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 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 1 1 [math]1[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 2 1 [math]C_2[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 4 1 [math]C_4[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 4 2 [math]C_2 \times C_2[/math] [math]\mathcal{O}[/math] 3(3) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Er82], [Li94] 8 1 [math]C_8[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 8 2 [math]C_4 \times C_2[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] 8 3 [math]D_8[/math] [math]\mathcal{O}[/math] 6(?) [math]k[/math] [math]k[/math] [Er87] See discussion in [EEKLS26] regarding Donovan over [math]\mathcal{O}[/math]. Principal blocks classified up to source algebra equivalence in [KoLa20] 8 4 [math]Q_8[/math] [math]\mathcal{O}[/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] [math]\mathcal{O}[/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]\mathcal{O}[/math] 5(?) [math]k[/math] [math]k[/math] [Er87] See discussion in [EEKLS26] regarding Donovan over [math]\mathcal{O}[/math]. 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. Donovan's conjecture known over [math]\mathcal{O}[/math] for [math]l(B) \neq 2[/math] by [La16] (see discussion in [EEKLS26]). 16 9 [math]Q_{16}[/math] [math]\mathcal{O}[/math] 6(?) [math]k[/math] [Er88a], [Er88b], [EEKLS26], [Ho97] Classified over [math]\mathcal{O}[/math] when [math]l(B)=3[/math] in [Ei16]. When [math]l(B)=2[/math] there are between 2 and 4 Morita equivalence classes over [math]k[/math]. Finite list of Morita equivalence classes over [math]\mathcal{O}[/math], with possible repetition, by [EEKLS26]. 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 6(6) No [Ku80], [KoLaSa23] Invariants known. Principal blocks classified up to source algebra equivalence in [KoLaSa23] 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]\mathcal{O}[/math] 5(?) [math]k[/math] [math]k[/math] [Er87] See discussion in [EEKLS26] regarding Donovan over [math]\mathcal{O}[/math]. Principal blocks classified up to source algebra equivalence in [KoLa20] 32 19 [math]SD_{32}[/math] [math]k[/math] Donovan's conjecture known over [math]\mathcal{O}[/math] for [math]l(B) \neq 2[/math] by [La16] (see discussion in [EEKLS26]). 32 20 [math]Q_{32}[/math] [math]\mathcal{O}[/math] [math]k[/math] [Er88a], [Er88b], [EEKLS26], [Ho97] Classified over [math]\mathcal{O}[/math] when [math]l(B)=3[/math] in [Ei16]. When [math]l(B)=2[/math] there are between 2 and 4 Morita equivalence classes over [math]k[/math]. Finite list of Morita equivalence classes over [math]\mathcal{O}[/math], with possible repetition, by [EEKLS26]. 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] 8(8) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [EL23] 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.

Fusion trivial groups of order 64 were determined by Pete Gautam.

[math]|D|=64[/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 64 1 [math]C_{64}[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 2 [math]C_8 \times C_8[/math] [math]\mathcal{O}[/math] 2(2) [math]\mathcal{O}[/math] [EKKS14] 64 3 SmallGroup(64,3) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 4 [math](C_2)^3:C_8[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 5 SmallGroup(64,5) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]C_8:C_8[/math]. Fusion trivial by [CG12], [Sa12b] 64 6 [math]D_8:C_8[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 7 [math]Q_8:C_8[/math] No 64 8 SmallGroup(64,8) No 64 9 SmallGroup(64,9) No 64 10 SmallGroup(64,10) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 11 SmallGroup(64,11) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 12 SmallGroup(64,12) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 13 SmallGroup(64,13) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 14 SmallGroup(64,14) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 15 SmallGroup(64,15) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]C_8:C_8[/math]. Fusion trivial by [CG12], [Sa12b] 64 16 SmallGroup(64,16) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]C_8:C_8[/math]. Fusion trivial by [CG12], [Sa12b] 64 17 SmallGroup(64,17) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 18 SmallGroup(64,18) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 19 SmallGroup(64,19) [math]\mathcal{O}[/math] Maxperm (as in the supplied data); no class count or completeness assertion supplied 64 20 SmallGroup(64,20) No 64 21 SmallGroup(64,21) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 22 SmallGroup(64,22) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 23 SmallGroup(64,23) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 24 SmallGroup(64,24) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]M_{16}:C_4[/math]. Fusion trivial 64 25 SmallGroup(64,25) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]M_{16}:C_4[/math]. Fusion trivial 64 26 [math]C_{16} \times C_4[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 27 SmallGroup(64,27) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]C_{16}:C_4[/math]. Fusion trivial by [CG12], [Sa12b] 64 28 SmallGroup(64,28) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]C_{16}:C_4[/math]. Fusion trivial by [CG12], [Sa12b] 64 29 [math](C_2)^2:C_{16}[/math] No 64 30 SmallGroup(64,30) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]M_{32}:C_2[/math]. Fusion trivial 64 31 SmallGroup(64,31) No [math]M_{32}:C_2[/math] 64 32 [math]C_2 \wr C_4[/math] No 64 33 SmallGroup(64,33) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 34 SmallGroup(64,34) No 64 35 SmallGroup(64,35) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 36 SmallGroup(64,36) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 37 SmallGroup(64,37) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 38 SmallGroup(64,38) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 39 SmallGroup(64,39) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 40 SmallGroup(64,40) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 41 SmallGroup(64,41) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 42 SmallGroup(64,42) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 43 SmallGroup(64,43) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 44 [math]C_4:C_{16}[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial by [CG12], [Sa12b] 64 45 SmallGroup(64,45) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 46 SmallGroup(64,46) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 47 SmallGroup(64,47) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 48 SmallGroup(64,48) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] [math]C_{16}:C_4[/math]. Fusion trivial by [CG12], [Sa12b] 64 49 SmallGroup(64,49) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 50 [math]C_{32} \times C_2[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 51 [math]M_6(2)[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 52 [math]D_{64}[/math] No 64 53 [math]SD_{64}[/math] No 64 54 [math]Q_{64}[/math] No 64 55 [math]C_4 \times C_4 \times C_4[/math] [math]\mathcal{O}[/math] 4(4) [math]\mathcal{O}[/math] [WZZ18], [EL18a] 64 56 SmallGroup(64,56) No 64 57 SmallGroup(64,57) No 64 58 [math]C_{4} \times (C_2 \times C_2):C_4[/math] No 64 59 [math]C_{4} \times (C_4:C_4)[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 60 SmallGroup(64,60) No 64 61 SmallGroup(64,61) No 64 62 SmallGroup(64,62) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 63 SmallGroup(64,63) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 64 SmallGroup(64,64) No 64 65 SmallGroup(64,65) No 64 66 SmallGroup(64,66) No 64 67 SmallGroup(64,67) No 64 68 SmallGroup(64,68) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 69 SmallGroup(64,69) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 70 SmallGroup(64,70) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 71 SmallGroup(64,71) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 72 SmallGroup(64,72) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 73 SmallGroup(64,73) No 64 74 [math](C_2)^3:Q_8[/math] No 64 75 SmallGroup(64,75) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 76 SmallGroup(64,76) No 64 77 SmallGroup(64,77) No 64 78 SmallGroup(64,78) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 79 SmallGroup(64,79) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 80 SmallGroup(64,80) No 64 81 SmallGroup(64,81) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 82 SmallGroup(64,82) [math]\mathcal{O}[/math] 6(6) [math]\mathcal{O}[/math] [Ea24] Suzuki [math]2[/math]-group of type A 64 83 [math]C_{8} \times C_4 \times C_2[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 84 [math]C_{2} \times (C_8:C_4)[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 85 [math]M_4(2) \times C_4[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 86 SmallGroup(64,86) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 87 [math]C_{2} \times (C_2 \times C_2):C_8[/math] No 64 88 SmallGroup(64,88) No 64 89 [math](C_8 \times C_2 \times C_2):C_2[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 90 [math]C_2 \times (C_2 \times C_2 \times C_2):C_4[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 91 SmallGroup(64,91) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 92 SmallGroup(64,92) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 93 SmallGroup(64,93) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 94 SmallGroup(64,94) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 95 [math]C_{2} \times (D_8:C_4)[/math] No 64 96 [math]C_{2} \times (Q_8:C_4)[/math] No 64 97 SmallGroup(64,97) No 64 98 SmallGroup(64,98) No 64 99 SmallGroup(64,99) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 100 SmallGroup(64,100) No 64 101 [math]C_{2} \times (C_4 \wr C_2)[/math] No 64 102 [math](C_4 \times C_4):(C_2 \times C_2)[/math] No 64 103 [math]C_{2} \times (C_4:C_8)[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 104 [math]C_{4}:M_4(2)[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 105 SmallGroup(64,105) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 106 SmallGroup(64,106) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 107 SmallGroup(64,107) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 108 SmallGroup(64,108) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 109 [math]M_4(2):C_4[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 110 SmallGroup(64,110) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 111 SmallGroup(64,111) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 112 SmallGroup(64,112) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 113 SmallGroup(64,113) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 114 SmallGroup(64,114) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 115 [math]D_8 \times C_8[/math] No 64 116 SmallGroup(64,116) No 64 117 SmallGroup(64,117) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 118 [math]D_{16} \times C_4[/math] No 64 119 [math]SD_{16} \times C_4[/math] No 64 120 [math]Q_{16} \times C_4[/math] No 64 121 [math]SD_{16}:C_4[/math] No 64 122 [math]Q_{16}:C_4[/math] No 64 123 [math]D_{16}:C_4[/math] [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 124 SmallGroup(64,124) No 64 125 SmallGroup(64,125) No 64 126 [math]Q_{8} \times C_8[/math] No Invariants known by [Sa14,9.28] 64 127 SmallGroup(64,127) [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 128 [math](C_2 \times C_2):D_{16}[/math] No 64 129 [math]Q_8:D_{8}[/math] No 64 130 [math]D_8:D_{8}[/math] No 64 131 No 64 132 No 64 133 No 64 134 [math]\operatorname{Syl}_2(M_{12})[/math] No 64 135 No 64 136 No 64 137 No 64 138 No 64 139 No 64 140 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 141 No 64 142 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 143 No 64 144 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 145 No 64 146 No 64 147 No 64 148 No 64 149 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 150 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 151 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 152 No 64 153 No 64 154 No 64 155 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 156 No 64 157 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 158 No 64 159 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 160 No 64 161 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 162 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 163 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 164 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 165 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 166 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 167 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 168 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 169 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 170 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 171 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 172 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 173 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 174 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 175 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 176 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 177 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 178 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 179 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 180 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 181 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 182 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 183 [math]C_{16} \times C_2^2[/math] [math]\mathcal{O}[/math] 3(3) [math]\mathcal{O}[/math] [EL18a] 64 184 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 185 No 64 186 No 64 187 No 64 188 No 64 189 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 190 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 191 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 192 [math]C_4^2 \times C_2^2[/math] [math]\mathcal{O}[/math] 8(8) [math]\mathcal{O}[/math] [EL23] Abelian 64 193 No 64 194 No 64 195 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 196 No 64 197 No 64 198 No 64 199 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 200 No 64 201 No 64 202 No 64 203 No 64 204 No 64 205 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 206 No 64 207 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 208 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 209 No 64 210 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 211 No 64 212 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 213 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 214 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 215 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 216 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 217 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 218 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 219 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 220 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 221 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 222 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 223 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 224 No 64 225 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 226 No 64 227 No 64 228 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 229 No 64 230 No 64 231 No 64 232 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 233 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 234 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 235 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 236 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 237 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 238 No 64 239 [math]Q_{8} \times Q_8[/math] [math]\mathcal{O}[/math] 64 240 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 241 No 64 242 [math]\operatorname{Syl}_2(L_3(4))[/math] No 64 243 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 244 [math]\mathcal{O}[/math] 1(1) [math]\mathcal{O}[/math] Fusion trivial 64 245 SmallGroup(64,245) [math]\mathcal{O}[/math] 3(3) [math]\mathcal{O}[/math] [Ea24] Suzuki [math]2[/math]-group of type B 64 246 [math]C_8 \times C_2^3[/math] [math]\mathcal{O}[/math] 8(8) [math]\mathcal{O}[/math] [EL23] 64 247 No 64 248 No 64 249 No 64 250 No 64 251 No 64 252 No 64 253 No 64 254 No 64 255 No 64 256 No 64 257 No 64 258 No 64 259 No 64 260 [math]C_4 \times C_2^4[/math] [math]\mathcal{O}[/math] No [EEL18] 64 261 No 64 262 No 64 263 No 64 264 No 64 265 No 64 266 [math]2_+^{1+4} * C_4[/math] No 64 267 [math](C_2)^6[/math] [math]\mathcal{O}[/math]  ?(81) No [Ar21], [EEL18] 81 known [math]\mathcal{O}[/math]-classes containing a principal block; full classification not known

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] Inertial quotients are consistent within classes 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]