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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|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 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|