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<Chapter Label="Chapter_Classification">
<Heading>Classification</Heading>
The <Emph>class number</Emph> <Math>k(G)</Math> of a group <Math>G</Math> is the number of conjugacy classes of <Math>G</Math>. In 1903, Landau proved in <Cite Key='land03-a' /> that for every <Math>n \in \mathbb{N}</Math>, there are only finitely many finite groups with exactly <Math>n</Math> conjugacy classes.
The &PACKAGENAME; package provides access to the finite groups with class number at most <Math>14</Math>.
<P/>
These groups were classified in the following papers:
<List>
<Item> <Math>k(G) \leq 5</Math>, by Miller in <Cite Key='mill11-a' /> and independently by Burnside in <Cite Key='burn11-a' /></Item>
<Item> <Math>k(G) = 6,7</Math>, by Poland in <Cite Key='pola68-a' /></Item>
<Item> <Math>k(G) = 8</Math>, by Kosvintsev in <Cite Key='kosv74-a' /></Item>
<Item> <Math>k(G) = 9</Math>, by Odincov and Starostin in <Cite Key='os76-a' /></Item>
<Item> <Math>k(G) = 10,11</Math>, by Vera López and Vera López in <Cite Key='ll85-a' /> (1) </Item>
<Item> <Math>k(G) = 12</Math>, by Vera López and Vera López in <Cite Key='ll86-a' /> (2) </Item>
<Item> <Math>k(G) = 13, 14</Math>, by Vera López and Sangroniz in <Cite Key='vs07-a' /></Item>
</List>
<P/>
<P/>
(1) In <Cite Key='ll85-a' />, three distinct groups of the form <Math>(C_5 \times C_5) \rtimes C_4</Math> order <Math>100</Math> with class number <Math>10</Math> are given.
However, only two such groups exist, being the ones with <C>IdClassNr</C> equal to <C>[10,25]</C> and <C>[10,26]</C>.
<P/>
(2) In <Cite Key='ll86-a' />, only 48 groups with class number 12 are listed. The three missing groups are provided in the appendix of <Cite Key='vs07-a' />.
These are the groups with <C>IdClassNr</C> equal to <C>[12,13]</C>, <C>[12,16]</C> and <C>[12,39]</C>.
</Chapter>
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