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Magma
magma: G := TransitiveGroup(42, 9);
Group action invariants
| Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3\times D_{14}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,23)(2,24)(3,19)(4,20)(5,21)(6,22)(7,18)(8,17)(9,13)(10,14)(11,16)(12,15)(25,40)(26,39)(27,41)(28,42)(29,37)(30,38)(31,32)(33,34)(35,36), (1,21,38,18,31,9,27,6,20,41,15,36,7,25,3,23,40,14,34,12,29,2,22,37,17,32,10,28,5,19,42,16,35,8,26,4,24,39,13,33,11,30) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $12$: $C_6\times C_2$ $14$: $D_{7}$ $28$: $D_{14}$ $42$: 21T3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 7: $D_{7}$
Degree 14: $D_{14}$
Degree 21: 21T3
Low degree siblings
42T9Siblings 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, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 7,42)( 8,41)( 9,37)(10,38)(11,40)(12,39)(13,35)(14,36)(15,31)(16,32)(17,34) (18,33)(19,30)(20,29)(21,25)(22,26)(23,28)(24,27)$ |
| $ 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 $ | $7$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,41)( 8,42)( 9,38)(10,37)(11,39)(12,40)(13,36)(14,35) (15,32)(16,31)(17,33)(18,34)(19,29)(20,30)(21,26)(22,25)(23,27)(24,28)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,11)( 8, 9,12)(13,15,17)(14,16,18)(19,21,23) (20,22,24)(25,28,30)(26,27,29)(31,34,35)(32,33,36)(37,39,41)(38,40,42)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3 $ | $7$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7,38,11,42,10,40)( 8,37,12,41, 9,39)(13,31,17,35,15,34) (14,32,18,36,16,33)(19,25,23,30,21,28)(20,26,24,29,22,27)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7, 9,11, 8,10,12)(13,16,17,14,15,18)(19,22,23,20,21,24) (25,27,30,26,28,29)(31,33,35,32,34,36)(37,40,41,38,39,42)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7,37,11,41,10,39)( 8,38,12,42, 9,40)(13,32,17,36,15,33) (14,31,18,35,16,34)(19,26,23,29,21,27)(20,25,24,30,22,28)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,11,10)( 8,12, 9)(13,17,15)(14,18,16)(19,23,21) (20,24,22)(25,30,28)(26,29,27)(31,35,34)(32,36,33)(37,41,39)(38,42,40)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3 $ | $7$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7,40,10,42,11,38)( 8,39, 9,41,12,37)(13,34,15,35,17,31) (14,33,16,36,18,32)(19,28,21,30,23,25)(20,27,22,29,24,26)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7,12,10, 8,11, 9)(13,18,15,14,17,16)(19,24,21,20,23,22) (25,29,28,26,30,27)(31,36,34,32,35,33)(37,42,39,38,41,40)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7,39,10,41,11,37)( 8,40, 9,42,12,38)(13,33,15,36,17,32) (14,34,16,35,18,31)(19,27,21,29,23,26)(20,28,22,30,24,25)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 7,17,24,27,34,42)( 2, 8,18,23,28,33,41)( 3,10,13,20,29,35,38) ( 4, 9,14,19,30,36,37)( 5,11,15,22,26,31,40)( 6,12,16,21,25,32,39)$ |
| $ 14, 14, 14 $ | $2$ | $14$ | $( 1, 8,17,23,27,33,42, 2, 7,18,24,28,34,41)( 3, 9,13,19,29,36,38, 4,10,14,20, 30,35,37)( 5,12,15,21,26,32,40, 6,11,16,22,25,31,39)$ |
| $ 42 $ | $2$ | $42$ | $( 1, 9,15,23,29,32,42, 4,11,18,20,25,34,37, 5, 8,13,21,27,36,40, 2,10,16,24, 30,31,41, 3,12,17,19,26,33,38, 6, 7,14,22,28,35,39)$ |
| $ 21, 21 $ | $2$ | $21$ | $( 1,10,15,24,29,31,42, 3,11,17,20,26,34,38, 5, 7,13,22,27,35,40) ( 2, 9,16,23,30,32,41, 4,12,18,19,25,33,37, 6, 8,14,21,28,36,39)$ |
| $ 21, 21 $ | $2$ | $21$ | $( 1,11,13,24,26,35,42, 5,10,17,22,29,34,40, 3, 7,15,20,27,31,38) ( 2,12,14,23,25,36,41, 6, 9,18,21,30,33,39, 4, 8,16,19,28,32,37)$ |
| $ 42 $ | $2$ | $42$ | $( 1,12,13,23,26,36,42, 6,10,18,22,30,34,39, 3, 8,15,19,27,32,38, 2,11,14,24, 25,35,41, 5, 9,17,21,29,33,40, 4, 7,16,20,28,31,37)$ |
| $ 21, 21 $ | $2$ | $21$ | $( 1,13,26,42,10,22,34, 3,15,27,38,11,24,35, 5,17,29,40, 7,20,31) ( 2,14,25,41, 9,21,33, 4,16,28,37,12,23,36, 6,18,30,39, 8,19,32)$ |
| $ 42 $ | $2$ | $42$ | $( 1,14,26,41,10,21,34, 4,15,28,38,12,24,36, 5,18,29,39, 7,19,31, 2,13,25,42, 9,22,33, 3,16,27,37,11,23,35, 6,17,30,40, 8,20,32)$ |
| $ 21, 21 $ | $2$ | $21$ | $( 1,15,29,42,11,20,34, 5,13,27,40,10,24,31, 3,17,26,38, 7,22,35) ( 2,16,30,41,12,19,33, 6,14,28,39, 9,23,32, 4,18,25,37, 8,21,36)$ |
| $ 42 $ | $2$ | $42$ | $( 1,16,29,41,11,19,34, 6,13,28,40, 9,24,32, 3,18,26,37, 7,21,35, 2,15,30,42, 12,20,33, 5,14,27,39,10,23,31, 4,17,25,38, 8,22,36)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,17,27,42, 7,24,34)( 2,18,28,41, 8,23,33)( 3,13,29,38,10,20,35) ( 4,14,30,37, 9,19,36)( 5,15,26,40,11,22,31)( 6,16,25,39,12,21,32)$ |
| $ 14, 14, 14 $ | $2$ | $14$ | $( 1,18,27,41, 7,23,34, 2,17,28,42, 8,24,33)( 3,14,29,37,10,19,35, 4,13,30,38, 9,20,36)( 5,16,26,39,11,21,31, 6,15,25,40,12,22,32)$ |
| $ 42 $ | $2$ | $42$ | $( 1,19,40,18,35,12,27, 4,22,41,13,32, 7,30, 5,23,38,16,34, 9,26, 2,20,39,17, 36,11,28, 3,21,42,14,31, 8,29, 6,24,37,15,33,10,25)$ |
| $ 21, 21 $ | $2$ | $21$ | $( 1,20,40,17,35,11,27, 3,22,42,13,31, 7,29, 5,24,38,15,34,10,26) ( 2,19,39,18,36,12,28, 4,21,41,14,32, 8,30, 6,23,37,16,33, 9,25)$ |
| $ 42 $ | $2$ | $42$ | $( 1,21,38,18,31, 9,27, 6,20,41,15,36, 7,25, 3,23,40,14,34,12,29, 2,22,37,17, 32,10,28, 5,19,42,16,35, 8,26, 4,24,39,13,33,11,30)$ |
| $ 21, 21 $ | $2$ | $21$ | $( 1,22,38,17,31,10,27, 5,20,42,15,35, 7,26, 3,24,40,13,34,11,29) ( 2,21,37,18,32, 9,28, 6,19,41,16,36, 8,25, 4,23,39,14,33,12,30)$ |
| $ 14, 14, 14 $ | $2$ | $14$ | $( 1,23,42,18,34, 8,27, 2,24,41,17,33, 7,28)( 3,19,38,14,35, 9,29, 4,20,37,13, 36,10,30)( 5,21,40,16,31,12,26, 6,22,39,15,32,11,25)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,24,42,17,34, 7,27)( 2,23,41,18,33, 8,28)( 3,20,38,13,35,10,29) ( 4,19,37,14,36, 9,30)( 5,22,40,15,31,11,26)( 6,21,39,16,32,12,25)$ |
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: | 84.12 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);