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Properties

Label 27T993
Degree $27$
Order $25920$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PSp(4,3)$

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Show commands: Magma

magma: G := TransitiveGroup(27, 993);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $993$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSp(4,3)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,20,25)(3,17,15,22)(4,8)(5,18,24,7)(6,19,26,9)(10,11,27,21)(13,14)(16,23), (1,16,24,4,5,8,7,21,25)(2,9,17,14,22,19,11,26,6)(3,20,10,12,27,18,15,13,23)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 9: None

Low degree siblings

36T12781, 40T14344, 40T14345, 45T666

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 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 $ $40$ $3$ $( 1, 7, 4)( 2,17,20)( 3,21,18)( 5,22,27)( 6,25,23)( 8,12,13)( 9,14,10) (11,24,16)(15,26,19)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 4, 7)( 2,20,17)( 3,18,21)( 5,27,22)( 6,23,25)( 8,13,12)( 9,10,14) (11,16,24)(15,19,26)$
$ 9, 9, 9 $ $2880$ $9$ $( 1,10,15, 7, 9,26, 4,14,19)( 2,18,25,17, 3,23,20,21, 6)( 5,11, 8,22,24,12,27, 16,13)$
$ 9, 9, 9 $ $2880$ $9$ $( 1,15, 9, 4,19,10, 7,26,14)( 2,25, 3,20, 6,18,17,23,21)( 5, 8,24,27,13,11,22, 12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $45$ $2$ $( 1,13)( 2,11)( 3, 6)( 4,12)( 5,15)( 7, 8)(16,20)(17,24)(18,23)(19,27)(21,25) (22,26)$
$ 4, 4, 4, 4, 4, 4, 1, 1, 1 $ $540$ $4$ $( 1,15,13, 5)( 2,25,11,21)( 3,20, 6,16)( 4,19,12,27)( 7,26, 8,22)(17,23,24,18)$
$ 6, 6, 6, 6, 3 $ $360$ $6$ $( 1,21,22,13,25,26)( 2, 7, 5,11, 8,15)( 3,17,27, 6,24,19)( 4,16,18,12,20,23) ( 9,10,14)$
$ 6, 6, 6, 6, 3 $ $360$ $6$ $( 1,26,25,13,22,21)( 2,15, 8,11, 5, 7)( 3,19,24, 6,27,17)( 4,23,20,12,18,16) ( 9,14,10)$
$ 12, 12, 3 $ $2160$ $12$ $( 1, 7,21, 5,22,11,13, 8,25,15,26, 2)( 3, 4,17,16,27,18, 6,12,24,20,19,23) ( 9,14,10)$
$ 12, 12, 3 $ $2160$ $12$ $( 1,11,26, 5,25, 7,13, 2,22,15,21, 8)( 3,18,19,16,24, 4, 6,23,27,20,17,12) ( 9,10,14)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $240$ $3$ $( 2,19, 4)( 3,16, 7)( 5,17,18)( 6,20, 8)(11,27,12)(15,24,23)$
$ 6, 6, 6, 2, 2, 2, 1, 1, 1 $ $720$ $6$ $( 1,13)( 2,12,19,11, 4,27)( 3, 8,16, 6, 7,20)( 5,23,17,15,18,24)(21,25)(22,26)$
$ 6, 6, 6, 2, 2, 2, 1, 1, 1 $ $720$ $6$ $( 1,13)( 2,27, 4,11,19,12)( 3,20, 7, 6,16, 8)( 5,24,18,15,17,23)(21,25)(22,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $480$ $3$ $( 1,25,22)( 2, 6,18)( 3,23,11)( 4, 8,17)( 5,19,20)( 7,24,12)( 9,10,14) (13,21,26)(15,27,16)$
$ 6, 6, 6, 6, 3 $ $1440$ $6$ $( 1,26,25,13,22,21)( 2,23, 6,11,18, 3)( 4,24, 8,12,17, 7)( 5,16,19,15,20,27) ( 9,14,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $270$ $2$ $( 1,14)( 3,11)( 5,20)( 6,18)( 7,12)( 8,17)( 9,22)(10,25)(13,21)(16,27)$
$ 6, 6, 3, 3, 2, 2, 2, 2, 1 $ $2160$ $6$ $( 1,14)( 2,19, 4)( 3,27, 7,11,16,12)( 5, 8,18,20,17, 6)( 9,22)(10,25)(13,21) (15,24,23)$
$ 5, 5, 5, 5, 5, 1, 1 $ $5184$ $5$ $( 1, 4, 6,11,16)( 2,12,25,18, 9)( 3,14,23,22,20)( 5,13,24,19, 7) (10,26,17,15,21)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $3240$ $4$ $( 1, 3,12,16)( 2,15,23,19)( 4,24)( 5,25,18,17)( 6, 8,20,10)( 7,27,14,11) ( 9,13)(21,22)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $25920=2^{6} \cdot 3^{4} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  25920.a
magma: IdentifyGroup(G);
 
Character table:

Size
2 P
3 P
5 P
Type

magma: CharacterTable(G);