[GAP Forum] Number of automorphisms in semidihedral group D_16

[Stefan Kohl]

Ranjit Ghumde asked:

> How many automorphisms are exists in Semidihedral group D_16?

In GAP, the semidihedral group (also known as quasidihedral group) of order 16
is SmallGroup(16,8):

gap> G := SmallGroup(16,8);
<pc group of size 16 with 4 generators>
gap> StructureDescription(G);
"QD16"

Its automorphism group can be determined with the operation AutomorphismGroup
as usual:

gap> A := AutomorphismGroup(G);
<group of size 16 with 4 generators>

Thus you see that the number of automorphisms of the semidihedral group
of order 16 is 16.

Actually, the automorphism group is a direct product of a cyclic group of order 2
and a dihedral group of order 8:

gap> StructureDescription(A);
"C2 x D8"

You can also list the automorphisms explicitly:

gap> AsList(A);
[ [ f1*f3, f2*f4 ] -> [ f1*f3, f2*f4 ], [ f1*f3, f2*f4 ] -> [ f1, f2*f4 ],
  [ f1*f3, f2*f4 ] -> [ f1*f4, f2 ], [ f1*f3, f2*f4 ] -> [ f1*f3*f4, f2 ],
  [ f1*f3, f2*f4 ] -> [ f1*f3*f4, f2*f3*f4 ], [ f1*f3, f2*f4 ] -> [ f1*f4, f2*f3 ],
  [ f1*f3, f2*f4 ] -> [ f1, f2*f3*f4 ], [ f1*f3, f2*f4 ] -> [ f1*f3, f2*f3 ],
  [ f1*f3, f2*f4 ] -> [ f1*f3, f2 ], [ f1*f3, f2*f4 ] -> [ f1, f2 ],
  [ f1*f3, f2*f4 ] -> [ f1*f4, f2*f4 ], [ f1*f3, f2*f4 ] -> [ f1*f3*f4, f2*f4 ],
  [ f1*f3, f2*f4 ] -> [ f1*f3*f4, f2*f3 ], [ f1*f3, f2*f4 ] -> [ f1*f4, f2*f3*f4 ],
  [ f1*f3, f2*f4 ] -> [ f1, f2*f3 ], [ f1*f3, f2*f4 ] -> [ f1*f3, f2*f3*f4 ] ]

Hope this helps,

    Stefan