Difference between revisions of "5 +^3"

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== Blocks with defect group <math>5_+^{1+2}</math> ==
 
== Blocks with defect group <math>5_+^{1+2}</math> ==
−
 
−
[[Image:under-construction.png|50px|left]]
 
  
 
The Morita equivalence classes of blocks with this defect group are classified in [[References#A|[AE23]]]. Blocks in the same Morita equivalence class have the same fusion, and so the same inertial quotient, and the same Külshammer-Puig class.
 
The Morita equivalence classes of blocks with this defect group are classified in [[References#A|[AE23]]]. Blocks in the same Morita equivalence class have the same fusion, and so the same inertial quotient, and the same Külshammer-Puig class.
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|[[M(125,4,15)]] || <math>B_0(kSU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||
 
|[[M(125,4,15)]] || <math>B_0(kSU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||
 
|-
 
|-
−
|[[M(125,3,16)]] || Faithful block of maximal defect of <math>k(3.SU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||
+
|[[M(125,3,16)]] || Faithful block of <math>k(3.SU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||
 
|-
 
|-
 
|[[M(125,3,17)]] || <math>k5_+^{1+2}:(C_4 \times C_2)</math> || 1 || 20 || 2 ||8 || <math>C_4 \times C_2</math> || || || SmallGroup(1000,89)
 
|[[M(125,3,17)]] || <math>k5_+^{1+2}:(C_4 \times C_2)</math> || 1 || 20 || 2 ||8 || <math>C_4 \times C_2</math> || || || SmallGroup(1000,89)
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|[[M(125,3,29)]] || <math>k5_+^{1+2}:(C_4 \times C_4)</math> || 1 || 25 || 4 ||16 || <math>C_4 \times C_4</math> || || || SmallGroup(2000,473)
 
|[[M(125,3,29)]] || <math>k5_+^{1+2}:(C_4 \times C_4)</math> || 1 || 25 || 4 ||16 || <math>C_4 \times C_4</math> || || || SmallGroup(2000,473)
 
|-
 
|-
−
|[[M(125,3,30)]] || Faithful block of <math>k5_+^{1+2}:(C_2.(C_4 \times C_4))</math>, in which <math>(C_2.(C_4 \times C_4))'</math> acts trivially || 1 || 13 || 4 ||4 || <math>C_4 \times C_4</math> || || || <math>C_2.(C_4 \times C_4</math> is SmallGroup(32,2)
+
|[[M(125,3,30)]] || Faithful block of <math>k5_+^{1+2}:(C_2.(C_4 \times C_4))</math>, in which <math>(C_2.(C_4 \times C_4))'</math> acts trivially || 1 || 13 || 4 ||4 || <math>C_4 \times C_4</math> || || || <math>C_2.(C_4 \times C_4)</math> is SmallGroup(32,2)
 
|-
 
|-
 
|[[M(125,3,31)]] || Faithful block of <math>k5_+^{1+2}:((C_4 \times C_4):C_4)</math>, in which <math>Z((C_4 \times C_4):C_4)</math> acts trivially || 1 || 10 || 4 ||1 || <math>C_4 \times C_4</math> || || || <math>(C_4 \times C_4):C_4</math> is SmallGroup(64,18)
 
|[[M(125,3,31)]] || Faithful block of <math>k5_+^{1+2}:((C_4 \times C_4):C_4)</math>, in which <math>Z((C_4 \times C_4):C_4)</math> acts trivially || 1 || 10 || 4 ||1 || <math>C_4 \times C_4</math> || || || <math>(C_4 \times C_4):C_4</math> is SmallGroup(64,18)
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|-
 
|-
 
|[[M(125,3,33)]] || <math>B_0(kSL_3(5))</math> || 1 || 25 || 4 ||24 || <math>C_4 \times C_4</math> || || ||  
 
|[[M(125,3,33)]] || <math>B_0(kSL_3(5))</math> || 1 || 25 || 4 ||24 || <math>C_4 \times C_4</math> || || ||  
 +
|-
 +
|[[M(125,3,34)]] || <math>k5_+^{1+2}:M_4(2)</math> || 1 || 13 || 4 ||10 || <math>M_4(2)</math> || || || SmallGroup(2000,474)
 +
|-
 +
|[[M(125,3,35)]] || <math>B_0(kSU_3(5).2)</math> || 1 || 13 || 4 ||10 || <math>M_4(2)</math> || || ||
 +
|-
 +
|[[M(125,3,36)]] || <math>B_0(HS)</math> || 1 || 13 || 4 ||10 || <math>M_4(2)</math> || || ||
 +
|-
 +
|[[M(125,3,37)]] || Faithful block of <math>k(2.HS)</math> || 1 || 13 || 4 ||10 || <math>M_4(2)</math> || || ||
 +
|-
 +
|[[M(125,3,38)]] || <math>k5_+^{1+2}:(C_4 \circ D_8)</math> || 1 || 16 || 10 ||10 || <math>C_4 \circ D_8</math> || || ||
 +
|-
 +
|[[M(125,3,39)]] || Faithful block of <math>k5_+^{1+2}:H</math>, in which <math>C_2 \times C_2 \cong Z \leq Z(H)</math> acts trivially (with <math>H \cong (C_2 \times C_2 \times C_2):D_8</math> and <math>H/Z \cong C_4 \circ D_8</math>) || 1 || 10 || 10 ||4 || <math>C_4 \circ D_8</math> || || || <math>H</math> is SmallGroup(64,73)
 +
|-
 +
|[[M(125,3,40)]] || <math>k5_+^{1+2}:(C_3:C_8)</math> || 1 || 13 || 6 ||12 || <math>C_3:C_8</math> || || ||
 +
|-
 +
|[[M(125,3,41)]] || <math>B_0(kMcL)</math> || 1 || 13 || 6 ||12 || <math>C_3:C_8</math> || || ||
 +
|-
 +
|[[M(125,3,42)]] || <math>k(3.McL)</math> || 1 || 13 || 6 ||12 || <math>C_3:C_8</math> || || ||
 +
|-
 +
|[[M(125,3,43)]] || <math>k5_+^{1+2}:C_{24}</math> || 1 || 25 || 6 ||24 || <math>C_{24}</math> || || ||
 +
|-
 +
|[[M(125,3,44)]] || <math>B_0(SU_3(5).3)</math> || 1 || 25 || 6 ||24 || <math>C_{24}</math> || || ||
 +
|-
 +
|[[M(125,3,45)]] || Faithful block of <math>kSU_3(5).3</math> || 1 || 25 || 6 ||24 || <math>C_{24}</math> || || ||
 +
|-
 +
|[[M(125,3,46)]] || <math>k5_+^{1+2}:SL_2(3)</math> || 1 || 8 || 28 ||7 || <math>C_{24}</math> || || ||
 +
|-
 +
|[[M(125,3,47)]] || <math>k5_+^{1+2}:(C_4 \times S_3)</math> || 1 || 16 || 12 ||12 || <math>C_4 \times S_3</math> || || ||
 +
|-
 +
|[[M(125,3,48)]] || Faithful block <math>k5_+^{1+2}:(C_8:S_3)</math>, in which <math>C_2 \cong Z \leq Z(C_8:S_3)</math> acts trivially (with <math>(C_8:S_3)/Z \cong C_4 \times S_3</math>) || 1 || 10 || 12 ||6 || <math>C_4 \times S_3</math> || || ||
 +
|-
 +
|[[M(125,3,49)]] || <math>k5_+^{1+2}:(C_4 \wr C_2)</math> || 1 || 20 || 5 ||14 || <math>C_4 \wr C_2</math> || || ||
 +
|-
 +
|[[M(125,3,50)]] || Faithful block of <math>k5_+^{1+2}:H</math>, in which <math>C_2 \cong Z \leq Z(H)</math> acts trivially (with <math>H=</math>SmallGroup(64,10) and <math>H/Z \cong C_4 \wr C_2</math>) || 1 || 11 || 5 ||5 || <math>C_4 \wr C_2</math> || || ||
 +
|-
 +
|[[M(125,3,51)]] || <math>B_0(kSL_3(5).2)</math> || 1 || 20 || 5 ||18 || <math>C_4 \wr C_2</math> || || ||
 +
|-
 +
|[[M(125,3,52)]] || <math>B_0(kHS.2)</math> || 1 || 20 || 5 ||18 || <math>C_4 \wr C_2</math> || || ||
 +
|-
 +
|[[M(125,3,53)]] || Faithful block of <math>k2.HS.2</math> || 1 || 20 || 5 ||18 || <math>C_4 \wr C_2</math> || || ||
 +
|-
 +
|[[M(125,3,54)]] || <math>B_0(kRu)</math> || 1 || 20 || 5 ||18 || <math>C_4 \wr C_2</math> || || ||
 +
|-
 +
|[[M(125,3,55)]] || Faithful block of <math>k2.Ru</math> || 1 || 20 || 5 ||18 || <math>C_4 \wr C_2</math> || || ||
 +
|-
 +
|[[M(125,3,56)]] || <math>k5_+^{1+2}:(C_8:S_3)</math> || 1 || 20 || 6 ||18 || <math>C_8:S_3</math> || || ||
 +
|-
 +
|[[M(125,3,57)]] || <math>B_0(kMcL.2)</math> || 1 || 20 || 6 ||18 || <math>C_8:S_3</math> || || ||
 +
|-
 +
|[[M(125,3,58)]] || <math>B_0(kCo_3)</math> || 1 || 20 || 6 ||18 || <math>C_8:S_3</math> || || ||
 +
|-
 +
|[[M(125,3,59)]] || <math>k5_+^{1+2}:(C_4 \circ SL_2(3))</math> || 1 || 16 || 14 ||14 || <math>C_4 \circ SL_2(3)</math> || || ||
 +
|-
 +
|[[M(125,3,60)]] || <math>k5_+^{1+2}:U_2(3)</math> || 1 || 20 || 7 ||16 || <math>U_2(3)</math> || || ||
 +
|-
 +
|[[M(125,3,61)]] || <math>B_0(kCo_2)</math> || 1 || 20 || 7 ||16 || <math>U_2(3)</math> || || ||
 +
|-
 +
|[[M(125,3,62)]] || <math>B_0(kTh)</math> || 1 || 20 || 7 ||20 || <math>U_2(3)</math> || || ||
 
|}
 
|}

Latest revision as of 09:36, 4 October 2023

Blocks with defect group [math]5_+^{1+2}[/math]

The Morita equivalence classes of blocks with this defect group are classified in [AE23]. Blocks in the same Morita equivalence class have the same fusion, and so the same inertial quotient, and the same Külshammer-Puig class.

Class Representative # lifts / [math]\mathcal{O}[/math] [math]k_0(B)[/math] [math]k_1(B)[/math] [math]l(B)[/math] Inertial quotient [math]{\rm Pic}_\mathcal{O}(B)[/math] [math]{\rm Pic}_k(B)[/math] Notes
M(125,3,1) [math]k5_+^{1+2}[/math] 1 20 4 1 [math]1[/math]
M(125,3,2) [math]k5_+^{1+2}:C_2[/math] 1 20 2 2 [math]C_2[/math] SmallGroup(250,5)
M(125,3,3) [math]k5_+^{1+2}:C_2[/math] 1 14 8 2 [math]C_2[/math] SmallGroup(250,8)
M(125,3,4) [math]k5_+^{1+2}:C_3[/math] 1 11 12 3 [math]C_3[/math] SmallGroup(375,2)
M(125,3,5) [math]k5_+^{1+2}:C_4[/math] 1 25 1 4 [math]C_4[/math] SmallGroup(500,17)
M(125,3,6) [math]k5_+^{1+2}:C_4[/math] 1 13 1 4 [math]C_4[/math] SmallGroup(500,21)
M(125,3,7) [math]k(C_5 \times C_5):SL_2(5)[/math] 1 13 1 5 [math]C_4[/math] Inertial quotient as in M(125,3,6)
M(125,3,8) [math]k5_+^{1+2}:C_4[/math] 1 10 4 4 [math]C_4[/math] SmallGroup(500,23)
M(125,3,9) [math]k5_+^{1+2}:C_4[/math] 1 10 16 4 [math]C_4[/math] SmallGroup(500,25)
M(125,3,10) [math]k5_+^{1+2}:(C_2 \times C_2)[/math] 1 16 4 4 [math]C_2 \times C_2[/math] SmallGroup(500,27)
M(125,3,11) Faithful block of [math]k5_+^{1+2}:Q_8[/math], in which [math]Z(Q_8)[/math] acts trivially 1 13 4 1 [math]C_2 \times C_2[/math] SmallGroup(1000,42)
M(125,3,12) [math]k5_+^{1+2}:C_6[/math] 1 10 24 6 [math]C_6[/math] SmallGroup(750,6)
M(125,3,13) [math]k5_+^{1+2}:S_3[/math] 1 13 6 3 [math]S_3[/math] SmallGroup(750,5)
M(125,3,14) [math]k5_+^{1+2}:C_8[/math] 1 11 2 8 [math]C_8[/math] SmallGroup(1000,86)
M(125,4,15) [math]B_0(kSU_3(5))[/math] 1 11 2 8 [math]C_8[/math]
M(125,3,16) Faithful block of [math]k(3.SU_3(5))[/math] 1 11 2 8 [math]C_8[/math]
M(125,3,17) [math]k5_+^{1+2}:(C_4 \times C_2)[/math] 1 20 2 8 [math]C_4 \times C_2[/math] SmallGroup(1000,89)
M(125,3,18) Faithful block of [math]k5_+^{1+2}:M_4(2)[/math], in which [math]M_4(2)'[/math] acts trivially 1 14 2 2 [math]C_2 \times C_2[/math] SmallGroup(2000,250)
M(125,3,20) [math]k5_+^{1+2}:(C_4 \times C_2)[/math] 1 14 8 8 [math]C_4 \times C_2[/math] SmallGroup(1000,91)
M(125,3,21) Faithful block of [math]k5_+^{1+2}:M_4(2)[/math], in which [math]M_4(2)'[/math] acts trivially 1 8 8 2 [math]C_2 \times C_2[/math] SmallGroup(2000,264)
M(125,3,22) [math]k5_+^{1+2}:D_8[/math] 1 14 8 5 [math]D_8[/math] SmallGroup(1000,92)
M(125,3,23) Faithful block of [math]k5_+^{1+2}:D_{16}[/math], in which [math]Z(D_{16})[/math] acts trivially 1 11 8 2 [math]D_8[/math]
M(125,3,24) [math]k5_+^{1+2}:Q_8[/math] 1 8 20 5 [math]Q_8[/math] SmallGroup(1000,93)
M(125,3,25) [math]k5_+^{1+2}:(C_3:C_4)[/math] 1 8 24 6 [math]C_3:C_4[/math] SmallGroup(1500,35)
M(125,3,26) [math]k5_+^{1+2}:C_{12}[/math] 1 14 12 12 [math]C_{12}[/math] SmallGroup(1500,36)
M(125,3,27) [math]k5_+^{1+2}:D_{12}[/math] 1 14 12 6 [math]D_{12}[/math] SmallGroup(1000,37)
M(125,3,28) Faithful block of [math]k5_+^{1+2}:D_{24}[/math], in which [math]Z(D_{24})[/math] acts trivially 1 10 4 1 [math]D_{12}[/math]
M(125,3,29) [math]k5_+^{1+2}:(C_4 \times C_4)[/math] 1 25 4 16 [math]C_4 \times C_4[/math] SmallGroup(2000,473)
M(125,3,30) Faithful block of [math]k5_+^{1+2}:(C_2.(C_4 \times C_4))[/math], in which [math](C_2.(C_4 \times C_4))'[/math] acts trivially 1 13 4 4 [math]C_4 \times C_4[/math] [math]C_2.(C_4 \times C_4)[/math] is SmallGroup(32,2)
M(125,3,31) Faithful block of [math]k5_+^{1+2}:((C_4 \times C_4):C_4)[/math], in which [math]Z((C_4 \times C_4):C_4)[/math] acts trivially 1 10 4 1 [math]C_4 \times C_4[/math] [math](C_4 \times C_4):C_4[/math] is SmallGroup(64,18)
M(125,3,32) [math]k(C_5 \times C_5):GL_2(5)[/math] 1 25 4 20 [math]C_4 \times C_4[/math]
M(125,3,33) [math]B_0(kSL_3(5))[/math] 1 25 4 24 [math]C_4 \times C_4[/math]
M(125,3,34) [math]k5_+^{1+2}:M_4(2)[/math] 1 13 4 10 [math]M_4(2)[/math] SmallGroup(2000,474)
M(125,3,35) [math]B_0(kSU_3(5).2)[/math] 1 13 4 10 [math]M_4(2)[/math]
M(125,3,36) [math]B_0(HS)[/math] 1 13 4 10 [math]M_4(2)[/math]
M(125,3,37) Faithful block of [math]k(2.HS)[/math] 1 13 4 10 [math]M_4(2)[/math]
M(125,3,38) [math]k5_+^{1+2}:(C_4 \circ D_8)[/math] 1 16 10 10 [math]C_4 \circ D_8[/math]
M(125,3,39) Faithful block of [math]k5_+^{1+2}:H[/math], in which [math]C_2 \times C_2 \cong Z \leq Z(H)[/math] acts trivially (with [math]H \cong (C_2 \times C_2 \times C_2):D_8[/math] and [math]H/Z \cong C_4 \circ D_8[/math]) 1 10 10 4 [math]C_4 \circ D_8[/math] [math]H[/math] is SmallGroup(64,73)
M(125,3,40) [math]k5_+^{1+2}:(C_3:C_8)[/math] 1 13 6 12 [math]C_3:C_8[/math]
M(125,3,41) [math]B_0(kMcL)[/math] 1 13 6 12 [math]C_3:C_8[/math]
M(125,3,42) [math]k(3.McL)[/math] 1 13 6 12 [math]C_3:C_8[/math]
M(125,3,43) [math]k5_+^{1+2}:C_{24}[/math] 1 25 6 24 [math]C_{24}[/math]
M(125,3,44) [math]B_0(SU_3(5).3)[/math] 1 25 6 24 [math]C_{24}[/math]
M(125,3,45) Faithful block of [math]kSU_3(5).3[/math] 1 25 6 24 [math]C_{24}[/math]
M(125,3,46) [math]k5_+^{1+2}:SL_2(3)[/math] 1 8 28 7 [math]C_{24}[/math]
M(125,3,47) [math]k5_+^{1+2}:(C_4 \times S_3)[/math] 1 16 12 12 [math]C_4 \times S_3[/math]
M(125,3,48) Faithful block [math]k5_+^{1+2}:(C_8:S_3)[/math], in which [math]C_2 \cong Z \leq Z(C_8:S_3)[/math] acts trivially (with [math](C_8:S_3)/Z \cong C_4 \times S_3[/math]) 1 10 12 6 [math]C_4 \times S_3[/math]
M(125,3,49) [math]k5_+^{1+2}:(C_4 \wr C_2)[/math] 1 20 5 14 [math]C_4 \wr C_2[/math]
M(125,3,50) Faithful block of [math]k5_+^{1+2}:H[/math], in which [math]C_2 \cong Z \leq Z(H)[/math] acts trivially (with [math]H=[/math]SmallGroup(64,10) and [math]H/Z \cong C_4 \wr C_2[/math]) 1 11 5 5 [math]C_4 \wr C_2[/math]
M(125,3,51) [math]B_0(kSL_3(5).2)[/math] 1 20 5 18 [math]C_4 \wr C_2[/math]
M(125,3,52) [math]B_0(kHS.2)[/math] 1 20 5 18 [math]C_4 \wr C_2[/math]
M(125,3,53) Faithful block of [math]k2.HS.2[/math] 1 20 5 18 [math]C_4 \wr C_2[/math]
M(125,3,54) [math]B_0(kRu)[/math] 1 20 5 18 [math]C_4 \wr C_2[/math]
M(125,3,55) Faithful block of [math]k2.Ru[/math] 1 20 5 18 [math]C_4 \wr C_2[/math]
M(125,3,56) [math]k5_+^{1+2}:(C_8:S_3)[/math] 1 20 6 18 [math]C_8:S_3[/math]
M(125,3,57) [math]B_0(kMcL.2)[/math] 1 20 6 18 [math]C_8:S_3[/math]
M(125,3,58) [math]B_0(kCo_3)[/math] 1 20 6 18 [math]C_8:S_3[/math]
M(125,3,59) [math]k5_+^{1+2}:(C_4 \circ SL_2(3))[/math] 1 16 14 14 [math]C_4 \circ SL_2(3)[/math]
M(125,3,60) [math]k5_+^{1+2}:U_2(3)[/math] 1 20 7 16 [math]U_2(3)[/math]
M(125,3,61) [math]B_0(kCo_2)[/math] 1 20 7 16 [math]U_2(3)[/math]
M(125,3,62) [math]B_0(kTh)[/math] 1 20 7 20 [math]U_2(3)[/math]
Retrieved from ‘http://wiki.manchester.ac.uk/blocks/index.php?title=5_%2B%5E3&oldid=1221’