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. Generic classifications for 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.

Blocks of defect zero

Defect group of size [math]1[/math]   [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]

Blocks for [math] p=2 [/math]

[math]2 \leq |D| \leq 8[/math]   [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 2 1 [math]C_2[/math] 1(1) [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] 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] 6(?) [math]k[/math] [Er87] 8 4 [math]Q_8[/math] [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

[math]|D|=16[/math]   [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 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]

Blocks for [math]p=3[/math]

[math]3 \leq |D| \leq 9[/math]   [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 3 1 [math]C_3[/math] 2(2) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 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]

Blocks for [math]p=5[/math]

[math]5 \leq |D| \leq 25[/math]   [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 5 1 [math]C_5[/math] 6(6) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 25 1 [math]C_{25}[/math] 6(6) No [math]\mathcal{O}[/math] Max 12 classes 25 2 [math]C_5 \times C_5[/math]

Blocks for [math]p\geq 7[/math]

[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 7 1 [math]C_7[/math] 14(14) No [math]\mathcal{O}[/math] Max 19 classes 11 1 [math]C_{11}[/math] No [math]\mathcal{O}[/math] 13 1 [math]C_{13}[/math] No [math]\mathcal{O}[/math] 17 1 [math]C_{17}[/math] No [math]\mathcal{O}[/math] 19 1 [math]C_{19}[/math] No [math]\mathcal{O}[/math] 23 1 [math]C_{23}[/math] No [math]\mathcal{O}[/math]