Binary octahedral group

Definition

The binary octahedral group is a binary von Dyck group with parameters $(4,3,2)$, i.e., it has the presentation:

$\langle x,y,z\mid x^{4}=y^{3}=z^{2}\rangle$.

Arithmetic functions

Function	Value	Explanation
order	48
exponent	24	Elements of order $2,3,4$.
derived length	4
nilpotency class	--	not a nilpotent group.
Frattini length	2	Frattini-free group: intersection of maximal subgroups is trivial.
minimum size of generating set	2
subgroup rank	2	--
max-length	5

Group properties

Property	Satisfied	Explanation	Comment
Abelian group	No
Nilpotent group	No
Metacyclic group	No
Supersolvable group	No
Solvable group	Yes	Length four.
T-group	No
HN-group	No
Monolithic group	Yes	The center of order two is the unique minimal normal subgroup.
One-headed group	Yes