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Magma
magma: G := TransitiveGroup(43, 8);
Group action invariants
| Degree $n$: | $43$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $F_{43}$ | ||
| 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: | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43), (1,3,9,27,38,28,41,37,25,32,10,30,4,12,36,22,23,26,35,19,14,42,40,34,16,5,15,2,6,18,11,33,13,39,31,7,21,20,17,8,24,29) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $7$: $C_7$ $14$: $C_{14}$ $21$: $C_{21}$ $42$: $C_{42}$ 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, 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 $ | $42$ | $43$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43)$ | |
| $ 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 $ | $43$ | $7$ | $( 2,12,36,42,22,17, 5)( 3,23,28,40,43,33, 9)( 4,34,20,38,21, 6,13) ( 7,24,39,32,41,11,25)( 8,35,31,30,19,27,29)(10,14,15,26,18,16,37)$ | |
| $ 21, 21, 1 $ | $43$ | $21$ | $( 2,10,39,42,26,11, 5,37,24,36,15,41,17,16, 7,12,14,32,22,18,25) ( 3,19,34,40, 8,21, 9,30, 4,28,29,38,33,31,13,23,27,20,43,35, 6)$ | |
| $ 21, 21, 1 $ | $43$ | $21$ | $( 2,26,24,17,14,25,42,37,41,12,18,39, 5,15, 7,22,10,11,36,16,32) ( 3, 8, 4,33,27, 6,40,30,38,23,35,34, 9,29,13,43,19,21,28,31,20)$ | |
| $ 21, 21, 1 $ | $43$ | $21$ | $( 2,15,25,36,18,24,22,37,32, 5,14,11,12,26, 7,42,16,39,17,10,41) ( 3,29, 6,28,35, 4,43,30,20, 9,27,21,23, 8,13,40,31,34,33,19,38)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $43$ | $3$ | $( 2,37, 7)( 3,30,13)( 4,23,19)( 5,16,25)( 6, 9,31)( 8,38,43)(10,24,12) (11,17,18)(14,39,36)(15,32,42)(20,40,29)(21,33,35)(22,26,41)(27,34,28)$ | |
| $ 21, 21, 1 $ | $43$ | $21$ | $( 2,18,32,12,16,41,36,37,11,42,10,25,22,14, 7,17,15,24, 5,26,39) ( 3,35,20,23,31,38,28,30,21,40,19, 6,43,27,13,33,29, 4, 9, 8,34)$ | |
| $ 21, 21, 1 $ | $43$ | $21$ | $( 2,16,11,22,15,39,12,37,25,17,26,32,36,10, 7, 5,18,41,42,14,24) ( 3,31,21,43,29,34,23,30, 6,33, 8,20,28,19,13, 9,35,38,40,27, 4)$ | |
| $ 21, 21, 1 $ | $43$ | $21$ | $( 2,14,41, 5,10,32,17,37,39,22,16,24,42,18, 7,36,26,25,12,15,11) ( 3,27,38, 9,19,20,33,30,34,43,31, 4,40,35,13,28, 8, 6,23,29,21)$ | |
| $ 21, 21, 1 $ | $43$ | $21$ | $( 2,39,26, 5,24,15,17, 7,14,22,25,10,42,11,37,36,41,16,12,32,18) ( 3,34, 8, 9, 4,29,33,13,27,43, 6,19,40,21,30,28,38,31,23,20,35)$ | |
| $ 21, 21, 1 $ | $43$ | $21$ | $( 2,11,15,12,25,26,36, 7,18,42,24,16,22,39,37,17,32,10, 5,41,14) ( 3,21,29,23, 6, 8,28,13,35,40, 4,31,43,34,30,33,20,19, 9,38,27)$ | |
| $ 21, 21, 1 $ | $43$ | $21$ | $( 2,41,10,17,39,16,42, 7,26,12,11,14, 5,32,37,22,24,18,36,25,15) ( 3,38,19,33,34,31,40,13, 8,23,21,27, 9,20,30,43, 4,35,28, 6,29)$ | |
| $ 21, 21, 1 $ | $43$ | $21$ | $( 2,24,14,42,41,18, 5, 7,10,36,32,26,17,25,37,12,39,15,22,11,16) ( 3, 4,27,40,38,35, 9,13,19,28,20, 8,33, 6,30,23,34,29,43,21,31)$ | |
| $ 21, 21, 1 $ | $43$ | $21$ | $( 2,25,18,22,32,14,12, 7,16,17,41,15,36,24,37, 5,11,26,42,39,10) ( 3, 6,35,43,20,27,23,13,31,33,38,29,28, 4,30, 9,21, 8,40,34,19)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $43$ | $3$ | $( 2, 7,37)( 3,13,30)( 4,19,23)( 5,25,16)( 6,31, 9)( 8,43,38)(10,12,24) (11,18,17)(14,36,39)(15,42,32)(20,29,40)(21,35,33)(22,41,26)(27,28,34)$ | |
| $ 21, 21, 1 $ | $43$ | $21$ | $( 2,32,16,36,11,10,22, 7,15, 5,39,18,12,41,37,42,25,14,17,24,26) ( 3,20,31,28,21,19,43,13,29, 9,34,35,23,38,30,40, 6,27,33, 4, 8)$ | |
| $ 42, 1 $ | $43$ | $42$ | $( 2, 4,10,28,39,29,42,38,26,33,11,31, 5,13,37,23,24,27,36,20,15,43,41,35,17, 6,16, 3, 7,19,12,34,14,40,32, 8,22,21,18, 9,25,30)$ | |
| $ 6, 6, 6, 6, 6, 6, 6, 1 $ | $43$ | $6$ | $( 2,38,37,43, 7, 8)( 3,32,30,42,13,15)( 4,26,23,41,19,22)( 5,20,16,40,25,29) ( 6,14, 9,39,31,36)(10,33,24,35,12,21)(11,27,17,34,18,28)$ | |
| $ 42, 1 $ | $43$ | $42$ | $( 2,20,18,23,32,31,12,38,16,28,41,30,36,21,37,40,11,19,42, 6,10,43,25,27,22, 13,14,33, 7,29,17, 4,15, 9,24, 8, 5,34,26, 3,39,35)$ | |
| $ 42, 1 $ | $43$ | $42$ | $( 2,13,16, 9,11,35,22,38,15,40,39,27,12, 4,37, 3,25,31,17,21,26,43,32,29,36, 34,10,23, 7,30, 5, 6,18,33,41, 8,42,20,14,28,24,19)$ | |
| $ 42, 1 $ | $43$ | $42$ | $( 2,21,14, 3,41,27, 5,38,10, 9,32,19,17,20,37,33,39,30,22,34,16,43,24,31,42, 4,18,40, 7,35,36,13,26,28,25, 8,12, 6,15,23,11,29)$ | |
| $ 42, 1 $ | $43$ | $42$ | $( 2, 6,26,40,24,30,17,38,14,23,25,35,42,34,37, 9,41,29,12,13,18,43,39,19, 5, 21,15,28, 7,31,22,20,10, 3,11, 8,36, 4,16,33,32,27)$ | |
| $ 42, 1 $ | $43$ | $42$ | $( 2,34,15,33,25,19,36,38,18, 3,24,29,22, 6,37,28,32,35, 5, 4,14,43,11,30,12, 20,26, 9, 7,27,42,21,16,23,39, 8,17,13,10,40,41,31)$ | |
| $ 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 $ | $43$ | $14$ | $( 2,40,17,23,42, 9,12,43, 5,28,22, 3,36,33)( 4,32, 6,24,38,25,34,41,13,39,21, 7,20,11)( 8,16,27,26,30,14,35,37,29,18,19,15,31,10)$ | |
| $ 42, 1 $ | $43$ | $42$ | $( 2,29,11,23,15, 6,12, 8,25,28,26,13,36,35, 7,40,18, 4,42,31,24,43,16,34,22, 30,39,33,37,20,17,19,32, 9,10,38, 5,27,41, 3,14,21)$ | |
| $ 42, 1 $ | $43$ | $42$ | $( 2,31,41,40,10,13,17, 8,39,23,16,21,42,27, 7, 9,26,20,12,30,11,43,14, 4, 5, 35,32,28,37, 6,22,29,24, 3,18,38,36,19,25,33,15,34)$ | |
| $ 42, 1 $ | $43$ | $42$ | $( 2,35,39, 3,26,34, 5, 8,24, 9,15, 4,17,29, 7,33,14,13,22,27,25,43,10, 6,42, 19,11,40,37,21,36,30,41,28,16,38,12,31,32,23,18,20)$ | |
| $ 42, 1 $ | $43$ | $42$ | $( 2,27,32,33,16, 4,36, 8,11, 3,10,20,22,31, 7,28,15,21, 5,19,39,43,18,13,12, 29,41, 9,37,34,42,35,25,23,14,38,17,30,24,40,26, 6)$ | |
| $ 42, 1 $ | $43$ | $42$ | $( 2,30,25, 9,18,21,22, 8,32,40,14,34,12,19, 7, 3,16, 6,17,35,41,43,15,20,36, 27,24,23,37,13, 5,31,11,33,26,38,42,29,39,28,10, 4)$ | |
| $ 42, 1 $ | $43$ | $42$ | $( 2,19,24,28,14,20,42, 8,41,33,18, 6, 5,30, 7,23,10,34,36,29,32,43,26,21,17, 31,25, 3,37, 4,12,27,39,40,15,38,22,35,11, 9,16,13)$ | |
| $ 6, 6, 6, 6, 6, 6, 6, 1 $ | $43$ | $6$ | $( 2, 8, 7,43,37,38)( 3,15,13,42,30,32)( 4,22,19,41,23,26)( 5,29,25,40,16,20) ( 6,36,31,39, 9,14)(10,21,12,35,24,33)(11,28,18,34,17,27)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $1806=2 \cdot 3 \cdot 7 \cdot 43$ | 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: | 1806.1 | magma: IdentifyGroup(G);
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| Character table: | 43 x 43 character table |
magma: CharacterTable(G);