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Magma
magma: G := TransitiveGroup(42, 34);
Group action invariants
| Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_7\times S_4$ | ||
| 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: | (1,5,4)(2,6,3)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,24,22)(20,23,21)(25,30,28)(26,29,27)(31,35,33)(32,36,34)(37,41,39)(38,42,40), (1,37,31,26,20,13,8,2,38,32,25,19,14,7)(3,41,34,29,22,17,9,6,39,36,27,24,15,11)(4,42,33,30,21,18,10,5,40,35,28,23,16,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $7$: $C_7$ $14$: $C_{14}$ $24$: $S_4$ $42$: 21T6 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: $S_4$
Degree 7: $C_7$
Degree 14: None
Degree 21: 21T6
Low degree siblings
28T31, 42T35Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(15,16)(17,18)(21,22)(23,24)(27,28)(29,30)(33,34) (35,36)(39,40)(41,42)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $4$ | $( 3, 5, 4, 6)( 9,12,10,11)(15,18,16,17)(21,24,22,23)(27,30,28,29)(33,36,34,35) (39,42,40,41)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 2)( 3, 5)( 4, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,24) (22,23)(25,26)(27,30)(28,29)(31,32)(33,36)(34,35)(37,38)(39,42)(40,41)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,11)( 8, 9,12)(13,16,17)(14,15,18)(19,21,24) (20,22,23)(25,27,30)(26,28,29)(31,34,35)(32,33,36)(37,40,41)(38,39,42)$ |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1, 7,14,19,25,32,38, 2, 8,13,20,26,31,37)( 3, 9,15,22,27,34,39) ( 4,10,16,21,28,33,40)( 5,11,18,24,30,36,42, 6,12,17,23,29,35,41)$ |
| $ 14, 14, 14 $ | $6$ | $14$ | $( 1, 7,14,19,25,32,38, 2, 8,13,20,26,31,37)( 3,11,15,24,27,36,39, 6, 9,17,22, 29,34,41)( 4,12,16,23,28,35,40, 5,10,18,21,30,33,42)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1, 8,14,20,25,31,38)( 2, 7,13,19,26,32,37)( 3, 9,15,22,27,34,39) ( 4,10,16,21,28,33,40)( 5,12,18,23,30,35,42)( 6,11,17,24,29,36,41)$ |
| $ 28, 7, 7 $ | $6$ | $28$ | $( 1, 8,14,20,25,31,38)( 2, 7,13,19,26,32,37)( 3,11,16,23,27,36,40, 5, 9,17,21, 30,34,41, 4,12,15,24,28,35,39, 6,10,18,22,29,33,42)$ |
| $ 21, 21 $ | $8$ | $21$ | $( 1, 9,17,20,27,36,38, 3,11,14,22,29,31,39, 6, 8,15,24,25,34,41) ( 2,10,18,19,28,35,37, 4,12,13,21,30,32,40, 5, 7,16,23,26,33,42)$ |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,13,25,37, 8,19,31, 2,14,26,38, 7,20,32)( 3,15,27,39, 9,22,34) ( 4,16,28,40,10,21,33)( 5,17,30,41,12,24,35, 6,18,29,42,11,23,36)$ |
| $ 14, 14, 14 $ | $6$ | $14$ | $( 1,13,25,37, 8,19,31, 2,14,26,38, 7,20,32)( 3,17,27,41, 9,24,34, 6,15,29,39, 11,22,36)( 4,18,28,42,10,23,33, 5,16,30,40,12,21,35)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,14,25,38, 8,20,31)( 2,13,26,37, 7,19,32)( 3,15,27,39, 9,22,34) ( 4,16,28,40,10,21,33)( 5,18,30,42,12,23,35)( 6,17,29,41,11,24,36)$ |
| $ 28, 7, 7 $ | $6$ | $28$ | $( 1,14,25,38, 8,20,31)( 2,13,26,37, 7,19,32)( 3,17,28,42, 9,24,33, 5,15,29,40, 12,22,36, 4,18,27,41,10,23,34, 6,16,30,39,11,21,35)$ |
| $ 21, 21 $ | $8$ | $21$ | $( 1,15,29,38, 9,24,31, 3,17,25,39,11,20,34, 6,14,27,41, 8,22,36) ( 2,16,30,37,10,23,32, 4,18,26,40,12,19,33, 5,13,28,42, 7,21,35)$ |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,19,38,13,31, 7,25, 2,20,37,14,32, 8,26)( 3,21,39,16,34,10,27, 4,22,40,15, 33, 9,28)( 5,23,42,18,35,12,30)( 6,24,41,17,36,11,29)$ |
| $ 14, 14, 14 $ | $6$ | $14$ | $( 1,19,38,13,31, 7,25, 2,20,37,14,32, 8,26)( 3,23,39,18,34,12,27, 5,22,42,15, 35, 9,30)( 4,24,40,17,33,11,28, 6,21,41,16,36,10,29)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,20,38,14,31, 8,25)( 2,19,37,13,32, 7,26)( 3,22,39,15,34, 9,27) ( 4,21,40,16,33,10,28)( 5,23,42,18,35,12,30)( 6,24,41,17,36,11,29)$ |
| $ 28, 7, 7 $ | $6$ | $28$ | $( 1,20,38,14,31, 8,25)( 2,19,37,13,32, 7,26)( 3,23,40,17,34,12,28, 6,22,42,16, 36, 9,30, 4,24,39,18,33,11,27, 5,21,41,15,35,10,29)$ |
| $ 21, 21 $ | $8$ | $21$ | $( 1,21,41,14,33,11,25, 4,24,38,16,36, 8,28, 6,20,40,17,31,10,29) ( 2,22,42,13,34,12,26, 3,23,37,15,35, 7,27, 5,19,39,18,32, 9,30)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,25, 8,31,14,38,20)( 2,26, 7,32,13,37,19)( 3,27, 9,34,15,39,22) ( 4,28,10,33,16,40,21)( 5,30,12,35,18,42,23)( 6,29,11,36,17,41,24)$ |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,25, 8,31,14,38,20)( 2,26, 7,32,13,37,19)( 3,28, 9,33,15,40,22, 4,27,10,34, 16,39,21)( 5,29,12,36,18,41,23, 6,30,11,35,17,42,24)$ |
| $ 28, 7, 7 $ | $6$ | $28$ | $( 1,25, 8,31,14,38,20)( 2,26, 7,32,13,37,19)( 3,29,10,35,15,41,21, 5,27,11,33, 18,39,24, 4,30, 9,36,16,42,22, 6,28,12,34,17,40,23)$ |
| $ 14, 14, 14 $ | $6$ | $14$ | $( 1,26, 8,32,14,37,20, 2,25, 7,31,13,38,19)( 3,29, 9,36,15,41,22, 6,27,11,34, 17,39,24)( 4,30,10,35,16,42,21, 5,28,12,33,18,40,23)$ |
| $ 21, 21 $ | $8$ | $21$ | $( 1,27,11,31,15,41,20, 3,29, 8,34,17,38,22, 6,25, 9,36,14,39,24) ( 2,28,12,32,16,42,19, 4,30, 7,33,18,37,21, 5,26,10,35,13,40,23)$ |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,31,20, 8,38,25,14)( 2,32,19, 7,37,26,13)( 3,33,22,10,39,28,15, 4,34,21, 9, 40,27,16)( 5,36,23,11,42,29,18, 6,35,24,12,41,30,17)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,31,20, 8,38,25,14)( 2,32,19, 7,37,26,13)( 3,34,22, 9,39,27,15) ( 4,33,21,10,40,28,16)( 5,35,23,12,42,30,18)( 6,36,24,11,41,29,17)$ |
| $ 28, 7, 7 $ | $6$ | $28$ | $( 1,31,20, 8,38,25,14)( 2,32,19, 7,37,26,13)( 3,35,21,11,39,30,16, 6,34,23,10, 41,27,18, 4,36,22,12,40,29,15, 5,33,24, 9,42,28,17)$ |
| $ 14, 14, 14 $ | $6$ | $14$ | $( 1,32,20, 7,38,26,14, 2,31,19, 8,37,25,13)( 3,35,22,12,39,30,15, 5,34,23, 9, 42,27,18)( 4,36,21,11,40,29,16, 6,33,24,10,41,28,17)$ |
| $ 21, 21 $ | $8$ | $21$ | $( 1,33,24, 8,40,29,14, 4,36,20,10,41,25,16, 6,31,21,11,38,28,17) ( 2,34,23, 7,39,30,13, 3,35,19, 9,42,26,15, 5,32,22,12,37,27,18)$ |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,37,31,26,20,13, 8, 2,38,32,25,19,14, 7)( 3,39,34,27,22,15, 9) ( 4,40,33,28,21,16,10)( 5,41,35,29,23,17,12, 6,42,36,30,24,18,11)$ |
| $ 14, 14, 14 $ | $6$ | $14$ | $( 1,37,31,26,20,13, 8, 2,38,32,25,19,14, 7)( 3,41,34,29,22,17, 9, 6,39,36,27, 24,15,11)( 4,42,33,30,21,18,10, 5,40,35,28,23,16,12)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,38,31,25,20,14, 8)( 2,37,32,26,19,13, 7)( 3,39,34,27,22,15, 9) ( 4,40,33,28,21,16,10)( 5,42,35,30,23,18,12)( 6,41,36,29,24,17,11)$ |
| $ 28, 7, 7 $ | $6$ | $28$ | $( 1,38,31,25,20,14, 8)( 2,37,32,26,19,13, 7)( 3,41,33,30,22,17,10, 5,39,36,28, 23,15,11, 4,42,34,29,21,18, 9, 6,40,35,27,24,16,12)$ |
| $ 21, 21 $ | $8$ | $21$ | $( 1,39,36,25,22,17, 8, 3,41,31,27,24,14, 9, 6,38,34,29,20,15,11) ( 2,40,35,26,21,18, 7, 4,42,32,28,23,13,10, 5,37,33,30,19,16,12)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $168=2^{3} \cdot 3 \cdot 7$ | 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: | 168.45 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);