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.

See this page for a description of the labelling conventions.

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

[math]1 \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 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] 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] [math]k[/math] [Er87] Principal blocks classified up to source algebra equivalence in [KoLa20] 8 4 [math]Q_8[/math] 3(3) [math]\mathcal{O}[/math] [math]k[/math] [Er88a], [Er88b], [HKL07], [Ei16] 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 Donovan (w.r.t)? 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] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 16 2 [math]C_4 \times C_4[/math] [math]\mathcal{O}[/math] 2(2) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [EKKS14] 16 3 MNA(2,1) No 3(?) No [Sa11] Block invariants known 16 4 [math]C_4:C_4[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [CG12], [Sa12b] 16 5 [math]C_8 \times C_2[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 16 6 [math]M_{16}[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [CG12], [Sa12b] 16 7 [math]D_{16}[/math] [math]k[/math] 5(?) [math]k[/math] [math]k[/math] [Er87] Principal blocks classified up to source algebra equivalence in [KoLa20] 16 8 [math]SD_{16}[/math] [math]k[/math] 7(?) [Er88c], [Er90b] Two other possible classes 16 9 [math]Q_{16}[/math] No 6(?) [math]k[/math] [Er88a], [Er88b], [Ho97] Two possibly infinite families when [math]l(B)=2[/math]. Classified over [math]\mathcal{O}[/math] when [math]l(B)=3[/math] in [Ei16]. Principal blocks classified up to source algebra equivalence in [KoLa20b] 16 10 [math]C_4 \times C_2 \times C_2[/math] [math]\mathcal{O}[/math] 3(3) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [EL18a] 16 11 [math]D_8 \times C_2[/math] No [Sa12] Block invariants known 16 12 [math]Q_8 \times C_2[/math] [math]\mathcal{O}[/math] 3(3) No [EL20] Block invariants known by [Sa13] 16 13 [math]D_8*C_4[/math] No 3(?) No [Sa13b] Block invariants known 16 14 [math](C_2)^4[/math] [math]\mathcal{O}[/math] 16(16) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Ea18]

The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [Sa14].

[math]|D|=32[/math]   [math]|D|[/math] SmallGroup Isotype Donovan (w.r.t)? Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes Complete (w.r.t.)? Derived equiv classes (w.r.t)? References Notes 32 1 [math]C_{32}[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 32 2 [math]MNA(2,2)[/math] [math]\mathcal{O}[/math] 2(2) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [EKS12] 32 3 [math]C_8 \times C_4[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 32 4 [math]C_8:C_4[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [CG12], [Sa12b] 32 5 [math]MNA(3,1)[/math] No [Sa11] Invariants known 32 6 [math]MNA(2,1):C_2[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Sa14] 32 7 SmallGroup(32,7) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Sa14] [math]M_{16}:C_2[/math] 32 8 [math]2.MNA(2,1)[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Sa14] 32 9 [math]D_8:C_4[/math] No [Sa14,10.23] Invariants known 32 10 [math]Q_8:C_4[/math] No [Sa14,10.25] Invariants known 32 11 [math]C_4 \wr C_2[/math] No [Ku80] Invariants known 32 12 [math]C_4:C_8[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [CG12], [Sa12b] 32 13 [math]C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [CG12], [Sa12b] 32 14 [math]C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [CG12], [Sa12b] 32 15 SmallGroup(32,15) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [CG12], [Sa12b] 32 16 [math]C_{16} \times C_2[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 32 17 [math]M_{32}[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [CG12], [Sa12b] 32 18 [math]D_{32}[/math] [math]k[/math] 5(?) [math]k[/math] [math]k[/math] [Er87] Principal blocks classified up to source algebra equivalence in [KoLa20] 32 19 [math]SD_{32}[/math] [math]k[/math] 32 20 [math]Q_{32}[/math] No [Er88a], [Er88b], [Ho97] Two possibly infinite families when [math]l(B)=2[/math]. Classified over [math]\mathcal{O}[/math] when [math]l(B)=3[/math] in [Ei16]. Principal blocks classified up to source algebra equivalence in [KoLa20b] 32 21 [math]C_4 \times C_4 \times C_2[/math] [math]\mathcal{O}[/math] 2(2) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [EKKS14] 32 22 [math]MNA(2,1) \times C_2[/math] No [Sa14,10.25] Invariants known 32 23 [math](C_4:C_4) \times C_2[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Sa14] 32 24 SmallGroup(32,24) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Sa14] 32 25 [math]D_8 \times C_4[/math] No [Sa14,9.7]

Invariants known

32 26 [math]Q_8 \times C_4[/math] [math]\mathcal{O}[/math] 3(3) No [EL20] Invariants known by [Sa14,9.28] 32 27 SmallGroup(32,27) No 32 28 SmallGroup(32,28) No [Sa14,13.11] Invariants known 32 29 SmallGroup(32,29) No [Sa14,13.11] Invariants known 32 30 SmallGroup(32,30) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Sa14] 32 31 SmallGroup(32,31) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Sa14] 32 32 SmallGroup(32,32) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Sa14] 32 33 SmallGroup(32,33) No [Sa14,13.12] Invariants partly known 32 34 SmallGroup(32,34) No [Sa14,13.12] Invariants partly known 32 35 [math]C_4:Q_8[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Sa14] 32 36 [math]C_8 \times C_2 \times C_2[/math] [math]\mathcal{O}[/math] 3(3) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [EL18a] 32 37 [math]M_{16} \times C_2[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Sa14] 32 38 [math]D_8*C_8[/math] No [Sa14,9.18] Invariants known 32 39 [math]D_{16} \times C_2[/math] No [Sa14,9.7] Invariants known 32 40 [math]SD_{16} \times C_2[/math] No [Sa14,9.37] Invariants known 32 41 [math]Q_{16} \times C_2[/math] No [Sa14,9.28] Invariants known 32 42 [math]D_{16}*C_4[/math] No [Sa14,9.18] Invariants known 32 43 SmallGroup(32,43) No 32 44 SmallGroup(32,44) No 32 45 [math]C_4 \times C_2 \times C_2 \times C_2[/math] [math]\mathcal{O}[/math] [Sa14, 13.9] Invariants known 32 46 [math]D_8 \times C_2 \times C_2[/math] No 32 47 [math]Q_8 \times C_2 \times C_2[/math] No 32 48 [math](D_8*C_4) \times C_2[/math] No 32 49 [math]D_8*D_8[/math] No [Sa13c] Invariants partly known 32 50 [math]D_8*Q_8[/math] No [Sa13c] Invariants partly known 32 51 [math](C_2)^5[/math] [math]\mathcal{O}[/math] 34 (34) [math]\mathcal{O}[/math] [Ar19] Derived eq. classes determined for 30 of the 34 Morita eq. classes.

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

[math]1 \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 1 1 [math]1[/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] 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 125[/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] 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] 125 1 [math]C_{125}[/math] 125 2 [math]C_{25} \times C_5[/math] 125 3 [math]5_+^{1+2}[/math] 62(62) [math]\mathcal{O}[/math] [AE23] 125 4 [math]5_-^{1+2}[/math] 125 5 [math]C_5 \times 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 1 1 [math]1[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 7 1 [math]C_7[/math] 14(14) No [math]\mathcal{O}[/math] Max 21 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]