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Magma
magma: G := TransitiveGroup(40, 14344);
Group action invariants
| Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $14344$ | 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: | (1,3,8,16,7)(2,6,13,19,17)(4,9,5,10,20)(11,23,36,26,35)(12,21,31,25,18)(14,27,29,22,34)(15,28,30,38,32)(24,37,40,39,33), (1,2,5,10,19)(3,4,7,15,28)(6,12,21,13,25)(8,16,29,38,14)(9,18,31,20,27)(11,22,33,37,34)(17,30,23,35,32)(24,26,39,40,36) | 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, 40T14345, 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 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $270$ | $2$ | $( 1, 6)( 2,16)( 3,23)( 4, 5)( 7,33)( 8,36)( 9,10)(11,26)(12,28)(13,27)(14,30) (15,29)(17,34)(18,38)(19,20)(21,25)(22,32)(24,39)(31,35)(37,40)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $3240$ | $4$ | $( 1,20, 6,19)( 2,17,16,34)( 3,38,23,18)( 4,15, 5,29)( 7,22,33,32)( 8,30,36,14) ( 9,24,10,39)(11,28,26,12)(13,25,27,21)(31,40,35,37)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $45$ | $2$ | $( 1,25)( 2,37)( 6,21)( 7,24)( 8,12)( 9,32)(10,22)(11,30)(13,19)(14,26)(16,40) (17,31)(20,27)(28,36)(33,39)(34,35)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $240$ | $3$ | $( 1, 2,21)( 4,23, 5)( 6,25,37)( 7,19,32)( 8,14,20)( 9,24,13)(10,30,17) (11,31,22)(12,26,27)(18,38,29)(28,33,34)(35,36,39)$ | |
| $ 6, 6, 6, 6, 6, 3, 3, 2, 1, 1 $ | $720$ | $6$ | $( 1,28, 2,33,21,34)( 3,15)( 4,29,23,18, 5,38)( 6,35,25,36,37,39) ( 7,14,19,20,32, 8)( 9,12,24,26,13,27)(10,17,30)(11,22,31)$ | |
| $ 6, 6, 6, 6, 6, 3, 3, 2, 1, 1 $ | $720$ | $6$ | $( 1,34,21,33, 2,28)( 3,15)( 4,38, 5,18,23,29)( 6,39,37,36,25,35) ( 7, 8,32,20,19,14)( 9,27,13,26,24,12)(10,30,17)(11,31,22)$ | |
| $ 6, 6, 6, 6, 6, 6, 2, 2 $ | $2160$ | $6$ | $( 1,39,21,36, 2,35)( 3,15)( 4, 7, 5,32,23,19)( 6,34,37,33,25,28) ( 8,18,20,29,14,38)( 9,27,13,26,24,12)(10,22,17,31,30,11)(16,40)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $40$ | $3$ | $( 1,21, 2)( 4,20,12)( 5,14,27)( 7,13,18)( 8,26,23)( 9,38,19)(10,17,30) (24,29,32)(28,34,33)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $40$ | $3$ | $( 1, 2,21)( 4,12,20)( 5,27,14)( 7,18,13)( 8,23,26)( 9,19,38)(10,30,17) (24,32,29)(28,33,34)$ | |
| $ 6, 6, 6, 6, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $360$ | $6$ | $( 1, 2,21)( 3,15)( 4, 9,20,38,12,19)( 5,13,14,18,27, 7)( 8,29,26,32,23,24) (10,33,17,28,30,34)(11,39)(22,36)(31,35)$ | |
| $ 6, 6, 6, 6, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $360$ | $6$ | $( 1,21, 2)( 3,15)( 4,19,12,38,20, 9)( 5, 7,27,18,14,13)( 8,24,23,32,26,29) (10,34,30,28,17,33)(11,39)(22,36)(31,35)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1, 1 $ | $540$ | $4$ | $( 3,31,15,35)( 4,27,38,13)( 5,19,18,20)( 6,25)( 7,12,14, 9)( 8,33,32,30) (10,26,28,24)(11,22,39,36)(16,37)(17,23,34,29)$ | |
| $ 12, 12, 4, 4, 3, 2, 2, 1 $ | $2160$ | $12$ | $( 1, 2,21)( 3,35,15,31)( 4, 7,19,27,12,18,38,14,20,13, 9, 5)( 6,25) ( 8,17,24,33,23,10,32,34,26,30,29,28)(11,36,39,22)(16,37)$ | |
| $ 12, 12, 4, 4, 3, 2, 2, 1 $ | $2160$ | $12$ | $( 1,21, 2)( 3,35,15,31)( 4,18, 9,27,20, 7,38, 5,12,13,19,14)( 6,25) ( 8,10,29,33,26,17,32,28,23,30,24,34)(11,36,39,22)(16,37)$ | |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $5184$ | $5$ | $( 1,28,23,31, 5)( 2,40,34,25,36)( 3,26,19,24,14)( 4,29,37,35,38) ( 6,16,39,20,12)( 7,15,13,21,33)( 8,27,10,18,30)( 9,22,17,11,32)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $480$ | $3$ | $( 1, 2,21)( 4,14,26)( 5, 8,12)( 6,37,25)( 7,24,38)( 9,18,32)(10,30,17) (11,22,31)(13,29,19)(20,27,23)(28,33,34)(35,39,36)$ | |
| $ 6, 6, 6, 6, 6, 3, 3, 2, 1, 1 $ | $1440$ | $6$ | $( 1,28, 2,33,21,34)( 3,15)( 4, 9,14,18,26,32)( 5,13, 8,29,12,19) ( 6,39,37,36,25,35)( 7,23,24,20,38,27)(10,17,30)(11,31,22)$ | |
| $ 9, 9, 9, 3, 3, 3, 3, 1 $ | $2880$ | $9$ | $( 1,26,33,15,36, 3,28,38,21)( 2,31,11)( 4,24,16)( 5, 6,13,39,23,20,32, 9,35) ( 7,10,25,37,30,14,12,40,29)( 8,22,19)(18,27,34)$ | |
| $ 9, 9, 9, 3, 3, 3, 3, 1 $ | $2880$ | $9$ | $( 1,33,36,28,21,26,15, 3,38)( 2,11,31)( 4,16,24)( 5,13,23,32,35, 6,39,20, 9) ( 7,25,30,12,29,10,37,14,40)( 8,19,22)(18,34,27)$ |
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);