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

[math]|D|=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] 3(?) [math]k[/math] [Er88a], [Er88b] Over [math]\mathcal{O}[/math] for [math]l(B) \neq 2[/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 27[/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] 27 1 [math]C_{27}[/math] 3(3) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 27 2 [math]C_9 \times C_3[/math] 27 3 [math]3_+^{1+2}[/math] 27 4 [math]3_-^{1+2}[/math] 27 5 [math]C_3 \times 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]   [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]