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Groupprops β

Finite group that is not isoclinic to a group of smaller order

Definition

A finite group is termed a finite group that is not isoclinic to a group of smaller order if there is no finite group of smaller order such that the two groups are isoclinic groups.

For every equivalence class of finite groups up to isocliny, there is a unique minimal order and a (not necessarily unique) representative of that order.

Examples

Here is a list, for various orders, of finite groups that are not isoclinic to groups of smaller orders. Note that any abelian group is isoclinic to the trivial group, so for any abelianness-forcing number, there are no such finite groups. Where such minimal groups are isoclinic to each other, we indicate this:

Order List of finite groups of that order not isoclinic to groups of smaller order Number of equivalence classes up to isocliny
1 trivial group 1
2 -- 0
3 -- 0
4 -- 0
5 -- 0
6 symmetric group:S3 1
7 -- 0
8 dihedral group:D8, quaternion group (both isoclinic to each other) 1
9 -- 0
10 dihedral group:D10 1
11 -- 0
12 alternating group:A4, (isoclinic pair: dihedral group:D12, dicyclic group:Dic12) 2