# Center of binary octahedral group

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z2 and the group is (up to isomorphism) binary octahedral group (see subgroup structure of binary octahedral group).
The subgroup is a normal subgroup and the quotient group is isomorphic to symmetric group:S4.
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## Arithmetic functions

Function Value Explanation
order of the whole group 48
order of the subgroup 2
index of the subgroup 24 Follows from Lagrange's theorem.
size of conjugacy class = index of normalizer 1 center is normal
number of conjugacy classes in automorphism class 1 center is characteristic

## Subgroup-defining functions

Subgroup-defining function What it means in general Why it takes this value
center set of elements that commute with every group element
third derived subgroup derived subgroup of derived subgroup of derived subgroup derived subgroup is isomorphic to SL(2,3), second derived subgroup is isomorphic to quaternion group
socle join of all minimal normal subgroups It is the unique minimal normal subgroup. The group is a monolithic group.
Frattini subgroup intersection of all maximal subgroups