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Magma
magma: G := TransitiveGroup(42, 27);
Group action invariants
Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{14}:A_4$ | ||
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: | (1,35,21,7,38,28,13)(2,36,22,8,37,27,14)(3,31,24,9,40,30,16,4,32,23,10,39,29,15)(5,33,20,12,42,25,17,6,34,19,11,41,26,18), (1,23,17,2,24,18)(3,19,13,4,20,14)(5,22,16,6,21,15)(7,31,42,8,32,41)(9,34,37,10,33,38)(11,36,40,12,35,39)(25,28,30,26,27,29) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $12$: $A_4$ $21$: $C_7:C_3$ $24$: $A_4\times C_2$ $42$: $(C_7:C_3) \times C_2$ $84$: 28T16 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: $A_4\times C_2$
Degree 7: $C_7:C_3$
Degree 14: None
Degree 21: 21T2
Low degree siblings
There are no siblings 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, 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 $ | $3$ | $2$ | $( 5, 6)(11,12)(17,18)(19,20)(25,26)(33,34)(41,42)$ |
$ 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 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(15,16)(17,18)(19,20)(23,24)(25,26)(29,30)(31,32) (33,34)(39,40)(41,42)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,16,26)( 8,15,25)( 9,18,27)(10,17,28)(11,13,29) (12,14,30)(19,37,31)(20,38,32)(21,40,34)(22,39,33)(23,41,36)(24,42,35)$ |
$ 6, 6, 6, 6, 6, 6, 6 $ | $28$ | $6$ | $( 1, 3, 5, 2, 4, 6)( 7,16,26, 8,15,25)( 9,18,28,10,17,27)(11,14,30,12,13,29) (19,38,32,20,37,31)(21,40,34,22,39,33)(23,41,35,24,42,36)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,26,16)( 8,25,15)( 9,27,18)(10,28,17)(11,29,13) (12,30,14)(19,31,37)(20,32,38)(21,34,40)(22,33,39)(23,36,41)(24,35,42)$ |
$ 6, 6, 6, 6, 6, 6, 6 $ | $28$ | $6$ | $( 1, 5, 4, 2, 6, 3)( 7,26,15, 8,25,16)( 9,27,18,10,28,17)(11,30,14,12,29,13) (19,32,38,20,31,37)(21,34,39,22,33,40)(23,36,41,24,35,42)$ |
$ 14, 7, 7, 7, 7 $ | $3$ | $14$ | $( 1, 7,13,21,28,35,38)( 2, 8,14,22,27,36,37)( 3, 9,16,23,29,31,40, 4,10,15,24, 30,32,39)( 5,11,17,20,26,34,42)( 6,12,18,19,25,33,41)$ |
$ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1, 7,13,21,28,35,38)( 2, 8,14,22,27,36,37)( 3, 9,16,23,29,31,40, 4,10,15,24, 30,32,39)( 5,12,17,19,26,33,42, 6,11,18,20,25,34,41)$ |
$ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1, 7,13,21,28,35,38)( 2, 8,14,22,27,36,37)( 3,10,16,24,29,32,40) ( 4, 9,15,23,30,31,39)( 5,11,17,20,26,34,42)( 6,12,18,19,25,33,41)$ |
$ 14, 7, 7, 7, 7 $ | $3$ | $14$ | $( 1, 7,13,21,28,35,38)( 2, 8,14,22,27,36,37)( 3,10,16,24,29,32,40) ( 4, 9,15,23,30,31,39)( 5,12,17,19,26,33,42, 6,11,18,20,25,34,41)$ |
$ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1, 8,13,22,28,36,38, 2, 7,14,21,27,35,37)( 3, 9,16,23,29,31,40, 4,10,15,24, 30,32,39)( 5,11,17,20,26,34,42)( 6,12,18,19,25,33,41)$ |
$ 14, 14, 14 $ | $3$ | $14$ | $( 1, 8,13,22,28,36,38, 2, 7,14,21,27,35,37)( 3, 9,16,23,29,31,40, 4,10,15,24, 30,32,39)( 5,12,17,19,26,33,42, 6,11,18,20,25,34,41)$ |
$ 14, 7, 7, 7, 7 $ | $3$ | $14$ | $( 1, 8,13,22,28,36,38, 2, 7,14,21,27,35,37)( 3,10,16,24,29,32,40) ( 4, 9,15,23,30,31,39)( 5,11,17,20,26,34,42)( 6,12,18,19,25,33,41)$ |
$ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1, 8,13,22,28,36,38, 2, 7,14,21,27,35,37)( 3,10,16,24,29,32,40) ( 4, 9,15,23,30,31,39)( 5,12,17,19,26,33,42, 6,11,18,20,25,34,41)$ |
$ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,21,38,13,35, 7,28)( 2,22,37,14,36, 8,27)( 3,23,40,15,32, 9,29, 4,24,39,16, 31,10,30)( 5,19,42,18,34,12,26, 6,20,41,17,33,11,25)$ |
$ 14, 7, 7, 7, 7 $ | $3$ | $14$ | $( 1,21,38,13,35, 7,28)( 2,22,37,14,36, 8,27)( 3,23,40,15,32, 9,29, 4,24,39,16, 31,10,30)( 5,20,42,17,34,11,26)( 6,19,41,18,33,12,25)$ |
$ 14, 7, 7, 7, 7 $ | $3$ | $14$ | $( 1,21,38,13,35, 7,28)( 2,22,37,14,36, 8,27)( 3,24,40,16,32,10,29) ( 4,23,39,15,31, 9,30)( 5,19,42,18,34,12,26, 6,20,41,17,33,11,25)$ |
$ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,21,38,13,35, 7,28)( 2,22,37,14,36, 8,27)( 3,24,40,16,32,10,29) ( 4,23,39,15,31, 9,30)( 5,20,42,17,34,11,26)( 6,19,41,18,33,12,25)$ |
$ 14, 14, 14 $ | $3$ | $14$ | $( 1,22,38,14,35, 8,28, 2,21,37,13,36, 7,27)( 3,23,40,15,32, 9,29, 4,24,39,16, 31,10,30)( 5,19,42,18,34,12,26, 6,20,41,17,33,11,25)$ |
$ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,22,38,14,35, 8,28, 2,21,37,13,36, 7,27)( 3,23,40,15,32, 9,29, 4,24,39,16, 31,10,30)( 5,20,42,17,34,11,26)( 6,19,41,18,33,12,25)$ |
$ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,22,38,14,35, 8,28, 2,21,37,13,36, 7,27)( 3,24,40,16,32,10,29) ( 4,23,39,15,31, 9,30)( 5,19,42,18,34,12,26, 6,20,41,17,33,11,25)$ |
$ 14, 7, 7, 7, 7 $ | $3$ | $14$ | $( 1,22,38,14,35, 8,28, 2,21,37,13,36, 7,27)( 3,24,40,16,32,10,29) ( 4,23,39,15,31, 9,30)( 5,20,42,17,34,11,26)( 6,19,41,18,33,12,25)$ |
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: | 168.53 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);