Incidence Geometry Atlas

This page provides an atlas for flag transitive, residually connected, thin geometries admitting trialities and no dualities. For every group $G$ in the category, we provide the number of geometries, the size of the residues, the time it took to run it on a computer. with specs X and for which groups of the category does the algorithm produce nothing. Group names are clickable and will link to a copy and pastable magma file which contains the triple of involutions and the geometries, ready to be used. A README on how to use and interpret these files is available here .

Here is a list of the families:

  1. Sporadic simple groups of order $\leq 10^{10}$ and others.
  2. Linear groups, including $\mathrm{PGL}(2,q)$ and $\mathrm{PSL}(2,q)$ for $1 \leq q \leq 500$ a power of a prime.
  3. The alternating and symmetric groups $\mathrm{A}_n$ and $\mathrm{S}_n$ for $1 \leq n \leq 16$.
  4. Marston Conder's list of orientable hypermaps of genus $1$ to $101$.
  5. Misc based on litterature.

    6. Cite as: “The full classification datasets are available at https://rdelaby.github.io/atlas.html.”

Sporadic groups.

Name Order # of geometries Size of residues (number) Time (s)
$\mathrm{J}_1$ 175560 11 6(2),7(1),10(5),11(1),15(1),19(1) 10.67
$\mathrm{J}_2$ 604800 6 7(1),10(3),15(2) 4.69
$\mathrm{T}$ 17971200 10 5(1),6(1),8(2),12(4),13(2) 274.73
$\mathrm{HS}$ 44352000 1 15(1) 12.59
$\mathrm{J}_3$ 50232960 9 9(2),10(2),12(2),15(2),17(1) 2345.42
$\mathrm{M}_{24}$ 244823040 9 6(2),11(2),12(5) 44.3
$\mathrm{He}$ 4030387200 64 6(5),8(5),10(4),12(6),15(8),17(19),21(17)
 60794.38
Groups that do not produce a geometry : $\mathrm{M}_{11}$, $\mathrm{M}_{12}$, $\mathrm{M}_{22}$, $\mathrm{M}_{23}$, $\mathrm{McL}$. $\mathrm{Ru}$ and $\mathrm{O'N}$ waiting room.

Linear groups.

Name Order # of geometries Size of residues (number) Time (s)
$\mathrm{PSL}(2,27)$ 9828 3 7(1),13(1),14(1) 0.73
$\mathrm{PGL}(2,27)$ 19656 2 13(1),14(1) 0.73
$\mathrm{PGL}(2,64)$ 262080 2 63(1),65(1) 47.19
$\mathrm{PSL}(2,125)$ 976500 9 9(1),21(2),31(2),62(3),63(1) 114.31
$\mathrm{PGL}(2,125)$ 1953000 17 7(1),9(1),31(3),62(5),63(7) 230.62
$\mathrm{PSU}(3,8)$ 5515776 2 7(1),9(1) 106.89
$\mathrm{PSL}(2,343)$ 20176632 28 9(1),19(1),43(3),57(4),86(5),171(7),172(7) 8502.40
$\mathrm{PGL}(2,343)$ 40353264 47 9(1),19(2),43(4),57(2),86(7),171(16),172(15) 10418.34
$\mathrm{PSU}(3,27)$ 282056445216 9 7(1),13(2),26(3),28(3) 1482100.6
There are no geometries for $\mathrm{PGL}(2,q)$, $\mathrm{PSL}(2,q)$, $\mathrm{P}\Gamma\mathrm{L}(2,q)$, $\mathrm{P}\Sigma\mathrm{L}(2,q)$ for $2 \leq q \leq 500$ a power of a prime except for the ones in the table. There are no geometries for $\mathrm{PSU}(3,q)$ with $2 \leq q \leq 27$ a power of a prime except for the ones in the table.

Symmetric and alternating groups.

Name Order # of geometries Size of residues (number) Time (s)
$\mathrm{A}_{10}$ $\frac{10!}{2}$ 6 8(2),9(1),12(1),15(1),21(1) 3.68
$\mathrm{S}_{10}$ $10!$ 2 10(1),12(1) 2.79
$\mathrm{A}_{11}$ $\frac{11!}{2}$ 3 8(1),20(1),21(1) 4.17
$\mathrm{S}_{11}$ $11!$ 6 9(1),11(4),14(1) 15.54
$\mathrm{A}_{12}$ $\frac{12!}{2}$ 1 14(1) 23.11
$\mathrm{S}_{12}$ $12!$ 7 8(1),9(1),10(2),11(1),21(1),35(1) 82.51
$\mathrm{A}_{13}$ $\frac{13!}{2}$ 27 6(1),11(4),12(1),13(13),14(1),15(3),18(1),20(1),21(1),30(1) 435.48
$\mathrm{S}_{13}$ $13!$ 30 6(3),8(2),9(5),10(2),12(6),15(1),18(1),24(2),28(2),30(3),35(3) 606.42
$\mathrm{A}_{14}$ $\frac{14!}{2}$ 37 6(2),7(1),8(3),9(1),10(1),11(1),12(5),13(4),18(1),20(3),24(4),33(5),45(6) 2137.6
$\mathrm{S}_{14}$ $14!$ 16 6(1),7(1),10(1),12(2),15(1),18(1),20(1),24(3),28(1),30(3),60(1) 1610.79
$\mathrm{A}_{15}$ $\frac{15!}{2}$ 65 6(3),7(3),8(2),10(2),12(5),20(3),21(1),22(1),24(8),28(3),30(5),33(7),35(1),36(4),40(4),
42(4),45(3),60(3),105(3) 		27749.76
$\mathrm{S}_{15}$ $15!$ 59 6(3),8(1),9(2),12(3),13(8),15(25),18(2),20(1),21(1),22(3),24(1),28(3),30(1),42(3),60(2) 26610.52
$\mathrm{A}_{16}$ $\frac{16!}{2}$ 135 6(5),7(3),8(13),9(2),10(4),12(10),14(1),15(3),18(4),20(6),21(1),22(2),24(4),28(6),30(18),
33(1),35(1),36(7),40(5),42(10),45(1),60(10),70(2),84(6),105(10) 		156502.04
$\mathrm{S}_{16}$ $16!$ 184 5(1),6(5),7(3),8(2),9(2),10(6),12(20),13(4),14(13),15(14),18(3),20(12),21(3),22(1),24(6)
30(13),33(5),35(1),39(21),42(3),45(4),55(17),60(4),63(20),70(1) 		304624.08
For $ 1 \leq n \leq 9$, there are no geometries admitting $\mathrm{A}_n$ or $\mathrm{S}_n$ as automorphism group.

Hypermaps (orientable hypermaps with genus $1 \leq g \leq 101$, non orientable hypermaps with genus $2 \leq g \leq 202$).

Name Order Marston Conder's index Size of residues (number)
$D_5 \wr C_2:C_2$ 400 RPH26.6 4(1)
$\mathrm{AGL}(2,3)$ 432 RPH55.89 6(1)
$C_4^3.C_2^3$ 512 RPH81.323 8(1)
$D_{10}^2.C_2^2$ 1600 RPH101.14 4(1)
The hypermaps data got extracted from Marston Conder's website. The non-orientable regular hypermaps of genus $2 \leq g \leq 202$ do not produce any geometries. The geometry over the group $C_4^3.C_2^3$ is the smallest known one with only $3 \times 32 = 96$ elements.

Misc based on litterature.

Name Order # of geometries Size of residues (number) Time (s)
$\mathrm{Sz}(8)$ 29120 4 5(1),7(1),13(2) 0.96
The geometry over $\mathrm{Sz}(8)$ has been predicted by D. Leemans, K. Stokes, P. Tranchida in Flag transitive geometries with trialities and no dualities coming from Suzuki groups.
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