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Magma
magma: G := TransitiveGroup(42, 12);
Group action invariants
| Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_3\times C_{14}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $14$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,18,27,37,11,22,31,6,16,26,41,10,20,35,3,13,30,39,8,24,34,2,17,28,38,12,21,32,5,15,25,42,9,19,36,4,14,29,40,7,23,33), (1,6)(2,5)(3,4)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,35)(32,36)(33,34)(37,41)(38,42)(39,40) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $7$: $C_7$ $12$: $D_{6}$ $14$: $C_{14}$ x 3 $28$: 28T2 $42$: 21T6 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $D_{6}$
Degree 7: $C_7$
Degree 14: $C_{14}$
Degree 21: 21T6
Low degree siblings
42T12Siblings are shown 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 $ | $1$ | $1$ | $()$ | |
| $ 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 $ | $3$ | $2$ | $( 3, 5)( 4, 6)( 9,11)(10,12)(15,18)(16,17)(21,23)(22,24)(27,30)(28,29)(33,35) (34,36)(39,42)(40,41)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 6)( 4, 5)( 7, 8)( 9,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,24) (22,23)(25,26)(27,29)(28,30)(31,32)(33,36)(34,35)(37,38)(39,41)(40,42)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,15,18)(14,16,17)(19,22,24) (20,21,23)(25,27,30)(26,28,29)(31,34,36)(32,33,35)(37,39,42)(38,40,41)$ | |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7, 9,12, 8,10,11)(13,16,18,14,15,17)(19,21,24,20,22,23) (25,28,30,26,27,29)(31,33,36,32,34,35)(37,40,42,38,39,41)$ | |
| $ 14, 14, 14 $ | $1$ | $14$ | $( 1, 7,14,19,25,32,38, 2, 8,13,20,26,31,37)( 3,10,16,22,27,33,40, 4, 9,15,21, 28,34,39)( 5,12,17,24,30,35,41, 6,11,18,23,29,36,42)$ | |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1, 7,14,19,25,32,38, 2, 8,13,20,26,31,37)( 3,12,16,24,27,35,40, 6, 9,18,21, 29,34,42)( 4,11,15,23,28,36,39, 5,10,17,22,30,33,41)$ | |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1, 8,14,20,25,31,38)( 2, 7,13,19,26,32,37)( 3, 9,16,21,27,34,40) ( 4,10,15,22,28,33,39)( 5,11,17,23,30,36,41)( 6,12,18,24,29,35,42)$ | |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1, 8,14,20,25,31,38)( 2, 7,13,19,26,32,37)( 3,11,16,23,27,36,40, 5, 9,17,21, 30,34,41)( 4,12,15,24,28,35,39, 6,10,18,22,29,33,42)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1, 9,17,20,27,36,38, 3,11,14,21,30,31,40, 5, 8,16,23,25,34,41) ( 2,10,18,19,28,35,37, 4,12,13,22,29,32,39, 6, 7,15,24,26,33,42)$ | |
| $ 42 $ | $2$ | $42$ | $( 1,10,17,19,27,35,38, 4,11,13,21,29,31,39, 5, 7,16,24,25,33,41, 2, 9,18,20, 28,36,37, 3,12,14,22,30,32,40, 6, 8,15,23,26,34,42)$ | |
| $ 14, 14, 14 $ | $1$ | $14$ | $( 1,13,25,37, 8,19,31, 2,14,26,38, 7,20,32)( 3,15,27,39, 9,22,34, 4,16,28,40, 10,21,33)( 5,18,30,42,11,24,36, 6,17,29,41,12,23,35)$ | |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1,13,25,37, 8,19,31, 2,14,26,38, 7,20,32)( 3,18,27,42, 9,24,34, 6,16,29,40, 12,21,35)( 4,17,28,41,10,23,33, 5,15,30,39,11,22,36)$ | |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,14,25,38, 8,20,31)( 2,13,26,37, 7,19,32)( 3,16,27,40, 9,21,34) ( 4,15,28,39,10,22,33)( 5,17,30,41,11,23,36)( 6,18,29,42,12,24,35)$ | |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,14,25,38, 8,20,31)( 2,13,26,37, 7,19,32)( 3,17,27,41, 9,23,34, 5,16,30,40, 11,21,36)( 4,18,28,42,10,24,33, 6,15,29,39,12,22,35)$ | |
| $ 42 $ | $2$ | $42$ | $( 1,15,30,37, 9,24,31, 4,17,26,40,12,20,33, 5,13,27,42, 8,22,36, 2,16,29,38, 10,23,32, 3,18,25,39,11,19,34, 6,14,28,41, 7,21,35)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,16,30,38, 9,23,31, 3,17,25,40,11,20,34, 5,14,27,41, 8,21,36) ( 2,15,29,37,10,24,32, 4,18,26,39,12,19,33, 6,13,28,42, 7,22,35)$ | |
| $ 14, 14, 14 $ | $1$ | $14$ | $( 1,19,38,13,31, 7,25, 2,20,37,14,32, 8,26)( 3,22,40,15,34,10,27, 4,21,39,16, 33, 9,28)( 5,24,41,18,36,12,30, 6,23,42,17,35,11,29)$ | |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1,19,38,13,31, 7,25, 2,20,37,14,32, 8,26)( 3,24,40,18,34,12,27, 6,21,42,16, 35, 9,29)( 4,23,39,17,33,11,28, 5,22,41,15,36,10,30)$ | |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,20,38,14,31, 8,25)( 2,19,37,13,32, 7,26)( 3,21,40,16,34, 9,27) ( 4,22,39,15,33,10,28)( 5,23,41,17,36,11,30)( 6,24,42,18,35,12,29)$ | |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,20,38,14,31, 8,25)( 2,19,37,13,32, 7,26)( 3,23,40,17,34,11,27, 5,21,41,16, 36, 9,30)( 4,24,39,18,33,12,28, 6,22,42,15,35,10,29)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,21,41,14,34,11,25, 3,23,38,16,36, 8,27, 5,20,40,17,31, 9,30) ( 2,22,42,13,33,12,26, 4,24,37,15,35, 7,28, 6,19,39,18,32,10,29)$ | |
| $ 42 $ | $2$ | $42$ | $( 1,22,41,13,34,12,25, 4,23,37,16,35, 8,28, 5,19,40,18,31,10,30, 2,21,42,14, 33,11,26, 3,24,38,15,36, 7,27, 6,20,39,17,32, 9,29)$ | |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,25, 8,31,14,38,20)( 2,26, 7,32,13,37,19)( 3,27, 9,34,16,40,21) ( 4,28,10,33,15,39,22)( 5,30,11,36,17,41,23)( 6,29,12,35,18,42,24)$ | |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,25, 8,31,14,38,20)( 2,26, 7,32,13,37,19)( 3,30, 9,36,16,41,21, 5,27,11,34, 17,40,23)( 4,29,10,35,15,42,22, 6,28,12,33,18,39,24)$ | |
| $ 14, 14, 14 $ | $1$ | $14$ | $( 1,26, 8,32,14,37,20, 2,25, 7,31,13,38,19)( 3,28, 9,33,16,39,21, 4,27,10,34, 15,40,22)( 5,29,11,35,17,42,23, 6,30,12,36,18,41,24)$ | |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1,26, 8,32,14,37,20, 2,25, 7,31,13,38,19)( 3,29, 9,35,16,42,21, 6,27,12,34, 18,40,24)( 4,30,10,36,15,41,22, 5,28,11,33,17,39,23)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,27,11,31,16,41,20, 3,30, 8,34,17,38,21, 5,25, 9,36,14,40,23) ( 2,28,12,32,15,42,19, 4,29, 7,33,18,37,22, 6,26,10,35,13,39,24)$ | |
| $ 42 $ | $2$ | $42$ | $( 1,28,11,32,16,42,20, 4,30, 7,34,18,38,22, 5,26, 9,35,14,39,23, 2,27,12,31, 15,41,19, 3,29, 8,33,17,37,21, 6,25,10,36,13,40,24)$ | |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,31,20, 8,38,25,14)( 2,32,19, 7,37,26,13)( 3,34,21, 9,40,27,16) ( 4,33,22,10,39,28,15)( 5,36,23,11,41,30,17)( 6,35,24,12,42,29,18)$ | |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,31,20, 8,38,25,14)( 2,32,19, 7,37,26,13)( 3,36,21,11,40,30,16, 5,34,23, 9, 41,27,17)( 4,35,22,12,39,29,15, 6,33,24,10,42,28,18)$ | |
| $ 14, 14, 14 $ | $1$ | $14$ | $( 1,32,20, 7,38,26,14, 2,31,19, 8,37,25,13)( 3,33,21,10,40,28,16, 4,34,22, 9, 39,27,15)( 5,35,23,12,41,29,17, 6,36,24,11,42,30,18)$ | |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1,32,20, 7,38,26,14, 2,31,19, 8,37,25,13)( 3,35,21,12,40,29,16, 6,34,24, 9, 42,27,18)( 4,36,22,11,39,30,15, 5,33,23,10,41,28,17)$ | |
| $ 42 $ | $2$ | $42$ | $( 1,33,23, 7,40,29,14, 4,36,19, 9,42,25,15, 5,32,21,12,38,28,17, 2,34,24, 8, 39,30,13, 3,35,20,10,41,26,16, 6,31,22,11,37,27,18)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,34,23, 8,40,30,14, 3,36,20, 9,41,25,16, 5,31,21,11,38,27,17) ( 2,33,24, 7,39,29,13, 4,35,19,10,42,26,15, 6,32,22,12,37,28,18)$ | |
| $ 14, 14, 14 $ | $1$ | $14$ | $( 1,37,31,26,20,13, 8, 2,38,32,25,19,14, 7)( 3,39,34,28,21,15, 9, 4,40,33,27, 22,16,10)( 5,42,36,29,23,18,11, 6,41,35,30,24,17,12)$ | |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1,37,31,26,20,13, 8, 2,38,32,25,19,14, 7)( 3,42,34,29,21,18, 9, 6,40,35,27, 24,16,12)( 4,41,33,30,22,17,10, 5,39,36,28,23,15,11)$ | |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,38,31,25,20,14, 8)( 2,37,32,26,19,13, 7)( 3,40,34,27,21,16, 9) ( 4,39,33,28,22,15,10)( 5,41,36,30,23,17,11)( 6,42,35,29,24,18,12)$ | |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,38,31,25,20,14, 8)( 2,37,32,26,19,13, 7)( 3,41,34,30,21,17, 9, 5,40,36,27, 23,16,11)( 4,42,33,29,22,18,10, 6,39,35,28,24,15,12)$ | |
| $ 42 $ | $2$ | $42$ | $( 1,39,36,26,21,18, 8, 4,41,32,27,24,14,10, 5,37,34,29,20,15,11, 2,40,35,25, 22,17, 7, 3,42,31,28,23,13, 9, 6,38,33,30,19,16,12)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,40,36,25,21,17, 8, 3,41,31,27,23,14, 9, 5,38,34,30,20,16,11) ( 2,39,35,26,22,18, 7, 4,42,32,28,24,13,10, 6,37,33,29,19,15,12)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $84=2^{2} \cdot 3 \cdot 7$ | 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: | 84.13 | magma: IdentifyGroup(G);
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| Character table: | 42 x 42 character table |
magma: CharacterTable(G);