(C2)^5
Blocks with defect group [math](C_2)^5[/math]
These were classified in [Ar19] using the CFSG. Each of the 34 [math]k[/math]-Morita equivalence classes lifts to an unique class over [math]\mathcal{O}[/math]. The known Picard groups were computed in [EL18c] or using the main theorem of [Liv19]
| Class | Representative | # lifts / [math]\mathcal{O}[/math] | [math]k(B)[/math] | [math]l(B)[/math] | Inertial quotients | [math]{\rm Pic}_\mathcal{O}(B)[/math] | [math]{\rm Pic}_k(B)[/math] | [math]{\rm mf_\mathcal{O}(B)}[/math] | [math]{\rm mf_k(B)}[/math] | Notes |
|---|---|---|---|---|---|---|---|---|---|---|
| M(32,51,1) | [math]k((C_2)^5)[/math] | 1 | 32 | 1 | [math]1[/math] | [math](C_2)^5:GL_5(2)[/math] | 1 | 1 | ||
| M(32,51,2) | [math]k(A_4 \times (C_2)^3)[/math] | 1 | 32 | 3 | [math]C_3[/math] | [math]((C_2)^3 : GL_3(2)) \times S_3[/math] | 1 | 1 | ||
| M(32,51,3) | [math]B_0(k(A_5 \times (C_2)^3))[/math] | 1 | 32 | 3 | [math]C_3[/math] | [math]((C_2)^3 : GL_3(2)) \times C_2[/math] | 1 | 1 | ||
| M(32,51,4) | [math]k(((C_2)^4 :C_3) \times C_2)[/math] | 1 | 16 | 3 | [math]C_3[/math] | 1 | 1 | The action of [math]C_3[/math] comes from the 5th power of a Singer cycle for [math]\mathbb{F}_{16}[/math] | ||
| M(32,51,5) | [math]k(((C_2)^4 :C_5) \times C_2)[/math] | 1 | 16 | 5 | [math]C_5[/math] | 1 | 1 | |||
| M(32,51,6) | [math]k(((C_2)^3:C_7) \times (C_2)^2)[/math] | 1 | 32 | 7 | [math]C_7[/math] | 1 | 1 | |||
| M(32,51,7) | [math]B_0(k(SL_2(8) \times (C_2)^2))[/math] | 1 | 32 | 7 | [math]C_7[/math] | 1 | 1 | |||
| M(32,51,8) | [math]k(A_4 \times A_4 \times C_2)[/math] | 1 | 32 | 9 | [math]C_3 \times C_3[/math] | 1 | 1 | |||
| M(32,51,9) | [math]B_0(k(A_4 \times A_5 \times C_2))[/math] | 1 | 32 | 9 | [math]C_3 \times C_3[/math] | 1 | 1 | |||
| M(32,51,10) | [math]B_0(k(A_5 \times A_5 \times C_2))[/math] | 1 | 32 | 9 | [math]C_3 \times C_3[/math] | 1 | 1 | |||
| M(32,51,11) | [math]k((C_2)^4 : C_{15} \times C_2)[/math] | 1 | 32 | 15 | [math]C_{15}[/math] | 1 | 1 | |||
| M(32,51,12) | [math]B_0(k(SL_2(16) \times C_2))[/math] | 1 | 32 | 15 | [math]C_{15}[/math] | 1 | 1 | |||
| M(32,51,13) | [math]k(((C_2)^3:C_7) \times A_4)[/math] | 1 | 32 | 21 | [math]C_{21}[/math] | 1 | 1 | |||
| M(32,51,14) | [math]B_0(k(((C_2)^3:C_7) \times A_5))[/math] | 1 | 32 | 21 | [math]C_{21}[/math] | 1 | 1 | |||
| M(32,51,15) | [math]B_0(k(SL_2(8) \times A_4))[/math] | 1 | 32 | 21 | [math]C_{21}[/math] | 1 | 1 | |||
| M(32,51,16) | [math]B_0(k(SL_2(8) \times A_5))[/math] | 1 | 32 | 21 | [math]C_{21}[/math] | 1 | 1 | |||
| M(32,51,17) | [math]k(((C_2)^3:(C_7:C_3)) \times (C_2)^2)[/math] | 1 | 32 | 5 | [math]C_7:C_3[/math] | 1 | 1 | |||
| M(32,51,18) | [math]B_0(k(J_1 \times (C_2)^2))[/math] | 1 | 32 | 5 | [math]C_7:C_3[/math] | 1 | 1 | |||
| M(32,51,19) | [math]B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))[/math] | 1 | 32 | 5 | [math]C_7:C_3[/math] | 1 | 1 | |||
| M(32,51,20) | [math]k((C_2)^5:(C_7:C_3))[/math] | 1 | 16 | 5 | [math]C_7:C_3[/math] | 1 | 1 | The action of the subgroup [math]C_3[/math] is as specified in M(32,51,4) above. | ||
| M(32,51,21) | [math]B_0(k((SL_2(8) \times (C_2)^2):C_3)[/math] | 1 | 16 | 5 | [math]C_7:C_3[/math] | 1 | 1 | The action of the subgroup [math]C_3[/math] is as specified in M(32,51,4) above. | ||
| M(32,51,22) | [math]k((C_2)^5:C_{31})[/math] | 1 | 32 | 31 | [math]C_{31}[/math] | 1 | 1 | |||
| M(32,51,23) | [math]B_0(SL_2(32))[/math] | 1 | 32 | 31 | [math]C_{31}[/math] | [math]C_{5}[/math] | 1 | 1 | ||
| M(32,51,24) | [math]k(((C_2)^3:(C_7:C_3)) \times A_4)[/math] | 1 | 32 | 15 | [math](C_7:C_3) \times C_3[/math] | 1 | 1 | |||
| M(32,51,25) | [math]B_0(k(((C_2)^3:(C_7:C_3)) \times A_5))[/math] | 1 | 32 | 15 | [math](C_7:C_3) \times C_3[/math] | 1 | 1 | |||
| M(32,51,26) | [math]B_0(k(J_1 \times A_4))[/math] | 1 | 32 | 15 | [math](C_7:C_3) \times C_3[/math] | 1 | 1 | |||
| M(32,51,27) | [math]B_0(k(J_1 \times A_5))[/math] | 1 | 32 | 15 | [math](C_7:C_3) \times C_3[/math] | 1 | 1 | |||
| M(32,51,28) | [math]B_0(k({\rm Aut}(SL_2(8)) \times A_4))[/math] | 1 | 32 | 15 | [math](C_7:C_3) \times C_3[/math] | 1 | 1 | |||
| M(32,51,29) | [math]B_0(k({\rm Aut}(SL_2(8)) \times A_5))[/math] | 1 | 32 | 15 | [math](C_7:C_3) \times C_3[/math] | 1 | 1 | |||
| M(32,51,30) | [math]k((C_2)^5:(C_{31}:C_5))[/math] | 1 | 16 | 11 | [math]C_{31}:C_5[/math] | 1 | 1 | |||
| M(32,51,31) | [math]B_0({\rm Aut}(SL_2(32)))[/math] | 1 | 16 | 11 | [math]C_{31}:C_5[/math] | 1 | 1 | |||
| M(32,51,32) | [math]b_2(k((C_2)^4 : 3^{1+2}_{+}) \times C_2)[/math] | 1 | 16 | 1 | [math]C_3 \times C_3[/math] | 1 | 1 | Non-principal block. Cannot be Morita equivalent to a principal block of any finite group. | ||
| M(32,51,33) | [math]b_2(k((C_2)^5 : (C_7 : 3^{1+2}_{+})))[/math] | 1 | 16 | 7 | [math](C_7:C_3) \times C_3[/math] | 1 | 1 | Non-principal block. Cannot be Morita equivalent to a principal block of any finite group. | ||
| M(32,51,34) | [math]b_2(k((SL_2(8) \times (C_2)^2) : 3^{1+2}_{+}))[/math] | 1 | 16 | 7 | [math](C_7:C_3) \times C_3[/math] | 1 | 1 | Non-principal block. Cannot be Morita equivalent to a principal block of any finite group. |
When considering any non-principal block above, considering [math]3^{1+2}_{-}[/math] instead of [math]3^{1+2}_{+}[/math] gives Morita equivalent blocks.
With the possible exception of M(32,51,5), M(32,51,11) and M(32,51,12), blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group).
With the possible exception of M(32,51,34) blocks are splendid derived equivalent to their Brauer correspondent. It is unknown whether the derived equivalence class of M(32,51,5) contains blocks with inertial quotient [math]C_7:C_3[/math] (as in M(32,51,11)). Hence, with the possible exceptions of M(32,51,5), M(32,51,11), M(32,51,12) and M(32,51,34), blocks with defect group [math](C_2)^5[/math] are derived equivalent if and only if they have the same inertial quotient (with the same action on the defect group) and the same number of simple modules. All the derived equivalences above also occur over [math]\mathcal{O}[/math].
