Classification by p-group

Classification of Morita equivalences for blocks with a given defect group

On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. Information on broad classes of p-groups can be found here.

We use the following notation for Morita equivalence classes of blocks of finite groups with respect to an algebraically closed field k.

[math]M(x,y,z)[/math] is a class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class.

Note that it is not known that the isomorphism class of a defect group is a Morita invariant, so it could be that [math]M(x,y1,z1)=M(x,y2,z2)[/math] for some [math](y1,z1) \neq (y2,z2)[/math].

Also, at present there is no known example of a k-Morita equivalence class of blocks which splits into more than one Morita equivalence class with respect to a complete discrete valuation ring. If such an example arises, then we will bring in more notation for classes with respect to the d.v.r.

[math]|D|[/math]	SmallGroup	Isotype	Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes	Complete (w.r.t.)?	Derived equiv classes (w.r.t)?	References	Notes
1	1	[math]1[/math]	1(1)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]
2	1	[math]C_2[/math]	1(1)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]
3	1	[math]C_3[/math]	2(2)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]
4	1	[math]C_4[/math]	1(1)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]
4	2	[math]C_2 \times C_2[/math]	3(3)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]	[Er82], [Li94]
5	1	[math]C_5[/math]	6(6)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]
7	1	[math]C_7[/math]	14(14)	No	[math]\mathcal{O}[/math]		Max 19 classes
8	1	[math]C_8[/math]	1(1)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]
8	2	[math]C_4 \times C_2[/math]	1(1)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]
8	3	[math]D_8[/math]	4(?)	[math]k[/math]
8	4	[math]Q_8[/math]	3(?)	[math]k[/math]
8	5	[math]C_2 \times C_2 \times C_2[/math]	8(8)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]	[Ea16]	Uses CFSG
9	1	[math]C_9[/math]	3(3)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]
9	2	[math]C_3 \times C_3[/math]
11	1	[math]C_{11}[/math]		No	[math]\mathcal{O}[/math]
13	1	[math]C_{13}[/math]		No	[math]\mathcal{O}[/math]
16	1	[math]C_{16}[/math]	1(1)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]
16	2	[math]C_4 \times C_4[/math]	2(2)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]	[EKKS14]
16	3	SmallGroup(16,3)				[Sa11]	Block invariants known
16	4	[math]C_4:C_4[/math]				[Sa12]	Block invariants known
16	5	[math]C_8 \times C_2[/math]	1(1)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]
16	6	[math]M_{16}[/math]				[Sa12b]	Block invariants known
16	7	[math]D_{16}[/math]
16	8	[math]SD_{16}[/math]
16	9	[math]Q_{16}[/math]
16	10	[math]C_4 \times C_2 \times C_2[/math]	3(3)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]	[EL18a]
16	11	[math]D_8 \times C_2[/math]				[Sa12]	Block invariants known
16	12	[math]Q_8 \times C_2[/math]				[Sa13]	Block invariants known
16	13	[math]D_8*C_4[/math]				[Sa13b]	Block invariants known
16	14	[math](C_2)^4[/math]	16(16)	[math]\mathcal{O}[/math]	[math]\mathcal{O}[/math]	[Ea18]