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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 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $45$ | $2$ | $( 1, 4)( 2,17)( 3,30)( 5, 9)( 6, 7)( 8,11)(13,27)(14,22)(16,21)(18,32)(20,24) (31,34)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $40$ | $3$ | $( 1, 3,34)( 2,21,11)( 4,30,31)( 5,14,24)( 6,13,32)( 7,27,18)( 8,17,16) ( 9,22,20)(10,23,26)(12,19,28)(15,33,35)(25,29,36)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $40$ | $3$ | $( 1,34, 3)( 2,11,21)( 4,31,30)( 5,24,14)( 6,32,13)( 7,18,27)( 8,16,17) ( 9,20,22)(10,26,23)(12,28,19)(15,35,33)(25,36,29)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 3 $ | $360$ | $6$ | $( 1,31, 3, 4,34,30)( 2, 8,21,17,11,16)( 5,20,14, 9,24,22)( 6,18,13, 7,32,27) (10,26,23)(12,28,19)(15,35,33)(25,36,29)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 3 $ | $360$ | $6$ | $( 1,30,34, 4, 3,31)( 2,16,11,17,21, 8)( 5,22,24, 9,14,20)( 6,27,32, 7,13,18) (10,23,26)(12,19,28)(15,33,35)(25,29,36)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $240$ | $3$ | $( 1, 2,24)( 3,21, 5)( 4,17,20)( 8,22,31)( 9,30,16)(10,29,33)(11,14,34) (12,28,19)(15,26,25)(23,36,35)$ | |
| $ 6, 6, 6, 3, 3, 3, 3, 2, 2, 2 $ | $720$ | $6$ | $( 1,24, 2)( 3, 5,21)( 4,29,17,33,20,10)( 6,18)( 7,13)( 8,15,22,26,31,25) ( 9,23,30,36,16,35)(11,34,14)(12,19,28)(27,32)$ | |
| $ 6, 6, 6, 3, 3, 3, 3, 2, 2, 2 $ | $720$ | $6$ | $( 1, 2,24)( 3,21, 5)( 4,10,20,33,17,29)( 6,18)( 7,13)( 8,25,31,26,22,15) ( 9,35,16,36,30,23)(11,14,34)(12,28,19)(27,32)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $480$ | $3$ | $( 1,14,21)( 2,34, 5)( 3,24,11)( 4,22,16)( 6,32,13)( 7,18,27)( 8,30,20) ( 9,17,31)(10,15,36)(23,33,25)(26,35,29)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 1, 1, 1 $ | $1440$ | $6$ | $( 1,14,21)( 2,34, 5)( 3,24,11)( 4,25,16,33,22,23)( 6,27,13,18,32, 7) ( 8,35,20,26,30,29)( 9,10,31,36,17,15)$ | |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2 $ | $540$ | $4$ | $( 1, 2,33,10)( 3,21,35,23)( 4,16)( 5, 7,36,32)( 6,14,27,25)( 8,30)( 9,12) (11,15,26,34)(13,24,18,29)(17,31)(19,22)(20,28)$ | |
| $ 12, 12, 6, 6 $ | $2160$ | $12$ | $( 1,18,30,17,34, 9, 4,32, 3, 2,31, 5)( 6, 8,21,20,14,13, 7,11,16,24,22,27) (10,23,29,36,19,28)(12,33,26,15,25,35)$ | |
| $ 12, 12, 6, 6 $ | $2160$ | $12$ | $( 1, 9,31,17, 3,18, 4, 5,34, 2,30,32)( 6,13,22,20,16, 8, 7,27,14,24,21,11) (10,28,19,36,29,23)(12,35,25,15,26,33)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $270$ | $2$ | $( 1,18)( 2, 7)( 3,36)( 5,33)( 6, 8)( 9,25)(11,22)(12,30)(13,35)(14,17)(19,26) (20,31)(23,29)(27,28)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1 $ | $3240$ | $4$ | $( 1,26,18,19)( 2,12, 7,30)( 3,23,36,29)( 4,21)( 5, 6,33, 8)( 9,28,25,27) (10,34)(11,20,22,31)(13,14,35,17)(15,16)$ | |
| $ 5, 5, 5, 5, 5, 5, 5, 1 $ | $5184$ | $5$ | $( 1,10,31,36, 3)( 2,23,13,28,26)( 4,30,21,12,24)( 5,25,19, 7,29) ( 6, 9,22,33,14)( 8,17,27,11,35)(16,34,20,18,32)$ | |
| $ 9, 9, 9, 9 $ | $2880$ | $9$ | $( 1,25,36, 2,13,16,28,21, 3)( 4,32,20,10,35,19,24, 8,17)( 5,12,34, 7,29, 6,31, 33, 9)(11,26,18,15,30,14,22,27,23)$ | |
| $ 9, 9, 9, 9 $ | $2880$ | $9$ | $( 1,36,13,28, 3,25, 2,16,21)( 4,20,35,24,17,32,10,19, 8)( 5,34,29,31, 9,12, 7, 6,33)(11,18,30,22,23,26,15,14,27)$ | |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1, 1 $ | $2160$ | $6$ | $( 1, 5,13)( 2,28,21,25, 7,31)( 3,16)( 6,23,30,29,17,32)( 8,20,14, 9,12,27) (10,24)(11,15,22)(18,19,33,34,35,26)$ |
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);