, 9)( 8,43,38)(10,12,24) (11,18,17)(14,36,39)(15,42,32)(20,29,40)(21,35,33)(22,41,26)(27,28,34)$ | |
$3^{14},1$ | $43$ | $3$ | $( 2,37, 7)( 3,30,13)( 4,23,19)( 5,16,25)( 6, 9,31)( 8,38,43)(10,24,12) (11,17,18)(14,39,36)(15,32,42)(20,40,29)(21,33,35)(22,26,41)(27,34,28)$ | |
$43$ | $3$ | $43$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43)$ | |
$43$ | $3$ | $43$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43, 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42)$ | |
$43$ | $3$ | $43$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40,43, 3, 6, 9,12,15,18,21,24,27,30, 33,36,39,42, 2, 5, 8,11,14,17,20,23,26,29,32,35,38,41)$ | |
$43$ | $3$ | $43$ | $( 1, 5, 9,13,17,21,25,29,33,37,41, 2, 6,10,14,18,22,26,30,34,38,42, 3, 7,11, 15,19,23,27,31,35,39,43, 4, 8,12,16,20,24,28,32,36,40)$ | |
$43$ | $3$ | $43$ | $( 1, 6,11,16,21,26,31,36,41, 3, 8,13,18,23,28,33,38,43, 5,10,15,20,25,30,35, 40, 2, 7,12,17,22,27,32,37,42, 4, 9,14,19,24,29,34,39)$ | |
$43$ | $3$ | $43$ | $( 1, 8,15,22,29,36,43, 7,14,21,28,35,42, 6,13,20,27,34,41, 5,12,19,26,33,40, 4,11,18,25,32,39, 3,10,17,24,31,38, 2, 9,16,23,30,37)$ | |
$43$ | $3$ | $43$ | $( 1,10,19,28,37, 3,12,21,30,39, 5,14,23,32,41, 7,16,25,34,43, 9,18,27,36, 2, 11,20,29,38, 4,13,22,31,40, 6,15,24,33,42, 8,17,26,35)$ | |
$43$ | $3$ | $43$ | $( 1,11,21,31,41, 8,18,28,38, 5,15,25,35, 2,12,22,32,42, 9,19,29,39, 6,16,26, 36, 3,13,23,33,43,10,20,30,40, 7,17,27,37, 4,14,24,34)$ | |
$43$ | $3$ | $43$ | $( 1,14,27,40,10,23,36, 6,19,32, 2,15,28,41,11,24,37, 7,20,33, 3,16,29,42,12, 25,38, 8,21,34, 4,17,30,43,13,26,39, 9,22,35, 5,18,31)$ | |
$43$ | $3$ | $43$ | $( 1,15,29,43,14,28,42,13,27,41,12,26,40,11,25,39,10,24,38, 9,23,37, 8,22,36, 7,21,35, 6,20,34, 5,19,33, 4,18,32, 3,17,31, 2,16,30)$ | |
$43$ | $3$ | $43$ | $( 1,20,39,15,34,10,29, 5,24,43,19,38,14,33, 9,28, 4,23,42,18,37,13,32, 8,27, 3,22,41,17,36,12,31, 7,26, 2,21,40,16,35,11,30, 6,25)$ | |
$43$ | $3$ | $43$ | $( 1,21,41,18,38,15,35,12,32, 9,29, 6,26, 3,23,43,20,40,17,37,14,34,11,31, 8, 28, 5,25, 2,22,42,19,39,16,36,13,33,10,30, 7,27, 4,24)$ | |
$43$ | $3$ | $43$ | $( 1,22,43,21,42,20,41,19,40,18,39,17,38,16,37,15,36,14,35,13,34,12,33,11,32, 10,31, 9,30, 8,29, 7,28, 6,27, 5,26, 4,25, 3,24, 2,23)$ | |
$43$ | $3$ | $43$ | $( 1,27,10,36,19, 2,28,11,37,20, 3,29,12,38,21, 4,30,13,39,22, 5,31,14,40,23, 6,32,15,41,24, 7,33,16,42,25, 8,34,17,43,26, 9,35,18)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $129=3 \cdot 43$ | magma: Order(G);
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Cyclic: | no | magma: IsCyclic(G);
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Abelian: | no | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 129.1 | magma: IdentifyGroup(G);
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Character table: |
1A | 3A1 | 3A-1 | 43A1 | 43A-1 | 43A2 | 43A-2 | 43A3 | 43A-3 | 43A4 | 43A-4 | 43A5 | 43A-5 | 43A9 | 43A-9 | 43A10 | 43A-10 | ||
Size | 1 | 43 | 43 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
3 P | 1A | 3A-1 | 3A1 | 43A9 | 43A4 | 43A3 | 43A-1 | 43A-9 | 43A-4 | 43A5 | 43A-10 | 43A10 | 43A1 | 43A-3 | 43A-2 | 43A-5 | 43A2 | |
43 P | 1A | 1A | 1A | 43A-5 | 43A1 | 43A-10 | 43A-9 | 43A5 | 43A-1 | 43A2 | 43A-4 | 43A4 | 43A9 | 43A10 | 43A-3 | 43A-2 | 43A3 | |
Type | ||||||||||||||||||
129.1.1a | R | |||||||||||||||||
129.1.1b1 | C | |||||||||||||||||
129.1.1b2 | C | |||||||||||||||||
129.1.3a1 | C | |||||||||||||||||
129.1.3a2 | C | |||||||||||||||||
129.1.3a3 | C | |||||||||||||||||
129.1.3a4 | C | |||||||||||||||||
129.1.3a5 | C | |||||||||||||||||
129.1.3a6 | C | |||||||||||||||||
129.1.3a7 | C | |||||||||||||||||
129.1.3a8 | C | |||||||||||||||||
129.1.3a9 | C | |||||||||||||||||
129.1.3a10 | C | |||||||||||||||||
129.1.3a11 | C | |||||||||||||||||
129.1.3a12 | C | |||||||||||||||||
129.1.3a13 | C | |||||||||||||||||
129.1.3a14 | C |
magma: CharacterTable(G);