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Properties

Label 18T912
Degree $18$
Order $663552$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times S_4^3.S_4$

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Show commands: Magma

magma: G := TransitiveGroup(18, 912);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $912$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times S_4^3.S_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,5,16)(2,6,15)(3,8,14)(4,7,13)(9,11,17,10,12,18), (7,8)(9,10)(11,15)(12,16), (1,6,17,9,2,5,18,10)(3,8,4,7)(11,14)(12,13)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$24$:  $S_4$ x 3, $S_3 \times C_2^2$
$48$:  $S_4\times C_2$ x 9
$96$:  $V_4^2:S_3$, 12T48 x 3
$192$:  12T100 x 3
$384$:  12T139
$1296$:  $S_3\wr S_3$
$2592$:  18T394
$5184$:  18T483
$10368$:  18T556
$82944$:  12T294
$165888$:  18T836
$331776$:  18T880

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: None

Degree 9: $S_3\wr S_3$

Low degree siblings

18T912 x 7, 24T20673 x 8, 36T33473 x 4, 36T33489 x 4, 36T33535 x 2, 36T33537 x 4, 36T33545 x 4, 36T33561 x 4, 36T33587 x 4, 36T33590 x 4, 36T33595 x 2, 36T33601 x 4, 36T33609 x 4, 36T33688 x 8, 36T33689 x 8, 36T33690 x 8, 36T33691 x 8, 36T33706 x 8, 36T33707 x 8, 36T33708 x 8, 36T33709 x 8, 36T33750 x 4, 36T33751 x 4, 36T33752 x 8, 36T33753 x 8, 36T33756 x 8, 36T33757 x 8, 36T33764 x 8, 36T33765 x 8, 36T33772 x 8, 36T33773 x 8, 36T33776 x 8, 36T33777 x 8, 36T33778 x 8, 36T33779 x 8, 36T33780 x 8, 36T33781 x 8, 36T33782 x 8, 36T33783 x 8, 36T33784 x 8, 36T33785 x 8, 36T33786 x 8, 36T33787 x 8, 36T33788 x 8, 36T33789 x 8, 36T33790 x 8, 36T33791 x 8, 36T33825 x 4

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

The 330 conjugacy class representatives for $C_2\times S_4^3.S_4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $663552=2^{13} \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  663552.i
magma: IdentifyGroup(G);
 
Character table:    330 x 330 character table

magma: CharacterTable(G);