LMFDB - Galois group: 18T650

Show commands: Magma

magma:G := TransitiveGroup(18, 650);

Group invariants

Abstract group:	$\He_3^2:S_3^2$	magma:IdentifyGroup(G);
Order:	$26244=2^{2} \cdot 3^{8}$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	yes	magma:IsSolvable(G);
Nilpotency class:	not nilpotent	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$18$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$650$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$-1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$1$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,7,13,2,8,14,3,9,15)(4,11,16,5,10,17,6,12,18)$, $(1,11,13,6,9,18,3,10,15,4,8,16,2,12,14,5,7,17)$, $(1,5,15,16,8,11)(2,6,13,17,9,10)(3,4,14,18,7,12)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$3$: $C_3$
$4$: $C_2^2$
$6$: $S_3$ x 3, $C_6$ x 3
$12$: $D_{6}$ x 3, $C_6\times C_2$
$18$: $S_3\times C_3$ x 3
$36$: $S_3^2$ x 3, $C_6\times S_3$ x 3
$108$: 12T70 x 3, 12T71
$324$: 12T130
$972$: 27T271 x 2
$2916$: 18T409
$8748$: 27T786 x 2

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$
$4$:	$C_2^2$
$6$:	$S_3$ x 3, $C_6$ x 3
$12$:	$D_{6}$ x 3, $C_6\times C_2$
$18$:	$S_3\times C_3$ x 3
$36$:	$S_3^2$ x 3, $C_6\times S_3$ x 3
$108$:	12T70 x 3, 12T71
$324$:	12T130
$972$:	27T271 x 2
$2916$:	18T409
$8748$:	27T786 x 2