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 maximal defect 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]