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Magma
magma: G := TransitiveGroup(42, 6);
Group action invariants
| Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $6$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_3\times C_7$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $42$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,21,37,16,32,9,25,3,19,40,14,34,7,27)(2,22,38,15,31,10,26,4,20,39,13,33,8,28)(5,24,42,18,36,12,30,6,23,41,17,35,11,29), (1,29,7,35,14,41,19,6,25,12,32,18,37,24)(2,30,8,36,13,42,20,5,26,11,31,17,38,23)(3,28,9,33,16,39,21,4,27,10,34,15,40,22) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $7$: $C_7$ $14$: $C_{14}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $S_3$
Degree 7: $C_7$
Degree 14: $C_{14}$
Degree 21: 21T6
Low degree siblings
21T6Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| 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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 5)( 4, 6)( 7, 8)( 9,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23) (22,24)(25,26)(27,30)(28,29)(31,32)(33,35)(34,36)(37,38)(39,41)(40,42)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4, 5)( 2, 3, 6)( 7,10,11)( 8, 9,12)(13,16,18)(14,15,17)(19,22,23) (20,21,24)(25,28,30)(26,27,29)(31,34,35)(32,33,36)(37,39,42)(38,40,41)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1, 7,14,19,25,32,37)( 2, 8,13,20,26,31,38)( 3, 9,16,21,27,34,40) ( 4,10,15,22,28,33,39)( 5,11,17,23,30,36,42)( 6,12,18,24,29,35,41)$ |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1, 8,14,20,25,31,37, 2, 7,13,19,26,32,38)( 3,11,16,23,27,36,40, 5, 9,17,21, 30,34,42)( 4,12,15,24,28,35,39, 6,10,18,22,29,33,41)$ |
| $ 21, 21 $ | $2$ | $21$ | $( 1,10,17,19,28,36,37, 4,11,14,22,30,32,39, 5, 7,15,23,25,33,42) ( 2, 9,18,20,27,35,38, 3,12,13,21,29,31,40, 6, 8,16,24,26,34,41)$ |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1,13,25,38, 7,20,32, 2,14,26,37, 8,19,31)( 3,17,27,42, 9,23,34, 5,16,30,40, 11,21,36)( 4,18,28,41,10,24,33, 6,15,29,39,12,22,35)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,14,25,37, 7,19,32)( 2,13,26,38, 8,20,31)( 3,16,27,40, 9,21,34) ( 4,15,28,39,10,22,33)( 5,17,30,42,11,23,36)( 6,18,29,41,12,24,35)$ |
| $ 21, 21 $ | $2$ | $21$ | $( 1,15,30,37,10,23,32, 4,17,25,39,11,19,33, 5,14,28,42, 7,22,36) ( 2,16,29,38, 9,24,31, 3,18,26,40,12,20,34, 6,13,27,41, 8,21,35)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,19,37,14,32, 7,25)( 2,20,38,13,31, 8,26)( 3,21,40,16,34, 9,27) ( 4,22,39,15,33,10,28)( 5,23,42,17,36,11,30)( 6,24,41,18,35,12,29)$ |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1,20,37,13,32, 8,25, 2,19,38,14,31, 7,26)( 3,23,40,17,34,11,27, 5,21,42,16, 36, 9,30)( 4,24,39,18,33,12,28, 6,22,41,15,35,10,29)$ |
| $ 21, 21 $ | $2$ | $21$ | $( 1,22,42,14,33,11,25, 4,23,37,15,36, 7,28, 5,19,39,17,32,10,30) ( 2,21,41,13,34,12,26, 3,24,38,16,35, 8,27, 6,20,40,18,31, 9,29)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,25, 7,32,14,37,19)( 2,26, 8,31,13,38,20)( 3,27, 9,34,16,40,21) ( 4,28,10,33,15,39,22)( 5,30,11,36,17,42,23)( 6,29,12,35,18,41,24)$ |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1,26, 7,31,14,38,19, 2,25, 8,32,13,37,20)( 3,30, 9,36,16,42,21, 5,27,11,34, 17,40,23)( 4,29,10,35,15,41,22, 6,28,12,33,18,39,24)$ |
| $ 21, 21 $ | $2$ | $21$ | $( 1,28,11,32,15,42,19, 4,30, 7,33,17,37,22, 5,25,10,36,14,39,23) ( 2,27,12,31,16,41,20, 3,29, 8,34,18,38,21, 6,26, 9,35,13,40,24)$ |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1,31,19, 8,37,26,14, 2,32,20, 7,38,25,13)( 3,36,21,11,40,30,16, 5,34,23, 9, 42,27,17)( 4,35,22,12,39,29,15, 6,33,24,10,41,28,18)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,32,19, 7,37,25,14)( 2,31,20, 8,38,26,13)( 3,34,21, 9,40,27,16) ( 4,33,22,10,39,28,15)( 5,36,23,11,42,30,17)( 6,35,24,12,41,29,18)$ |
| $ 21, 21 $ | $2$ | $21$ | $( 1,33,23, 7,39,30,14, 4,36,19,10,42,25,15, 5,32,22,11,37,28,17) ( 2,34,24, 8,40,29,13, 3,35,20, 9,41,26,16, 6,31,21,12,38,27,18)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,37,32,25,19,14, 7)( 2,38,31,26,20,13, 8)( 3,40,34,27,21,16, 9) ( 4,39,33,28,22,15,10)( 5,42,36,30,23,17,11)( 6,41,35,29,24,18,12)$ |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1,38,32,26,19,13, 7, 2,37,31,25,20,14, 8)( 3,42,34,30,21,17, 9, 5,40,36,27, 23,16,11)( 4,41,33,29,22,18,10, 6,39,35,28,24,15,12)$ |
| $ 21, 21 $ | $2$ | $21$ | $( 1,39,36,25,22,17, 7, 4,42,32,28,23,14,10, 5,37,33,30,19,15,11) ( 2,40,35,26,21,18, 8, 3,41,31,27,24,13, 9, 6,38,34,29,20,16,12)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $42=2 \cdot 3 \cdot 7$ | 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: | 42.3 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);