For 
,
the semidihedral group of group order 
, denoted 
, is the finite group
with group presentation
 |
It is also called the quasidihedral group and denoted 
(Bautista Serrano et al. 2023).
Equivalently, 
is the semidirect product ![C_(2^(n-1))×AdjustmentBox[│, BoxMargins -> {{-0.27, 0.13913}, {-0.5, 0.5}}]C_2](/images/equations/SemidihedralGroup/Inline6.svg)
in which the nonidentity element of 
maps a generator 
of the cyclic
group 
to 
.
The smallest member of the family is 
, which is 
in the GAP Small Groups Library. It contains
subgroups isomorphic to the order-8 dihedral
group 
and the quaternion group 
. It is not isomorphic to the order-16 dihedral group 
: conjugation by 
maps 
to 
in 
, but to 
in 
(Bautista Serrano et al. 2023).
The Möbius-Kantor graph is a Cayley graph of 
(García-Marco and Knauer 2022).
