5 +^3
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] |
