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Magma
magma: G := TransitiveGroup(40, 5);
Group action invariants
| Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $5$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_5\times Q_8$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $40$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,25,9,33,18,3,27,12,35,19,2,26,10,34,17,4,28,11,36,20)(5,30,15,40,22,7,31,14,38,23,6,29,16,39,21,8,32,13,37,24), (1,8,9,13,18,24,27,30,35,40,2,7,10,14,17,23,28,29,36,39)(3,6,12,16,19,21,26,32,34,37,4,5,11,15,20,22,25,31,33,38) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $5$: $C_5$ $8$: $Q_8$ $10$: $C_{10}$ x 3 $20$: 20T3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 5: $C_5$
Degree 8: $Q_8$
Degree 10: $C_{10}$ x 3
Degree 20: 20T3
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, 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 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,24,22,23) (25,27,26,28)(29,32,30,31)(33,35,34,36)(37,40,38,39)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1, 5, 9,15,18,22,27,31,35,38, 2, 6,10,16,17,21,28,32,36,37)( 3, 8,12,13,19, 24,26,30,34,40, 4, 7,11,14,20,23,25,29,33,39)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1, 7, 9,14,18,23,27,29,35,39, 2, 8,10,13,17,24,28,30,36,40)( 3, 5,12,15,19, 22,26,31,34,38, 4, 6,11,16,20,21,25,32,33,37)$ |
| $ 10, 10, 10, 10 $ | $1$ | $10$ | $( 1, 9,18,27,35, 2,10,17,28,36)( 3,12,19,26,34, 4,11,20,25,33)( 5,15,22,31,38, 6,16,21,32,37)( 7,14,23,29,39, 8,13,24,30,40)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,10,18,28,35)( 2, 9,17,27,36)( 3,11,19,25,34)( 4,12,20,26,33) ( 5,16,22,32,38)( 6,15,21,31,37)( 7,13,23,30,39)( 8,14,24,29,40)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,11,17,26,35, 3, 9,20,28,34, 2,12,18,25,36, 4,10,19,27,33)( 5,13,21,29,38, 7,15,24,32,39, 6,14,22,30,37, 8,16,23,31,40)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,13,27,40,10,23,36, 8,18,30, 2,14,28,39, 9,24,35, 7,17,29)( 3,16,26,37,11, 22,33, 6,19,32, 4,15,25,38,12,21,34, 5,20,31)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,15,27,38,10,21,36, 5,18,31, 2,16,28,37, 9,22,35, 6,17,32)( 3,13,26,40,11, 23,33, 8,19,30, 4,14,25,39,12,24,34, 7,20,29)$ |
| $ 10, 10, 10, 10 $ | $1$ | $10$ | $( 1,17,35, 9,28, 2,18,36,10,27)( 3,20,34,12,25, 4,19,33,11,26)( 5,21,38,15,32, 6,22,37,16,31)( 7,24,39,14,30, 8,23,40,13,29)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,18,35,10,28)( 2,17,36, 9,27)( 3,19,34,11,25)( 4,20,33,12,26) ( 5,22,38,16,32)( 6,21,37,15,31)( 7,23,39,13,30)( 8,24,40,14,29)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,19,36,12,28, 3,17,33,10,25, 2,20,35,11,27, 4,18,34, 9,26)( 5,23,37,14,32, 7,21,40,16,30, 6,24,38,13,31, 8,22,39,15,29)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,21, 2,22)( 3,23, 4,24)( 5,28, 6,27)( 7,26, 8,25)( 9,32,10,31)(11,30,12,29) (13,33,14,34)(15,36,16,35)(17,38,18,37)(19,39,20,40)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,23, 2,24)( 3,22, 4,21)( 5,26, 6,25)( 7,27, 8,28)( 9,29,10,30)(11,32,12,31) (13,36,14,35)(15,34,16,33)(17,40,18,39)(19,38,20,37)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,25, 9,33,18, 3,27,12,35,19, 2,26,10,34,17, 4,28,11,36,20)( 5,30,15,40,22, 7,31,14,38,23, 6,29,16,39,21, 8,32,13,37,24)$ |
| $ 10, 10, 10, 10 $ | $1$ | $10$ | $( 1,27,10,36,18, 2,28, 9,35,17)( 3,26,11,33,19, 4,25,12,34,20)( 5,31,16,37,22, 6,32,15,38,21)( 7,29,13,40,23, 8,30,14,39,24)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,28,10,35,18)( 2,27, 9,36,17)( 3,25,11,34,19)( 4,26,12,33,20) ( 5,32,16,38,22)( 6,31,15,37,21)( 7,30,13,39,23)( 8,29,14,40,24)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,29,17, 7,35,24, 9,39,28,14, 2,30,18, 8,36,23,10,40,27,13)( 3,31,20, 5,34, 21,12,38,25,15, 4,32,19, 6,33,22,11,37,26,16)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,31,17, 5,35,21, 9,38,28,15, 2,32,18, 6,36,22,10,37,27,16)( 3,30,20, 8,34, 23,12,40,25,13, 4,29,19, 7,33,24,11,39,26,14)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,33,27,19,10, 4,36,25,18,12, 2,34,28,20, 9, 3,35,26,17,11)( 5,40,31,23,16, 8,37,30,22,14, 6,39,32,24,15, 7,38,29,21,13)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,35,28,18,10)( 2,36,27,17, 9)( 3,34,25,19,11)( 4,33,26,20,12) ( 5,38,32,22,16)( 6,37,31,21,15)( 7,39,30,23,13)( 8,40,29,24,14)$ |
| $ 10, 10, 10, 10 $ | $1$ | $10$ | $( 1,36,28,17,10, 2,35,27,18, 9)( 3,33,25,20,11, 4,34,26,19,12)( 5,37,32,21,16, 6,38,31,22,15)( 7,40,30,24,13, 8,39,29,23,14)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,37,36,32,28,21,17,16,10, 6, 2,38,35,31,27,22,18,15, 9, 5)( 3,39,33,29,25, 23,20,14,11, 7, 4,40,34,30,26,24,19,13,12, 8)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,39,36,29,28,23,17,14,10, 7, 2,40,35,30,27,24,18,13, 9, 8)( 3,38,33,31,25, 22,20,15,11, 5, 4,37,34,32,26,21,19,16,12, 6)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $40=2^{3} \cdot 5$ | 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: | $2$ | ||
| Label: | 40.11 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);