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Properties

Label 18T879
Degree $18$
Order $331776$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times S_4^3.A_4$

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Show commands: Magma

magma: G := TransitiveGroup(18, 879);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $879$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times S_4^3.A_4$
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:  (1,4,17)(2,3,18)(7,8)(9,10)(11,16,14,12,15,13), (1,6,14)(2,5,13)(3,9,15,4,10,16)(7,12,18,8,11,17), (1,14,5)(2,13,6)(3,15,10,17,11,8)(4,16,9,18,12,7)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $A_4$ x 5, $C_6\times C_2$
$24$:  $A_4\times C_2$ x 15
$48$:  $C_2^2 \times A_4$ x 5, $C_2^4:C_3$
$96$:  12T56 x 3
$192$:  12T90
$648$:  $S_3 \wr C_3 $
$1296$:  18T283
$2592$:  18T399
$5184$:  18T472
$41472$:  12T292
$82944$:  18T765
$165888$:  18T838

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: None

Degree 9: $S_3 \wr C_3 $

Low degree siblings

18T879 x 23, 24T19613 x 24, 36T27991 x 24, 36T28003 x 24, 36T28029 x 12, 36T28047 x 12, 36T28059 x 12, 36T28158 x 48, 36T28159 x 48, 36T28166 x 24, 36T28167 x 24, 36T28175 x 24, 36T28176 x 24, 36T28390 x 24, 36T28391 x 24, 36T28392 x 24, 36T28393 x 24, 36T28394 x 48, 36T28395 x 48, 36T28402 x 24, 36T28403 x 24, 36T28404 x 24, 36T28405 x 24, 36T28406 x 48, 36T28407 x 48, 36T28425 x 24, 36T28426 x 24, 36T28427 x 24, 36T28428 x 24, 36T28429 x 24, 36T28430 x 24, 36T28431 x 24, 36T28432 x 24, 36T28433 x 48, 36T28434 x 48, 36T28435 x 48, 36T28436 x 48, 36T28437 x 48, 36T28438 x 48, 36T28439 x 48, 36T28440 x 48, 36T28441 x 48, 36T28442 x 48, 36T28443 x 48, 36T28444 x 48, 36T28519 x 12

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

The 360 conjugacy class representatives for $C_2\times S_4^3.A_4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $331776=2^{12} \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  331776.j
magma: IdentifyGroup(G);
 
Character table:    360 x 360 character table

magma: CharacterTable(G);