Conjectural classification of purely nonprincipal Morita classes with defect group (C2)6(C_2)^6

Let D=(C2)6D=(C_2)^6 and consider purely nonprincipal blocks with defect group DD. For each group in the following list, there is one relevant Morita equivalence class of nonprincipal blocks: ((C2)43+1+2)×(C2)2((C_2)^4\rtimes 3^{1+2}_+)\times(C_2)^2, ((C2)43+1+2)×A4((C_2)^4\rtimes 3^{1+2}_+)\times A_4, ((C2)43+1+2)×A5((C_2)^4\rtimes 3^{1+2}_+)\times A_5, ((C2)5(C73+1+2))×C2((C_2)^5\rtimes(C_7\rtimes3^{1+2}_+))\times C_2, ((SL2(8)×(C2)2)3+1+2)×C2((\operatorname{SL}_2(8)\times(C_2)^2)\rtimes3^{1+2}_+)\times C_2, (C2)6(C73+1+2)2(C_2)^6\rtimes(C_7\rtimes3^{1+2}_+)_2, (C2)6(C5×3+1+2)(C_2)^6\rtimes(C_5\times3^{1+2}_+), (C2)67+1+2(C_2)^6\rtimes7^{1+2}_+, (C2)63+1+2(C_2)^6\rtimes3^{1+2}_+, (C2)6((C3)23+1+2)(C_2)^6\rtimes((C_3)^2\rtimes3^{1+2}_+), the two classes labelled b1b_1 and b2b_2 for (C2)6SmallGroup(729,122)(C_2)^6\rtimes\operatorname{SmallGroup}(729,122), (C2)6SmallGroup(1029,12)(C_2)^6\rtimes\operatorname{SmallGroup}(1029,12), (C2)6((C7×C7)3+1+2)(C_2)^6\rtimes((C_7\times C_7)\rtimes3^{1+2}_+), and SL2(8)23+1+2\operatorname{SL}_2(8)^2\rtimes3^{1+2}_+. Purely nonprincipal classification conjecture. The purely nonprincipal Morita equivalence classes of blocks with defect group (C2)6(C_2)^6 are exactly these classes, and every block of a finite group with defect group (C2)6(C_2)^6 is Morita equivalent either to one listed above or to a block in the list of Theorem 1.1. The source proposes this as a conjectural complete classification; it remains open.

Sources & referencesView supporting material

Primary source

Cesare Giulio Ardito, “Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64”, arXiv:2010.07629 (2020).

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