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Magma
magma: G := TransitiveGroup(41, 8);
Group action invariants
| Degree $n$: | $41$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $F_{41}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,6,36,11,25,27,39,29,10,19,32,28,4,24,21,3,18,26,33,34,40,35,5,30,16,14,2,12,31,22,9,13,37,17,20,38,23,15,8,7), (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) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $5$: $C_5$ $8$: $C_8$ $10$: $C_{10}$ $20$: 20T1 $40$: $C_{40}$ Resolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
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
| Label | 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 $ | $1$ | $1$ | $()$ | |
| $ 41 $ | $40$ | $41$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$ | |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,11,19,17,38)( 3,21,37,33,34)( 4,31,14, 8,30)( 5,41,32,24,26) ( 6,10, 9,40,22)( 7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23)$ | |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,19,38,11,17)( 3,37,34,21,33)( 4,14,30,31, 8)( 5,32,26,41,24) ( 6, 9,22,10,40)( 7,27,18,20,15)(12,35,39,29,13)(16,25,23,28,36)$ | |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,38,17,19,11)( 3,34,33,37,21)( 4,30, 8,14,31)( 5,26,24,32,41) ( 6,22,40, 9,10)( 7,18,15,27,20)(12,39,13,35,29)(16,23,36,25,28)$ | |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,17,11,38,19)( 3,33,21,34,37)( 4, 8,31,30,14)( 5,24,41,26,32) ( 6,40,10,22, 9)( 7,15,20,18,27)(12,13,29,39,35)(16,36,28,23,25)$ | |
| $ 10, 10, 10, 10, 1 $ | $41$ | $10$ | $( 2,26,11, 5,19,41,17,32,38,24)( 3,10,21, 9,37,40,33,22,34, 6)( 4,35,31,13,14, 39, 8,12,30,29)( 7,28,20,25,27,36,15,23,18,16)$ | |
| $ 10, 10, 10, 10, 1 $ | $41$ | $10$ | $( 2, 5,17,24,11,41,38,26,19,32)( 3, 9,33, 6,21,40,34,10,37,22)( 4,13, 8,29,31, 39,30,35,14,12)( 7,25,15,16,20,36,18,28,27,23)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $41$ | $2$ | $( 2,41)( 3,40)( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32)(12,31) (13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)$ | |
| $ 10, 10, 10, 10, 1 $ | $41$ | $10$ | $( 2,24,38,32,17,41,19, 5,11,26)( 3, 6,34,22,33,40,37, 9,21,10)( 4,29,30,12, 8, 39,14,13,31,35)( 7,16,18,23,15,36,27,25,20,28)$ | |
| $ 10, 10, 10, 10, 1 $ | $41$ | $10$ | $( 2,32,19,26,38,41,11,24,17, 5)( 3,22,37,10,34,40,21, 6,33, 9)( 4,12,14,35,30, 39,31,29, 8,13)( 7,23,27,28,18,36,20,16,15,25)$ | |
| $ 20, 20, 1 $ | $41$ | $20$ | $( 2,37,26,40,11,33, 5,22,19,34,41, 6,17, 3,32,10,38,21,24, 9)( 4,27,35,36,31, 15,13,23,14,18,39,16, 8, 7,12,28,30,20,29,25)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $ | $41$ | $4$ | $( 2,33,41,10)( 3,24,40,19)( 4,15,39,28)( 5, 6,38,37)( 7,29,36,14)( 8,20,35,23) ( 9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)$ | |
| $ 20, 20, 1 $ | $41$ | $20$ | $( 2,34,24,22,38,33,32,40,17,37,41, 9,19,21, 5,10,11, 3,26, 6)( 4,18,29,23,30, 15,12,36, 8,27,39,25,14,20,13,28,31, 7,35,16)$ | |
| $ 20, 20, 1 $ | $41$ | $20$ | $( 2,21,32, 6,19,33,26, 9,38, 3,41,22,11,37,24,10,17,34, 5,40)( 4,20,12,16,14, 15,35,25,30, 7,39,23,31,27,29,28, 8,18,13,36)$ | |
| $ 20, 20, 1 $ | $41$ | $20$ | $( 2, 3, 5, 9,17,33,24, 6,11,21,41,40,38,34,26,10,19,37,32,22)( 4, 7,13,25, 8, 15,29,16,31,20,39,36,30,18,35,28,14,27,12,23)$ | |
| $ 20, 20, 1 $ | $41$ | $20$ | $( 2,40, 5,34,17,10,24,37,11,22,41, 3,38, 9,26,33,19, 6,32,21)( 4,36,13,18, 8, 28,29,27,31,23,39, 7,30,25,35,15,14,16,12,20)$ | |
| $ 20, 20, 1 $ | $41$ | $20$ | $( 2,22,32,37,19,10,26,34,38,40,41,21,11, 6,24,33,17, 9, 5, 3)( 4,23,12,27,14, 28,35,18,30,36,39,20,31,16,29,15, 8,25,13, 7)$ | |
| $ 20, 20, 1 $ | $41$ | $20$ | $( 2, 6,26, 3,11,10, 5,21,19, 9,41,37,17,40,32,33,38,22,24,34)( 4,16,35, 7,31, 28,13,20,14,25,39,27, 8,36,12,15,30,23,29,18)$ | |
| $ 20, 20, 1 $ | $41$ | $20$ | $( 2, 9,24,21,38,10,32, 3,17, 6,41,34,19,22, 5,33,11,40,26,37)( 4,25,29,20,30, 28,12, 7, 8,16,39,18,14,23,13,15,31,36,35,27)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $ | $41$ | $4$ | $( 2,10,41,33)( 3,19,40,24)( 4,28,39,15)( 5,37,38, 6)( 7,14,36,29)( 8,23,35,20) ( 9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2, 7,37,12,26,28,40,30,11,20,33,29, 5,25,22, 4,19,27,34,35,41,36, 6,31,17, 15, 3,13,32,23,10,14,38,18,21,39,24,16, 9, 8)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,20,34,13,24,28,22,31,38, 7,33,35,32,16,40, 4,17,18,37,29,41,23, 9,30,19, 15,21,12, 5,36,10, 8,11,27, 3,39,26,25, 6,14)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,27,21,29,32,28, 6, 8,19,18,33,13,26,36, 9, 4,38,20, 3,12,41,16,22,14,11, 15,37,35,24,25,10,30,17, 7,34,39, 5,23,40,31)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,18, 3,35, 5,28, 9,14,17,27,33,12,24,23, 6, 4,11, 7,21,13,41,25,40, 8,38, 15,34,29,26,16,10,31,19,20,37,39,32,36,22,30)$ | |
| $ 8, 8, 8, 8, 8, 1 $ | $41$ | $8$ | $( 2,15,33,39,41,28,10, 4)( 3,29,24,36,40,14,19, 7)( 5,16, 6,30,38,27,37,13) ( 8,17,20,21,35,26,23,22)( 9,31,11,18,34,12,32,25)$ | |
| $ 8, 8, 8, 8, 8, 1 $ | $41$ | $8$ | $( 2,28,33, 4,41,15,10,39)( 3,14,24, 7,40,29,19,36)( 5,27, 6,13,38,16,37,30) ( 8,26,20,22,35,17,23,21)( 9,12,11,25,34,31,32,18)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,25, 3, 8, 5,15, 9,29,17,16,33,31,24,20, 6,39,11,36,21,30,41,18,40,35,38, 28,34,14,26,27,10,12,19,23,37, 4,32, 7,22,13)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,36,37,31,26,15,40,13,11,23,33,14, 5,18,22,39,19,16,34, 8,41, 7, 6,12,17, 28, 3,30,32,20,10,29,38,25,21, 4,24,27, 9,35)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,16,21,14,32,15, 6,35,19,25,33,30,26, 7, 9,39,38,23, 3,31,41,27,22,29,11, 28,37, 8,24,18,10,13,17,36,34, 4, 5,20,40,12)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,23,34,30,24,15,22,12,38,36,33, 8,32,27,40,39,17,25,37,14,41,20, 9,13,19, 28,21,31, 5, 7,10,35,11,16, 3, 4,26,18, 6,29)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,12,40,20, 5, 4,34,36,17,13,10,18,24, 8,37,28,11,29,22,27,41,31, 3,23,38, 39, 9, 7,26,30,33,25,19,35, 6,15,32,14,21,16)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,29, 6,18,26, 4, 3,16,11,35,10, 7, 5,31,21,28,19,13, 9,20,41,14,37,25,17, 39,40,27,32, 8,33,36,38,12,22,15,24,30,34,23)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,35, 9,27,24, 4,21,25,38,29,10,20,32,30, 3,28,17,12, 6, 7,41, 8,34,16,19, 39,22,18, 5,14,33,23,11,13,40,15,26,31,37,36)$ | |
| $ 8, 8, 8, 8, 8, 1 $ | $41$ | $8$ | $( 2,39,10,15,41, 4,33,28)( 3,36,19,29,40, 7,24,14)( 5,30,37,16,38,13, 6,27) ( 8,21,23,17,35,22,20,26)( 9,18,32,31,34,25,11,12)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,13,22, 7,32, 4,37,23,19,12,10,27,26,14,34,28,38,35,40,18,41,30,21,36,11, 39, 6,20,24,31,33,16,17,29, 9,15, 5, 8, 3,25)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,30,22,36,32,39,37,20,19,31,10,16,26,29,34,15,38, 8,40,25,41,13,21, 7,11, 4, 6,23,24,12,33,27,17,14, 9,28, 5,35, 3,18)$ | |
| $ 8, 8, 8, 8, 8, 1 $ | $41$ | $8$ | $( 2, 4,10,28,41,39,33,15)( 3, 7,19,14,40,36,24,29)( 5,13,37,27,38,30, 6,16) ( 8,22,23,26,35,21,20,17)( 9,25,32,12,34,18,11,31)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,31,40,23, 5,39,34, 7,17,30,10,25,24,35,37,15,11,14,22,16,41,12, 3,20,38, 4, 9,36,26,13,33,18,19, 8, 6,28,32,29,21,27)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2, 8, 9,16,24,39,21,18,38,14,10,23,32,13, 3,15,17,31, 6,36,41,35,34,27,19, 4,22,25, 5,29,33,20,11,30,40,28,26,12,37, 7)$ | |
| $ 40, 1 $ | $41$ | $40$ | $( 2,14, 6,25,26,39, 3,27,11, 8,10,36, 5,12,21,15,19,30, 9,23,41,29,37,18,17, 4,40,16,32,35,33, 7,38,31,22,28,24,13,34,20)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $1640=2^{3} \cdot 5 \cdot 41$ | 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: | 1640.47 | magma: IdentifyGroup(G);
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| Character table: | 41 x 41 character table |
magma: CharacterTable(G);