# Difference between revisions of "(C2)^5"

## Blocks with defect group $(C_2)^5$

These were classified in [Ar19] using the CFSG. Each of the 34 $k$-Morita equivalence classes lifts to an unique class over $\mathcal{O}$. The known Picard groups were computed in [EL18c] or using the main theorem of [Liv19]

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(32,51,1) $k((C_2)^5)$ 1 32 1 $1$ $(C_2)^5:GL_5(2)$ 1 1
M(32,51,2) $k(A_4 \times (C_2)^3)$ 1 32 3 $C_3$ $((C_2)^3 : GL_3(2)) \times S_3$ 1 1
M(32,51,3) $B_0(k(A_5 \times (C_2)^3))$ 1 32 3 $C_3$ $((C_2)^3 : GL_3(2)) \times C_2$ 1 1
M(32,51,4) $k(((C_2)^4 :C_3) \times C_2)$ 1 16 3 $C_3$ 1 1 The action of $C_3$ comes from the 5th power of a Singer cycle for $\mathbb{F}_{16}$
M(32,51,5) $k(((C_2)^4 :C_5) \times C_2)$ 1 16 5 $C_5$ 1 1
M(32,51,6) $k(((C_2)^3:C_7) \times (C_2)^2)$ 1 32 7 $C_7$ 1 1
M(32,51,7) $B_0(k(SL_2(8) \times (C_2)^2))$ 1 32 7 $C_7$ 1 1
M(32,51,8) $k(A_4 \times A_4 \times C_2)$ 1 32 9 $C_3 \times C_3$ 1 1
M(32,51,9) $B_0(k(A_4 \times A_5 \times C_2))$ 1 32 9 $C_3 \times C_3$ 1 1
M(32,51,10) $B_0(k(A_5 \times A_5 \times C_2))$ 1 32 9 $C_3 \times C_3$ 1 1
M(32,51,11) $k((C_2)^4 : C_{15} \times C_2)$ 1 32 15 $C_{15}$ 1 1
M(32,51,12) $B_0(k(SL_2(16) \times C_2))$ 1 32 15 $C_{15}$ 1 1
M(32,51,13) $k(((C_2)^3:C_7) \times A_4)$ 1 32 21 $C_{21}$ 1 1
M(32,51,14) $B_0(k(((C_2)^3:C_7) \times A_5))$ 1 32 21 $C_{21}$ 1 1
M(32,51,15) $B_0(k(SL_2(8) \times A_4))$ 1 32 21 $C_{21}$ 1 1
M(32,51,16) $B_0(k(SL_2(8) \times A_5))$ 1 32 21 $C_{21}$ 1 1
M(32,51,17) $k(((C_2)^3:(C_7:C_3)) \times (C_2)^2)$ 1 32 5 $C_7:C_3$ 1 1
M(32,51,18) $B_0(k(J_1 \times (C_2)^2))$ 1 32 5 $C_7:C_3$ 1 1
M(32,51,19) $B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))$ 1 32 5 $C_7:C_3$ 1 1
M(32,51,20) $k((C_2)^5:(C_7:C_3))$ 1 16 5 $C_7:C_3$ 1 1 The action of the subgroup $C_3$ is as specified in M(32,51,4) above.
M(32,51,21) $B_0(k((SL_2(8) \times (C_2)^2):C_3)$ 1 16 5 $C_7:C_3$ 1 1 The action of the subgroup $C_3$ is as specified in M(32,51,4) above.
M(32,51,22) $k((C_2)^5:C_{31})$ 1 32 31 $C_{31}$ 1 1
M(32,51,23) $B_0(SL_2(32))$ 1 32 31 $C_{31}$ $C_{5}$ 1 1
M(32,51,24) $k(((C_2)^3:(C_7:C_3)) \times A_4)$ 1 32 15 $(C_7:C_3) \times C_3$ 1 1
M(32,51,25) $B_0(k(((C_2)^3:(C_7:C_3)) \times A_5))$ 1 32 15 $(C_7:C_3) \times C_3$ 1 1
M(32,51,26) $B_0(k(J_1 \times A_4))$ 1 32 15 $(C_7:C_3) \times C_3$ 1 1
M(32,51,27) $B_0(k(J_1 \times A_5))$ 1 32 15 $(C_7:C_3) \times C_3$ 1 1
M(32,51,28) $B_0(k({\rm Aut}(SL_2(8)) \times A_4))$ 1 32 15 $(C_7:C_3) \times C_3$ 1 1
M(32,51,29) $B_0(k({\rm Aut}(SL_2(8)) \times A_5))$ 1 32 15 $(C_7:C_3) \times C_3$ 1 1
M(32,51,30) $k((C_2)^5:(C_{31}:C_5))$ 1 16 11 $C_{31}:C_5$ 1 1
M(32,51,31) $B_0({\rm Aut}(SL_2(32)))$ 1 16 11 $C_{31}:C_5$ 1 1
M(32,51,32) $b_2(k((C_2)^4 : 3^{1+2}_{+}) \times C_2)$ 1 16 1 $C_3 \times C_3$ 1 1 Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.
M(32,51,33) $b_2(k((C_2)^5 : (C_7 : 3^{1+2}_{+})))$ 1 16 7 $(C_7:C_3) \times C_3$ 1 1 Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.
M(32,51,34) $b_2(k((SL_2(8) \times (C_2)^2) : 3^{1+2}_{+}))$ 1 16 7 $(C_7:C_3) \times C_3$ 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 $(C_2)^5$, then it also has defect group $(C_2)^5$.

Blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group), as shown in [Ar19] and [AS20]

All blocks with defect group $(C_2)^5$ are derived equivalent to their Brauer correspondent, and each derived equivalence class is determined by the number of simple modules and the inertial quotient (and its action on the defect group) [AS20].

It is unknown whether these derived equivalences are splendid, because we cannot say anything about the sources of the Morita equivalences determined in [Ar19].