# C3xC3

## Blocks with defect group $C_3 \times C_3$

Source algebra equivalence classes of principal blocks with this defect group have been classified by Koshitani in [Ko03] using the CFSG. This accounts for classes M(9,2,1) to M(9,2,22). Note that it does not follow from the results in [Ko03] that there is a unique $\mathcal{O}$-Morita equivalence class for each of these $k$-Morita equivalence classes as they may also contain non-principal blocks. Some Picard groups calculated in [Mar].

The numerical invariants of arbitrary blocks with defect group $C_3 \times C_3$ were calculated for all inertial quotients except $C_8$ and $Q_8$ by Kiyota in [Ki84], and these cases remain an open problem. Kiyota's calculations do not involve the CFSG.

CLASSIFICATION INCOMPLETE
Class Representative # lifts / $\mathcal{O}$ $k(B)$ $l(B)$ Inertial quotients ${\rm Pic}_\mathcal{O}(B)$ ${\rm Pic}_k(B)$ ${\rm mf_\mathcal{O}(B)}$ ${\rm mf_k(B)}$ Notes
M(9,2,1) $k(C_3 \times C_3)$ 1 9 1 $1$ $(C_3 \times C_3):GL_2(3)$ 1 1
M(9,2,2) $k(S_3 \times C_3)$ 9 2 $C_2$ $C_2 \times S_3$ 1 1
M(9,2,3) $k(C_3 \times C_3):C_2$ 6 2 $C_2$ (power of Singer cycle) $C_8$ 1 1 SmallGroup(18,4)
M(9,2,4) $k(S_3 \times S_3)$ 9 4 $C_2 \times C_2$ $D_8$ 1 1
M(9,2,5) $k(C_3 \times C_3):C_4$ 6 4 $C_4$ $C_2 \times D_8$ 1 1
M(9,2,6) $B_0(kA_6)$ 6 4 $C_4$ $C_2 \times C_2$ 1 1
M(9,2,7) $B_0(kA_7)$ 6 4 $C_4$ $C_2 \times C_2$ 1 1
M(9,2,8) $k(C_3 \times C_3):C_8$ 9 8 $C_8$ $S_4$ 1 1
M(9,2,9) $B_0(kPGL_2(9))$ 9 8 $C_8$ $C_2 \times C_2$ 1 1
M(9,2,10) $k(C_3 \times C_3):D_8$ 9 5 $D_8$ $D_8$ 1 1
M(9,2,11) $B_0(kA_8)$ 9 5 $D_8$ $C_2$ 1 1
M(9,2,12) $B_0(S_6)$ 9 5 $D_8$ $C_2 \times C_2$ 1 1
M(9,2,13) $B_0(S_7)$ 9 5 $D_8$ $C_2 \times C_2$ 1 1
M(9,2,14) $k(C_3 \times C_3):Q_8$ 6 5 $Q_8$ $S_4$ 1 1
M(9,2,15) $B_0(kM_{22})$ 6 5 $Q_8$ $C_2 \times C_2$ 1 1 Also $M_{10}$
M(9,2,16) $B_0(kPSL_3(4))$ 6 5 $Q_8$ 1 1
M(9,2,17) $k(C_3 \times C_3):SD_{16}$ 9 7 $SD_{16}$ $C_2 \times C_2$ 1 1
M(9,2,18) $B_0(kM_{11})$ 9 7 $SD_{16}$ 1 1
M(9,2,19) $B_0(kHS)$ 9 7 $SD_{16}$ 1 1
M(9,2,20) $B_0(kM_{23})$ 9 7 $SD_{16}$ $1$ 1 1
M(9,2,21) $B_0(kPSL_3(4).2_3)$ 9 7 $SD_{16}$ 1 1 Extension by graph automorphism
M(9,2,22) $B_0(k{\rm Aut}(S_6))$ 9 7 $SD_{16}$ $C_2 \times C_2$ 1 1
M(9,2,23) Faithful block of $k((C_3 \times C_3):Q_8)$, in which $Z(Q_8)$ acts trivially 6 1 $C_2 \times C_2$ 1 1 SmallGroup(72,24)
M(9,2,24) Faithful block of $k((C_3 \times C_3):SD_{16})$, in which $Z(SD_{16})$ acts trivially 6 2 $D_8$ 1 1
M(9,2,25) $B_{10}(k(4.M_{22}))$ 6 5 $Q_8$ 1 1
M(9,2,26) $B_{7}(k(2.HS))$ 9 5 $D_8$ 1 1
M(9,2,27) $B_{2}(k(HS))$ 9 7 $SD_{16}$ 1 1
$B_3(kCo_1)$ 9 5 $D_8$ 1 1
$B_6(kJ_4)$ 9 5 $D_8$ 1 1
$B_2(kFi_{24}')$ 6 4 $C_4$ 1 1
Block of $kFi_{24}'.2$ covering $B_2(kFi_{24}')$ 9 5 $D_8$ 1 1
$B_3(k(HS.2))$ 9 5 $D_8$ 1 1
$B_{12}(k(2.HS.2))$ 9 7 $SD_{16}$ 1 1
$B_6(k(2.M_{22}))$ 6 5 1 1
$B_9(k(2.M_{22}.2))$ 9 7 $SD_{16}$ 1 1
$B_6(kB)$ 9 5 $D_8$ 1 1
$B_2(kB)$ 9 7 $D_8$ 1 1
$B_3(kB)$ 9 7 $D_8$ 1 1

Some open problems:

• Determine whether $B_6(k(2.M_{22}))$ is Morita equivalent to $(C_3 \times C_3):Q_8$.
• Determine whether $B_0(kA_8)$ is Morita equivalent to $B_3(k(HS.2))$.
• Determine whether $B_3(kCo_1), B_6(kJ_4), B_2(kFi_{24}'), B_2(kB), B_3(kB), B_6(kB)$ are in Morita equivalence classes not listed in the table.