(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.

If a block of a finite group is Morita equivalent to another block with defect group [math](C_2)^5[/math], then it also has defect group [math](C_2)^5[/math].

With the possible exceptions 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).

All blocks with defect group [math](C_2)^5[/math] are splendid derived equivalent to their Brauer correspondent, with the possible exceptions of:

• Blocks in M(32,51,34).
• Possible blocks in M(32,51,5) with inertial quotient [math]C_7:C_3[/math] (acting as in M(32,51,20)).
• Possible blocks in M(32,51,11) or M(32,51,12) with inertial quotient [math](C_7:C_3) \times C_3[/math].

At the moment it is unknown whether M(32,51,5) contains blocks with inertial quotient [math]C_7:C_3[/math] (as in M(32,51,20)), and whether M(32,51,11) and M(32,51,12) contain blocks with inertial quotient [math]C_7:C_3 \times C_3[/math].

All the derived equivalences above are splendid, and also occur over [math]\mathcal{O}[/math].