LMFDB - Galois group: 27T993

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

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{27}$	$1$	$1$	$0$	$()$
2A	$2^{12},1^{3}$	$45$	$2$	$12$	$( 1, 2)( 4, 5)( 7, 8)(10,14)(11,13)(12,15)(16,20)(17,19)(18,21)(22,26)(23,25)(24,27)$
2B	$2^{10},1^{7}$	$270$	$2$	$10$	$( 1,24)( 2,27)( 4,25)( 5,23)( 7, 8)(11,19)(12,21)(13,17)(15,18)(22,26)$
3A1	$3^{9}$	$40$	$3$	$18$	$( 1, 3, 2)( 4,18,20)( 5,10,15)( 6,23,25)( 7,21,17)( 8,13,12)( 9,26,22)(11,16,24)(14,19,27)$
3A-1	$3^{9}$	$40$	$3$	$18$	$( 1, 2, 3)( 4,20,18)( 5,15,10)( 6,25,23)( 7,17,21)( 8,12,13)( 9,22,26)(11,24,16)(14,27,19)$
3B	$3^{6},1^{9}$	$240$	$3$	$12$	$( 1,27,11)( 2,19,24)( 3,14,16)( 4,23,15)( 5,18,25)( 6,10,20)$
3C	$3^{9}$	$480$	$3$	$18$	$( 1,11,27)( 2,24,19)( 3,16,14)( 4, 6, 5)( 7,13,22)( 8,26,17)( 9,21,12)(10,18,23)(15,20,25)$
4A	$4^{6},1^{3}$	$540$	$4$	$18$	$( 1,21, 2,18)( 4,13, 5,11)( 7,26, 8,22)(10,16,14,20)(12,27,15,24)(17,23,19,25)$
4B	$4^{5},2^{3},1$	$3240$	$4$	$18$	$( 1, 4,24,25)( 2, 5,27,23)( 3, 6)( 7,26, 8,22)(10,20)(11,21,19,12)(13,18,17,15)(14,16)$
5A	$5^{5},1^{2}$	$5184$	$5$	$20$	$( 1, 3,27, 5, 4)( 2,24,22, 6, 7)( 8,26,25, 9,23)(10,12,19,21,20)(13,15,14,16,18)$
6A1	$6^{4},3$	$360$	$6$	$22$	$( 1, 5, 3,10, 2,15)( 4,27,18,14,20,19)( 6,24,23,11,25,16)( 7,22,21, 9,17,26)( 8,12,13)$
6A-1	$6^{4},3$	$360$	$6$	$22$	$( 1,15, 2,10, 3, 5)( 4,19,20,14,18,27)( 6,16,25,11,23,24)( 7,26,17, 9,21,22)( 8,13,12)$
6B1	$6^{3},2^{3},1^{3}$	$720$	$6$	$18$	$( 1,15,27, 4,11,23)( 2,25,19, 5,24,18)( 3,20,14, 6,16,10)(12,13)(17,21)(22,26)$
6B-1	$6^{3},2^{3},1^{3}$	$720$	$6$	$18$	$( 1,23,11, 4,27,15)( 2,18,24, 5,19,25)( 3,10,16, 6,14,20)(12,13)(17,21)(22,26)$
6C	$6^{4},3$	$1440$	$6$	$22$	$( 1,22,11, 7,27,13)( 2,17,24, 8,19,26)( 3,12,16, 9,14,21)( 4, 5, 6)(10,25,18,15,23,20)$
6D	$6^{2},3^{2},2^{4},1$	$2160$	$6$	$18$	$( 1,19,27,24,11, 2)( 3,16,14)( 4,18,23,25,15, 5)( 6,20,10)( 7, 8)(12,21)(13,17)(22,26)$
9A1	$9^{3}$	$2880$	$9$	$24$	$( 1,25,21, 3, 6,17, 2,23, 7)( 4,22,14,18, 9,19,20,26,27)( 5,12,11,10, 8,16,15,13,24)$
9A-1	$9^{3}$	$2880$	$9$	$24$	$( 1,17,25, 2,21,23, 3, 7, 6)( 4,19,22,20,14,26,18,27, 9)( 5,16,12,15,11,13,10,24, 8)$
12A1	$12^{2},3$	$2160$	$12$	$24$	$( 1, 4, 5,27, 3,18,10,14, 2,20,15,19)( 6,22,24,21,23, 9,11,17,25,26,16, 7)( 8,13,12)$
12A-1	$12^{2},3$	$2160$	$12$	$24$	$( 1,18,15,27, 2, 4,10,19, 3,20, 5,14)( 6, 9,16,21,25,22,11, 7,23,26,24,17)( 8,12,13)$

Malle's constant $a(G)$: $1/10$

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: