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Magma
magma: G := TransitiveGroup(42, 44);
Group action invariants
| Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_3\times F_7$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,11,26,41,32,18)(2,12,25,42,31,17)(3,9,28,39,34,15)(4,10,27,40,33,16)(5,7,29,38,35,14)(6,8,30,37,36,13)(19,23)(20,24)(21,22), (1,21,13,3,20,16)(2,22,14,4,19,15)(5,23,17)(6,24,18)(7,33,38,9,31,39)(8,34,37,10,32,40)(11,36,41)(12,35,42)(25,27)(26,28) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $36$: $C_6\times S_3$ $42$: $F_7$ $84$: $F_7 \times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $D_{6}$
Degree 7: $F_7$
Degree 14: $F_7$
Degree 21: 21T15
Low degree siblings
21T15, 42T43, 42T45, 42T52Siblings 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 $ | $7$ | $3$ | $( 7,14,25)( 8,13,26)( 9,15,27)(10,16,28)(11,18,30)(12,17,29)(19,38,31) (20,37,32)(21,40,34)(22,39,33)(23,42,35)(24,41,36)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 7,25,14)( 8,26,13)( 9,27,15)(10,28,16)(11,30,18)(12,29,17)(19,31,38) (20,32,37)(21,34,40)(22,33,39)(23,35,42)(24,36,41)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3, 5)( 4, 6)( 9,11)(10,12)(15,18)(16,17)(21,23)(22,24)(27,30)(28,29)(33,36) (34,35)(39,41)(40,42)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 1, 1 $ | $21$ | $6$ | $( 3, 5)( 4, 6)( 7,14,25)( 8,13,26)( 9,18,27,11,15,30)(10,17,28,12,16,29) (19,38,31)(20,37,32)(21,42,34,23,40,35)(22,41,33,24,39,36)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 1, 1 $ | $21$ | $6$ | $( 3, 5)( 4, 6)( 7,25,14)( 8,26,13)( 9,30,15,11,27,18)(10,29,16,12,28,17) (19,31,38)(20,32,37)(21,35,40,23,34,42)(22,36,39,24,33,41)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $7$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,20,14,37,25,32)( 8,19,13,38,26,31)( 9,21,15,40,27,34) (10,22,16,39,28,33)(11,23,18,42,30,35)(12,24,17,41,29,36)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $7$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,32,25,37,14,20)( 8,31,26,38,13,19)( 9,34,27,40,15,21) (10,33,28,39,16,22)(11,35,30,42,18,23)(12,36,29,41,17,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,37)( 8,38)( 9,40)(10,39)(11,42)(12,41)(13,31)(14,32) (15,34)(16,33)(17,36)(18,35)(19,26)(20,25)(21,27)(22,28)(23,30)(24,29)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $21$ | $6$ | $( 1, 2)( 3, 6)( 4, 5)( 7,20,14,37,25,32)( 8,19,13,38,26,31)( 9,23,15,42,27,35) (10,24,16,41,28,36)(11,21,18,40,30,34)(12,22,17,39,29,33)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $21$ | $6$ | $( 1, 2)( 3, 6)( 4, 5)( 7,32,25,37,14,20)( 8,31,26,38,13,19)( 9,35,27,42,15,23) (10,36,28,41,16,24)(11,34,30,40,18,21)(12,33,29,39,17,22)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $21$ | $2$ | $( 1, 2)( 3, 6)( 4, 5)( 7,37)( 8,38)( 9,42)(10,41)(11,40)(12,39)(13,31)(14,32) (15,35)(16,36)(17,33)(18,34)(19,26)(20,25)(21,30)(22,29)(23,27)(24,28)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12)(13,16,17)(14,15,18)(19,22,24) (20,21,23)(25,27,30)(26,28,29)(31,33,36)(32,34,35)(37,40,42)(38,39,41)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,15,30)( 8,16,29)( 9,18,25)(10,17,26)(11,14,27) (12,13,28)(19,39,36)(20,40,35)(21,42,32)(22,41,31)(23,37,34)(24,38,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,27,18)( 8,28,17)( 9,30,14)(10,29,13)(11,25,15) (12,26,16)(19,33,41)(20,34,42)(21,35,37)(22,36,38)(23,32,40)(24,31,39)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $14$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7,21,18,37,27,35)( 8,22,17,38,28,36)( 9,23,14,40,30,32) (10,24,13,39,29,31)(11,20,15,42,25,34)(12,19,16,41,26,33)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $14$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7,34,30,37,15,23)( 8,33,29,38,16,24)( 9,35,25,40,18,20) (10,36,26,39,17,19)(11,32,27,42,14,21)(12,31,28,41,13,22)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $14$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7,40,11,37, 9,42)( 8,39,12,38,10,41)(13,33,17,31,16,36) (14,34,18,32,15,35)(19,28,24,26,22,29)(20,27,23,25,21,30)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 8,13,20,26,32,37)( 2, 7,14,19,25,31,38)( 3,10,16,21,28,34,40) ( 4, 9,15,22,27,33,39)( 5,12,17,23,29,35,42)( 6,11,18,24,30,36,41)$ |
| $ 14, 14, 7, 7 $ | $18$ | $14$ | $( 1, 8,13,20,26,32,37)( 2, 7,14,19,25,31,38)( 3,12,16,23,28,35,40, 5,10,17,21, 29,34,42)( 4,11,15,24,27,36,39, 6, 9,18,22,30,33,41)$ |
| $ 21, 21 $ | $12$ | $21$ | $( 1,10,17,20,28,35,37, 3,12,13,21,29,32,40, 5, 8,16,23,26,34,42) ( 2, 9,18,19,27,36,38, 4,11,14,22,30,31,39, 6, 7,15,24,25,33,41)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $252=2^{2} \cdot 3^{2} \cdot 7$ | 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: | 252.26 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);