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Magma
magma: G := TransitiveGroup(42, 17);
Group action invariants
| Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $17$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3\times F_7$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,23,33,27,10,42)(2,24,34,28,9,41)(3,20,35,30,12,37)(4,19,36,29,11,38)(5,21,31,26,7,39)(6,22,32,25,8,40)(13,14)(15,16)(17,18), (1,14,29,41,8,19,33,6,18,28,39,12,24,32,3,15,26,38,10,21,35)(2,13,30,42,7,20,34,5,17,27,40,11,23,31,4,16,25,37,9,22,36) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ x 4 $6$: $C_6$ x 4 $9$: $C_3^2$ $18$: $C_6 \times C_3$ $42$: $F_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 7: $F_7$
Degree 14: $F_7$
Degree 21: 21T9
Low degree siblings
21T9 x 3, 42T17 x 2Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 7,13,25)( 8,14,26)( 9,16,27)(10,15,28)(11,17,30)(12,18,29)(19,38,35) (20,37,36)(21,39,32)(22,40,31)(23,42,34)(24,41,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 7,25,13)( 8,26,14)( 9,27,16)(10,28,15)(11,30,17)(12,29,18)(19,35,38) (20,36,37)(21,32,39)(22,31,40)(23,34,42)(24,33,41)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $7$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,21,13,39,25,32)( 8,22,14,40,26,31)( 9,24,16,41,27,33) (10,23,15,42,28,34)(11,19,17,38,30,35)(12,20,18,37,29,36)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $7$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,32,25,39,13,21)( 8,31,26,40,14,22)( 9,33,27,41,16,24) (10,34,28,42,15,23)(11,35,30,38,17,19)(12,36,29,37,18,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,39)( 8,40)( 9,41)(10,42)(11,38)(12,37)(13,32)(14,31) (15,34)(16,33)(17,35)(18,36)(19,30)(20,29)(21,25)(22,26)(23,28)(24,27)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7, 9,11)( 8,10,12)(13,16,17)(14,15,18)(19,21,24) (20,22,23)(25,27,30)(26,28,29)(31,34,36)(32,33,35)(37,40,42)(38,39,41)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,16,30)( 8,15,29)( 9,17,25)(10,18,26)(11,13,27) (12,14,28)(19,39,33)(20,40,34)(21,41,35)(22,42,36)(23,37,31)(24,38,32)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,27,17)( 8,28,18)( 9,30,13)(10,29,14)(11,25,16) (12,26,15)(19,32,41)(20,31,42)(21,33,38)(22,34,37)(23,36,40)(24,35,39)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7,24,17,39,27,35)( 8,23,18,40,28,36)( 9,19,13,41,30,32) (10,20,14,42,29,31)(11,21,16,38,25,33)(12,22,15,37,26,34)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7,33,30,39,16,19)( 8,34,29,40,15,20)( 9,35,25,41,17,21) (10,36,26,42,18,22)(11,32,27,38,13,24)(12,31,28,37,14,23)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7,41,11,39, 9,38)( 8,42,12,40,10,37)(13,33,17,32,16,35) (14,34,18,31,15,36)(19,25,24,30,21,27)(20,26,23,29,22,28)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,19,16,39,30,33)( 8,20,15,40,29,34)( 9,21,17,41,25,35) (10,22,18,42,26,36)(11,24,13,38,27,32)(12,23,14,37,28,31)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,35,27,39,17,24)( 8,36,28,40,18,23)( 9,32,30,41,13,19) (10,31,29,42,14,20)(11,33,25,38,16,21)(12,34,26,37,15,22)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,38, 9,39,11,41)( 8,37,10,40,12,42)(13,35,16,32,17,33) (14,36,15,31,18,34)(19,27,21,30,24,25)(20,28,22,29,23,26)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 6, 3)( 2, 5, 4)( 7,11, 9)( 8,12,10)(13,17,16)(14,18,15)(19,24,21) (20,23,22)(25,30,27)(26,29,28)(31,36,34)(32,35,33)(37,42,40)(38,41,39)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 6, 3)( 2, 5, 4)( 7,17,27)( 8,18,28)( 9,13,30)(10,14,29)(11,16,25) (12,15,26)(19,41,32)(20,42,31)(21,38,33)(22,37,34)(23,40,36)(24,39,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 6, 3)( 2, 5, 4)( 7,30,16)( 8,29,15)( 9,25,17)(10,26,18)(11,27,13) (12,28,14)(19,33,39)(20,34,40)(21,35,41)(22,36,42)(23,31,37)(24,32,38)$ |
| $ 21, 21 $ | $6$ | $21$ | $( 1, 8,18,24,26,35,41, 6,12,15,21,29,33,39, 3,10,14,19,28,32,38) ( 2, 7,17,23,25,36,42, 5,11,16,22,30,34,40, 4, 9,13,20,27,31,37)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1,10,15,24,28,33,41)( 2, 9,16,23,27,34,42)( 3,12,18,19,29,35,38) ( 4,11,17,20,30,36,37)( 5, 7,13,22,25,31,40)( 6, 8,14,21,26,32,39)$ |
| $ 21, 21 $ | $6$ | $21$ | $( 1,12,14,24,29,32,41, 3, 8,15,19,26,33,38, 6,10,18,21,28,35,39) ( 2,11,13,23,30,31,42, 4, 7,16,20,25,34,37, 5, 9,17,22,27,36,40)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $126=2 \cdot 3^{2} \cdot 7$ | 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: | 126.7 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);