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Magma
magma: G := TransitiveGroup(40, 14345);
Group action invariants
| Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $14345$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $\PSp(4,3)$ | ||
| 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: | (1,2,4,7,10)(3,6,12,21,20)(5,9,16,27,38)(8,14,24,32,34)(11,19,23,31,17)(13,22,30,40,37)(15,25,28,18,29)(26,33,39,36,35), (1,3,4,7,12)(2,5,10,18,22)(6,9,16,27,38)(8,15,26,36,37)(11,20,30,40,34)(13,23,19,29,17)(14,25,28,21,31)(24,33,39,32,35) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 5: None
Degree 8: None
Degree 10: None
Degree 20: None
Low degree siblings
27T993, 36T12781, 40T14344, 45T666Siblings are shown 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 $ | $1$ | $1$ | $()$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $40$ | $3$ | $( 1, 5,13)( 2,28,11)( 3, 6,16)( 4, 9,27)( 7,23,22)( 8,39,17)(10,18,38) (12,30,14)(15,40,21)(19,35,36)(20,32,25)(26,34,31)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $40$ | $3$ | $( 1,13, 5)( 2,11,28)( 3,16, 6)( 4,27, 9)( 7,22,23)( 8,17,39)(10,38,18) (12,14,30)(15,21,40)(19,36,35)(20,25,32)(26,31,34)$ | |
| $ 9, 9, 9, 9, 3, 1 $ | $2880$ | $9$ | $( 1,22,14, 5, 7,12,13,23,30)( 2,39,21,28,17,15,11, 8,40)( 3,10,31, 6,18,26,16, 38,34)( 4,35,25, 9,36,20,27,19,32)(24,37,33)$ | |
| $ 9, 9, 9, 9, 3, 1 $ | $2880$ | $9$ | $( 1,14, 7,13,30,22, 5,12,23)( 2,21,17,11,40,39,28,15, 8)( 3,31,18,16,34,10, 6, 26,38)( 4,25,36,27,32,35, 9,20,19)(24,33,37)$ | |
| $ 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, 1, 1 $ | $45$ | $2$ | $( 1,16)( 2,29)( 3,31)( 4,40)( 7, 9)(11,24)(13,26)(14,19)(15,23)(17,30)(28,33) (35,39)$ | |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2 $ | $540$ | $4$ | $( 1,28,16,33)( 2,39,29,35)( 3, 4,31,40)( 5, 8)( 6,27)( 7,24, 9,11)(10,18) (12,32)(13,30,26,17)(14,23,19,15)(20,21)(22,36)(25,34)(37,38)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1 $ | $360$ | $6$ | $( 1,24,31,16,11, 3)( 2,14,17,29,19,30)( 4,28, 9,40,33, 7)( 5,36,10)( 8,22,18) (13,35,15,26,39,23)(20,25,37)(21,34,38)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1 $ | $360$ | $6$ | $( 1, 3,11,16,31,24)( 2,30,19,29,17,14)( 4, 7,33,40, 9,28)( 5,10,36)( 8,18,22) (13,23,39,26,15,35)(20,37,25)(21,38,34)$ | |
| $ 12, 12, 6, 6, 2, 2 $ | $2160$ | $12$ | $( 1, 4,24,28,31, 9,16,40,11,33, 3, 7)( 2,26,14,39,17,23,29,13,19,35,30,15) ( 5,18,36, 8,10,22)( 6,27)(12,32)(20,38,25,21,37,34)$ | |
| $ 12, 12, 6, 6, 2, 2 $ | $2160$ | $12$ | $( 1, 9, 3,28,11, 4,16, 7,31,33,24,40)( 2,23,30,39,19,26,29,15,17,35,14,13) ( 5,22,10, 8,36,18)( 6,27)(12,32)(20,34,37,21,25,38)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $240$ | $3$ | $( 1,11,31)( 2,33,13)( 3,16,24)( 4,15,17)( 5,37,34)( 7,35,14)( 8,18,22) ( 9,39,19)(10,25,21)(12,27,32)(20,38,36)(23,30,40)(26,29,28)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $270$ | $2$ | $( 1,24)( 2,10)( 3,11)( 4,34)( 5,15)( 8,32)( 9,36)(12,18)(13,21)(16,31)(17,37) (19,38)(20,39)(22,27)(23,28)(25,33)(26,30)(29,40)$ | |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $5184$ | $5$ | $( 1, 7,24,30,26)( 2, 9,20,21,12)( 3, 8,34,17,22)( 4,32,11,27,37) ( 5,38,19,15,14)( 6,28,33,25,23)(10,18,13,39,36)(16,35,31,40,29)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ | $480$ | $3$ | $( 2, 8,27)( 4,19,18)( 6, 9,17)( 7,37,30)(10,16,20)(11,35,40)(12,22,33) (14,23,24)(15,39,32)(21,36,26)(25,31,38)$ | |
| $ 6, 6, 6, 3, 3, 3, 3, 3, 2, 2, 2, 1 $ | $1440$ | $6$ | $( 1,34)( 2,23, 8,24,27,14)( 3, 5)( 4,18,19)( 6,36, 9,26,17,21)( 7,30,37) (10,20,16)(11,22,35,33,40,12)(15,32,39)(25,38,31)(28,29)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 3, 1 $ | $720$ | $6$ | $( 1, 9,33,39,16,25)( 2,37,11)( 3,34,13)( 4,17,38)( 5,20,31,15,24,36) ( 6,22,29,14,26,32)( 7,30,40,35,27, 8)(10,21,19)(12,18,23)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 3, 1 $ | $720$ | $6$ | $( 1,25,16,39,33, 9)( 2,11,37)( 3,13,34)( 4,38,17)( 5,36,24,15,31,20) ( 6,32,26,14,29,22)( 7, 8,27,35,40,30)(10,19,21)(12,23,18)$ | |
| $ 6, 6, 6, 6, 6, 6, 3, 1 $ | $2160$ | $6$ | $( 1,36,17,22,40,34)( 2, 7,21,13,35,10)( 3, 9,27,12,29,24)( 4,32,30,37,16,20) ( 5,19,23,15,28,38)( 6,14,33)( 8,18,26,31,11,39)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 1 $ | $3240$ | $4$ | $( 1,24, 4,10)( 2,32,34,14)( 3, 9,15,13)( 5, 6,33,30)(11,37,25,36)(12,26) (16,27,31,29)(17,35,21,18)(19,23,38,40)(20,22,39,28)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $25920=2^{6} \cdot 3^{4} \cdot 5$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 25920.a | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| Type |
magma: CharacterTable(G);