LMFDB - Galois group: 38T46

Show commands: Magma

magma:G := TransitiveGroup(38, 46);

Group invariants

Abstract group:	$C_{19}^2:(C_9\times D_{18})$	magma:IdentifyGroup(G);
Order:	$116964=2^{2} \cdot 3^{4} \cdot 19^{2}$	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 $n$:	$38$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$46$	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:	$(1,7,15,13,4,11,14,18,17,3,16,8,10,19,12,9,5,6)(21,28,27,38,31,32)(22,36,34,37,23,25)(24,33,29,35,26,30)$, $(1,27,9,34,19,38,3,24,2,35,15,25,17,22,10,23,6,29)(4,32,8,26,13,28,5,21,14,36,11,31,12,20,18,30,16,33)(7,37)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$3$: $C_3$
$4$: $C_2^2$
$6$: $S_3$, $C_6$ x 3
$9$: $C_9$
$12$: $D_{6}$, $C_6\times C_2$
$18$: $S_3\times C_3$, $D_{9}$, $C_{18}$ x 3
$36$: $C_6\times S_3$, $D_{18}$, 36T2
$54$: $C_9\times S_3$, 18T19
$108$: 36T63, 36T69
$162$: 18T74
$324$: 36T461

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$
$4$:	$C_2^2$
$6$:	$S_3$, $C_6$ x 3
$9$:	$C_9$
$12$:	$D_{6}$, $C_6\times C_2$
$18$:	$S_3\times C_3$, $D_{9}$, $C_{18}$ x 3
$36$:	$C_6\times S_3$, $D_{18}$, 36T2
$54$:	$C_9\times S_3$, 18T19
$108$:	36T63, 36T69
$162$:	18T74
$324$:	36T461