The class number \(k(G)\) of a group \(G\) is the number of conjugacy classes of \(G\). In 1903, Landau proved in [Lan03] that for every \(n \in \mathbb{N}\), there are only finitely many finite groups with exactly \(n\) conjugacy classes. The SmallClassNr package provides access to the finite groups with class number at most \(14\).
These groups were classified in the following papers:
\(k(G) \leq 5\), by Miller in [Mil11] and independently by Burnside in [Bur11]
\(k(G) = 6,7\), by Poland in [Pol68]
\(k(G) = 8\), by Kosvintsev in [Kos74]
\(k(G) = 9\), by Odincov and Starostin in [OS76]
\(k(G) = 10,11\), by Vera López and Vera López in [VLVL85] (1)
\(k(G) = 12\), by Vera López and Vera López in [VLVL86] (2)
\(k(G) = 13, 14\), by Vera López and Sangroniz in [VLS07]
(1) In [VLVL85], three distinct groups of the form \((C_5 \times C_5) \rtimes C_4\) order \(100\) with class number \(10\) are given. However, only two such groups exist, being the ones with IdClassNr
equal to [10,25]
and [10,26]
.
(2) In [VLVL86], only 48 groups with class number 12 are listed. The three missing groups are provided in the appendix of [VLS07]. These are the groups with IdClassNr
equal to [12,13]
, [12,16]
and [12,39]
.
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