# Miscallaneous results

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## Blocks with basic algebras of dimension at most 12

In [Li18] Markus Linckelmann calculated the $k$-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9.

Dimension Class Defect group Representative $\dim_k(Z(A))$ $l(A)$ Notes
1 M(1,1,1) $1$ $k1$ 1 1 Blocks of defect zero
2 M(2,1,1) $C_2$ $kC_2$ 2 1
3 M(3,1,1) $C_3$ $kC_3$ 3 1
4 M(4,1,1) $C_4$ $kC_4$ 4 1
4 M(4,2,1) $C_2 \times C_2$ $k(C_2 \times C_2)$ 4 1
5 M(5,1,1) $C_5$ $kC_5$ 5 1
6 M(3,1,2) $C_3$ $kS_3$ 3 2
7 M(5,1,3) $C_5$ $B_0(kA_5)$ 4 2
7 M(7,1,1) $C_7$ $kC_7$ 7 1
8 M(8,1,1) $C_8$ $kC_8$ 8 1
8 M(8,2,1) $C_4 \times C_2$ $k(C_4 \times C_2)$ 8 1
8 M(8,3,1) $D_8$ $kD_8$ 5 1
8 M(8,4,1) $Q_8$ $kQ_8$ 5 1
8 M(8,5,1) $C_2 \times C_2 \times C_2$ $k(C_2 \times C_2 \times C_2)$ 8 1
8 M(7,1,3) $C_7$ $B_0(kPSL_2(13))$ 5 2
9 M(9,1,1) $C_9$ $kC_9$ 9 1
9 M(9,1,3) $C_9$ $B_0(kPSL_2(8))$ 6 2
9 M(9,2,1) $C_3 \times C_3$ $k(C_3 \times C_3)$ 9 1
9 $C_3 \times C_3$  ? 6 2 Unknown
10 M(5,1,2) $C_5$ $kD_{10}$ 4 2
10 M(11,1,3) $C_{11}$ $B_0(kPSL_2(32))$ 7 2
11 M(8,3,3) $D_8$ $kS_4$ 5 2
11 M(7,1,6) $C_7$ $B_0(kA_7)$ 5 3
11 M(11,1,1) $C_{11}$ $kC_{11}$ 11 1
11 M(13,1,3) $C_{13}$ $B_0(kPSL_2(25))$ 8 2
12 M(4,2,3) $C_2 \times C_2$ $kA_4$ 4 3