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Properties

Label 16T1823
Degree $16$
Order $32768$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^4.C_2\wr D_4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 1823);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1823$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4^4.C_2\wr D_4$
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:  (1,2)(3,4)(5,14,6,13), (15,16), (1,4,6,15,2,3,5,16)(7,9,11,13)(8,10,12,14), (1,5,2,6)(9,13)(10,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $D_{4}$ x 28, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 42, $C_2^4$
$32$:  $C_2^2 \wr C_2$ x 28, $C_2^2 \times D_4$ x 7
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 6, 16T105 x 7
$128$:  $C_2 \wr C_2\wr C_2$ x 12, 16T241 x 3, 16T245 x 3, 16T325
$256$:  16T509 x 6, 32T4223 x 3
$512$:  16T819 x 3, 16T907, 16T919 x 3
$1024$:  32T40151 x 3
$2048$:  16T1340
$4096$:  32T317640
$8192$:  16T1719
$16384$:  32T815463

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_2 \wr C_2\wr C_2$

Low degree siblings

16T1823 x 63, 32T1123296 x 32, 32T1123297 x 32, 32T1123298 x 32, 32T1123299 x 32, 32T1123300 x 32, 32T1123301 x 32, 32T1123302 x 32, 32T1123303 x 32, 32T1123304 x 32, 32T1123305 x 32, 32T1123306 x 32, 32T1123307 x 32, 32T1123308 x 32, 32T1123309 x 32, 32T1123310 x 32, 32T1123311 x 32, 32T1123312 x 32, 32T1123313 x 32, 32T1123314 x 32, 32T1123315 x 32, 32T1123316 x 32, 32T1123317 x 32, 32T1123318 x 32, 32T1123319 x 32, 32T1123320 x 32, 32T1123321 x 32, 32T1123322 x 32, 32T1123323 x 32, 32T1123324 x 32, 32T1123325 x 32, 32T1123326 x 32, 32T1123327 x 32, 32T1123328 x 32, 32T1123329 x 32, 32T1123330 x 32, 32T1123331 x 32, 32T1123332 x 32, 32T1123333 x 32, 32T1123334 x 32, 32T1123335 x 32, 32T1123336 x 32, 32T1123337 x 32, 32T1123338 x 32, 32T1123339 x 32, 32T1123340 x 32, 32T1123341 x 32, 32T1123342 x 32, 32T1123343 x 32, 32T1123344 x 32, 32T1123345 x 32, 32T1123346 x 32, 32T1123347 x 32, 32T1123348 x 32, 32T1123349 x 32, 32T1123350 x 32, 32T1123351 x 32, 32T1123352 x 32, 32T1123353 x 32, 32T1123354 x 32, 32T1123355 x 32, 32T1123356 x 32, 32T1123357 x 32, 32T1123358 x 32, 32T1124904 x 32, 32T1130094 x 16, 32T1130117 x 16, 32T1130614 x 16, 32T1237410 x 16, 32T1237501 x 16, 32T1415705 x 16, 32T1415720 x 16, 32T1468669 x 16, 32T1468670 x 16

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

The 230 conjugacy class representatives for $C_4^4.C_2\wr D_4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $32768=2^{15}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $8$
Label:  32768.cr
magma: IdentifyGroup(G);
 
Character table:    230 x 230 character table

magma: CharacterTable(G);