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Magma
magma: G := TransitiveGroup(36, 12781);
Group action invariants
| Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $12781$ | 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: | (1,2,4,8,16)(3,6,7,14,25)(5,11,21,32,35)(9,17,18,28,29)(10,19,30,22,33)(12,13,20,24,31)(15,27,34,36,23), (2,3,5,10,18)(4,8,15,26,30)(6,12,17,25,22)(7,13,23,24,29)(9,14,19,27,31)(11,20,16,28,34)(21,32,35,33,36) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: None
Degree 4: None
Degree 6: None
Degree 9: None
Degree 12: None
Degree 18: None
Low degree siblings
27T993, 40T14344, 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 $ | $1$ | $1$ | $()$ | |
| $ 5, 5, 5, 5, 5, 5, 5, 1 $ | $5184$ | $5$ | $( 1, 4,11,24,32)( 2,14,34,17,20)( 3,28,27,19,33)( 5,10,29,12,21) ( 7,31,18, 8,22)( 9,15,26,36,16)(13,23,25,30,35)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $270$ | $2$ | $( 2,24)( 3,10)( 4,14)( 5,29)( 7,18)( 8,31)( 9,26)(11,17)(12,28)(15,30)(16,25) (20,34)(21,33)(35,36)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1 $ | $3240$ | $4$ | $( 2,18,24, 7)( 3,11,10,17)( 4,36,14,35)( 5,20,29,34)( 6,23)( 8,21,31,33) ( 9,30,26,15)(12,25,28,16)(13,22)(19,27)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $240$ | $3$ | $( 1,28,23)( 2,36, 6)( 3,16,14)( 4,29,25)( 7,17,15)( 9,18,34)(11,21,30) (13,31,32)(19,35,24)(20,33,26)$ | |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1, 1 $ | $2160$ | $6$ | $( 1,32,28,13,23,31)( 2, 6,36)( 3, 4,16,29,14,25)( 5,10)( 7,20,17,33,15,26) ( 8,12)( 9,30,18,11,34,21)(19,24,35)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $45$ | $2$ | $( 1, 4)( 3,31)( 5, 8)( 9,11)(10,12)(13,14)(16,32)(18,21)(22,27)(23,25)(28,29) (30,34)$ | |
| $ 6, 6, 6, 3, 3, 3, 3, 2, 2, 2 $ | $720$ | $6$ | $( 1,25,28, 4,23,29)( 2, 6,36)( 3,13,16,31,14,32)( 5, 8)( 7,15,17) ( 9,30,18,11,34,21)(10,12)(19,24,35)(20,26,33)(22,27)$ | |
| $ 6, 6, 6, 3, 3, 3, 3, 2, 2, 2 $ | $720$ | $6$ | $( 1,29,23, 4,28,25)( 2,36, 6)( 3,32,14,31,16,13)( 5, 8)( 7,17,15) ( 9,21,34,11,18,30)(10,12)(19,35,24)(20,33,26)(22,27)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $480$ | $3$ | $( 1, 9,14)( 3,28,18)( 4,11,13)( 5,10,27)( 7,19,33)( 8,12,22)(15,24,20) (16,23,34)(17,35,26)(21,31,29)(25,30,32)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $40$ | $3$ | $( 1,16,18)( 2, 6,36)( 3, 9,23)( 4,32,21)( 5,27,10)( 7,20,35)( 8,22,12) (11,25,31)(13,30,29)(14,34,28)(15,26,19)(17,33,24)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $40$ | $3$ | $( 1,18,16)( 2,36, 6)( 3,23, 9)( 4,21,32)( 5,10,27)( 7,35,20)( 8,12,22) (11,31,25)(13,29,30)(14,28,34)(15,19,26)(17,24,33)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 3 $ | $360$ | $6$ | $( 1,18,16)( 2,36, 6)( 3,23, 9)( 4,19,32,15,21,26)( 5,22,27,12,10, 8) ( 7,30,20,29,35,13)(11,33,25,24,31,17)(14,28,34)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 3 $ | $360$ | $6$ | $( 1,16,18)( 2, 6,36)( 3, 9,23)( 4,26,21,15,32,19)( 5, 8,10,12,27,22) ( 7,13,35,29,20,30)(11,17,31,24,25,33)(14,34,28)$ | |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2 $ | $540$ | $4$ | $( 1,23, 4,25)( 2,26)( 3,14,31,13)( 5,32, 8,16)( 6, 7)( 9,27,11,22) (10,29,12,28)(15,20)(17,35)(18,30,21,34)(19,33)(24,36)$ | |
| $ 12, 12, 6, 6 $ | $2160$ | $12$ | $( 1,13,30,23, 3,21, 4,14,34,25,31,18)( 2,24, 6,26,36, 7)( 5, 9,12,32,27,28, 8, 11,10,16,22,29)(15,19,35,20,33,17)$ | |
| $ 12, 12, 6, 6 $ | $2160$ | $12$ | $( 1,21,31,23,34,13, 4,18, 3,25,30,14)( 2, 7,36,26, 6,24)( 5,28,22,32,10, 9, 8, 29,27,16,12,11)(15,17,33,20,35,19)$ | |
| $ 9, 9, 9, 9 $ | $2880$ | $9$ | $( 1, 6,30,22,34,24,13, 9,18)( 2,16,23,11,17,26,21, 8,10)( 3,35,29,20,36,12,32, 19,33)( 4,31,28,14, 5,15,27, 7,25)$ | |
| $ 9, 9, 9, 9 $ | $2880$ | $9$ | $( 1,30,34,13,18, 6,22,24, 9)( 2,23,17,21,10,16,11,26, 8)( 3,29,36,32,33,35,20, 12,19)( 4,28, 5,27,25,31,14,15, 7)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 1, 1, 1 $ | $1440$ | $6$ | $( 1,14, 9)( 3,18,28)( 4,20,11,15,13,24)( 5, 8,10,12,27,22)( 7,31,19,29,33,21) (16,34,23)(17,32,35,25,26,30)$ |
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);