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Properties

Label 18T951
Degree $18$
Order $3359232$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^6.S_4^2:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(18, 951);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $951$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^6.S_4^2:D_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,13,9,2,14,8)(3,15,7)(4,18)(5,16,6,17)(10,11), (1,14,2,13)(3,15)(4,12)(5,10)(6,11)(7,8,9), (1,10,14,16,9,5,3,11,15,18,7,6)(2,12,13,17,8,4)
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
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$
$72$:  $C_3^2:D_4$
$144$:  12T77
$288$:  12T125
$1152$:  $S_4\wr C_2$
$2304$:  12T235
$4608$:  12T260

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:D_4$

Degree 9: None

Low degree siblings

24T22970, 36T47917, 36T47918, 36T47919, 36T47920, 36T47921, 36T47922, 36T47923, 36T47924, 36T47925, 36T47926, 36T47927, 36T47928, 36T47929, 36T47930, 36T47931, 36T47932, 36T47933, 36T47934, 36T47935, 36T47936, 36T47937, 36T47938, 36T47939, 36T47940, 36T47941, 36T47942, 36T47943, 36T47944, 36T47945, 36T47946, 36T47947, 36T47961, 36T48005

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

There are 275 conjugacy classes of elements. Data not shown.

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $3359232=2^{9} \cdot 3^{8}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  3359232.a
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);