Show commands:
Magma
magma: G := TransitiveGroup(40, 47);
Group action invariants
| Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $47$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_{40}:C_2$ | ||
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,32,18,5,36,23,9,38,26,16,4,29,20,8,34,22,11,40,28,13,2,31,17,6,35,24,10,37,25,15,3,30,19,7,33,21,12,39,27,14), (1,33)(2,34)(3,36)(4,35)(5,29)(6,30)(7,31)(8,32)(9,25)(10,26)(11,28)(12,27)(13,21)(14,22)(15,24)(16,23)(17,20)(18,19)(39,40) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ $10$: $D_{5}$ $16$: $QD_{16}$ $20$: $D_{10}$ $40$: $D_{20}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 5: $D_{5}$
Degree 8: $QD_{16}$
Degree 10: $D_{10}$
Degree 20: $D_{20}$
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $20$ | $2$ | $( 3, 4)( 5,40)( 6,39)( 7,37)( 8,38)( 9,33)(10,34)(11,35)(12,36)(13,32)(14,31) (15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,32,30,31)(33,35,34,36)(37,40,38,39)$ |
| $ 40 $ | $2$ | $40$ | $( 1, 5, 9,16,20,22,28,31,35,37, 3, 7,12,14,18,23,26,29,34,40, 2, 6,10,15,19, 21,27,32,36,38, 4, 8,11,13,17,24,25,30,33,39)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $20$ | $4$ | $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,39,10,40)(11,37,12,38)(13,36,14,35)(15,33,16,34) (17,31,18,32)(19,29,20,30)(21,25,22,26)(23,28,24,27)$ |
| $ 40 $ | $2$ | $40$ | $( 1, 6, 9,15,20,21,28,32,35,38, 3, 8,12,13,18,24,26,30,34,39, 2, 5,10,16,19, 22,27,31,36,37, 4, 7,11,14,17,23,25,29,33,40)$ |
| $ 40 $ | $2$ | $40$ | $( 1, 7,10,13,20,23,27,30,35,40, 4, 5,12,15,17,22,26,32,33,37, 2, 8, 9,14,19, 24,28,29,36,39, 3, 6,11,16,18,21,25,31,34,38)$ |
| $ 40 $ | $2$ | $40$ | $( 1, 8,10,14,20,24,27,29,35,39, 4, 6,12,16,17,21,26,31,33,38, 2, 7, 9,13,19, 23,28,30,36,40, 3, 5,11,15,18,22,25,32,34,37)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1, 9,20,28,35, 3,12,18,26,34, 2,10,19,27,36, 4,11,17,25,33)( 5,16,22,31,37, 7,14,23,29,40, 6,15,21,32,38, 8,13,24,30,39)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,10,20,27,35, 4,12,17,26,33, 2, 9,19,28,36, 3,11,18,25,34)( 5,15,22,32,37, 8,14,24,29,39, 6,16,21,31,38, 7,13,23,30,40)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,11,19,26,35)( 2,12,20,25,36)( 3, 9,17,27,34)( 4,10,18,28,33) ( 5,13,21,29,37)( 6,14,22,30,38)( 7,16,24,32,40)( 8,15,23,31,39)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,12,19,25,35, 2,11,20,26,36)( 3,10,17,28,34, 4, 9,18,27,33)( 5,14,21,30,37, 6,13,22,29,38)( 7,15,24,31,40, 8,16,23,32,39)$ |
| $ 40 $ | $2$ | $40$ | $( 1,13,27,40,12,22,33, 8,19,29, 3,16,25,38,10,23,35, 5,17,32, 2,14,28,39,11, 21,34, 7,20,30, 4,15,26,37, 9,24,36, 6,18,31)$ |
| $ 40 $ | $2$ | $40$ | $( 1,14,27,39,12,21,33, 7,19,30, 3,15,25,37,10,24,35, 6,17,31, 2,13,28,40,11, 22,34, 8,20,29, 4,16,26,38, 9,23,36, 5,18,32)$ |
| $ 40 $ | $2$ | $40$ | $( 1,15,28,38,12,24,34, 5,19,31, 4,14,25,40, 9,21,35, 8,18,30, 2,16,27,37,11, 23,33, 6,20,32, 3,13,26,39,10,22,36, 7,17,29)$ |
| $ 40 $ | $2$ | $40$ | $( 1,16,28,37,12,23,34, 6,19,32, 4,13,25,39, 9,22,35, 7,18,29, 2,15,27,38,11, 24,33, 5,20,31, 3,14,26,40,10,21,36, 8,17,30)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,17,36,10,26, 3,20,33,11,27, 2,18,35, 9,25, 4,19,34,12,28)( 5,24,38,15,29, 7,22,39,13,32, 6,23,37,16,30, 8,21,40,14,31)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,18,36, 9,26, 4,20,34,11,28, 2,17,35,10,25, 3,19,33,12,27)( 5,23,38,16,29, 8,22,40,13,31, 6,24,37,15,30, 7,21,39,14,32)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,19,35,11,26)( 2,20,36,12,25)( 3,17,34, 9,27)( 4,18,33,10,28) ( 5,21,37,13,29)( 6,22,38,14,30)( 7,24,40,16,32)( 8,23,39,15,31)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,20,35,12,26, 2,19,36,11,25)( 3,18,34,10,27, 4,17,33, 9,28)( 5,22,37,14,29, 6,21,38,13,30)( 7,23,40,15,32, 8,24,39,16,31)$ |
| $ 8, 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,21, 3,24, 2,22, 4,23)( 5,27, 7,25, 6,28, 8,26)( 9,32,12,30,10,31,11,29) (13,34,16,36,14,33,15,35)(17,40,20,38,18,39,19,37)$ |
| $ 8, 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,22, 3,23, 2,21, 4,24)( 5,28, 7,26, 6,27, 8,25)( 9,31,12,29,10,32,11,30) (13,33,16,35,14,34,15,36)(17,39,20,37,18,40,19,38)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $80=2^{4} \cdot 5$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 80.6 | magma: IdentifyGroup(G);
|
| Character table: not available. |
magma: CharacterTable(G);