View source for Q8xC2

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== Blocks with defect group <math>Q_8 \times C_2</math> ==

The invariants <math>k(B)</math>, <math>k_i(B)</math> and <math>l(B)</math> are determined in [[References|[Sa13]]]. There is as yet no classification of blocks with these defect groups, but Donovan's conjecture is known to hold over <math>\mathcal{O}</math> [[References|[EL20]]].

'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''

<!--{| class="wikitable"
|-
! scope="col"| Class
! scope="col"| Representative
! scope="col"| # lifts / <math>\mathcal{O}</math>
! scope="col"| <math>k(B)</math>
! scope="col"| <math>l(B)</math>
! scope="col"| Inertial quotients
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math>
! scope="col"| <math>{\rm Pic}_k(B)</math>
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math>
! scope="col"| <math>{\rm mf_k(B)}</math>
! scope="col"| Notes

|-
|[[M(16,9,1)]] || <math>kSD_{16}</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 ||
|-
|[[M(16,9,2)]] || <math>B_0(k \tilde{S}_5)</math><ref>This is the double cover SmallGroup(240,89)</ref> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal A})</math>
|-
|[[M(16,9,3)]] || <math>B_0(k \tilde{S}_4)</math><ref>This is the double cover SmallGroup(48,28)</ref> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal B})_1</math>
|-
|[[M(16,9,4)]] || <math>B_0(kSL_2(9))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math>
|-
|[[M(16,9,5)]] || <math>B_0(k(2.A_7))</math> || 1 ||10 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal B})</math>
|-
|[[M(16,9,6)]] || <math>B_0(kSL_2(7))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal K})</math>
|}

[[M(16,9,2)]] and [[M(16,9,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97]]], in which it is further proved that ''all'' blocks with defect group <math>Q_{16}</math> and two simple modules are derived equivalent (irrespective of the unknown cases in the classification).
 
[[M(16,9,4)]], [[M(16,9,5)]] and [[M(16,9,6)]] are derived equivalent over <math>\mathcal{O}</math> by [[References|[Ei16]]]<ref>This result was obtained over <math>k</math> in [[References|[Ho97]]]</ref>.-->

== Notes ==

<references />