A Classification of BLT-sets of Small Order

by Anton Betten, 2009

Introduction

For each of the listed field orders q, the following classification is complete. The stabilizer (denoted as "Aut") is calculated in P Gamma O(5,q), a group of order h q4 (q4-1)(q2-1) where q = ph.

BLT-sets of order 3

Line |Aut| induced kernel orbit structure common name
1 192 24 8 (4) Linear
More detailed information can be found here:
BLT set #1

BLT-sets of order 5

Line |Aut| induced kernel orbit structure common name
1 1440 120 12 (6) Linear
2 720 720 1 (6) Fi
More detailed information can be found here:
BLT set #1
BLT set #2

BLT-sets of order 7

Line |Aut| induced kernel orbit structure common name
1 5376 336 16 (8) Linear
2 384 384 1 (8) Fi, K2
More detailed information can be found here:
BLT set #1
BLT set #2

BLT-sets of order 9

Line |Aut| induced kernel orbit structure common name
1 28800 1440 20 (10) Linear, kernel is Dihedral
2 5760 1440 4 (10) K1, kernel is cyclic
3 400 400 1 (10) Fi, Mondello
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
The polynomial used to create GF(9) is X2 + 2X + 2. If x denotes a root, the field element a+bx (a,b in 0,1,2) is encoded as integer a+3b. An element of the semilinear group is denoted as a pair (A,i) where A is a matrix and i is an integer denoting the power of the field automorphism that is to be applied (i in 0,1).

BLT-sets of order 11

Line |Aut| induced kernel orbit structure common name
1 31680 1320 24 (12) Linear
2 288 288 1 (12) Fi
3 144 144 1 (12) DCH, Mondello
4 1320 1320 1 (12) FTWKB
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4

BLT-sets of order 13

Line |Aut| induced kernel orbit structure common name
1 61152 2184 28 (14) Linear
2 392 392 1 (14) Fi
3 48 48 1 (12,2) K2
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3

BLT-sets of order 17

The classification of BLT sets of order 17 took around 3 minutes CPU-time on a MAC PowerBook.
Line |Aut| induced kernel orbit structure common name comment
1 176256 4896 36 (18) Linear
2 648 648 1 (18) Fi
3 144 144 1 (12,6) DCH1/2
4 24 24 1 (12,6) PR
5 32 32 1 (16,2) K2
6 4896 4896 1 (18) FTWKB
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6

BLT-sets of order 19

The classification of BLT sets of order 19 took around 5 minutes CPU-time on a MAC PowerBook.
Line |Aut| induced kernel orbit structure common name
1 273600 6840 40 (20) Linear
2 800 800 1 (20) Fi
3 16 16 1 (82,22) PR
4 40 40 1 (20) PR, Mondello
5 20 20 1 (20) PR
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5

BLT-sets of order 23

The classification of BLT sets of order 23 took around 40 minutes CPU-time on a MAC PowerBook.
Line |Aut| induced kernel orbit structure common name
1 582912 12144 48 (24) Linear
2 1152 1152 1 (24) Fi
3 1152 1152 1 (24) PR
4 16 16 1 (82,42) PR
5 6 6 1 (63,32) PR
6 44 44 1 (22,2) K2
7 72 72 1 (18,6) DCH
8 24 24 1 (24) PR
9 12144 12144 1 (24) FTWKB
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6
BLT set #7
BLT set #8
BLT set #9

BLT-sets of order 25

The classification of BLT sets of order 25 took around 2 hours CPU-time on an Intel Xeon desktop.
Line |Aut| induced kernel orbit structure common name
1 1622400 31200 52 (26) Linear
2 124800 31200 4 (26) K1
3 2704 2704 1 (26) Fi
4 16 16 1 (16,8,2) PR
5 8 8 1 (83,2) PR
6 100 100 1 (25,1) K3
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6
The polynomial used to create GF(25) is X2-X-3. If x denotes a root, the field element a+bx (a,b in 0,1,2,3,4) is encoded as integer a+5b. An element of the semilinear group is denoted as a pair (A,i) where A is a matrix and i is an integer denoting the power of the field automorphism that is to be applied (i in 0,1).

BLT-sets of order 27

The classification of BLT sets of order 27 took around 4 hours CPU-time on an Intel Xeon desktop.
Line |Aut| induced kernel orbit structure common name comment
1 3302208 58968 56 (28) Linear
2 4704 4704 1 (28) Fi
3 117936 58968 2 (28) K1
4 648 648 1 (27,1) G/GJT
5 156 156 1 (26,2) K2
6 6 6 1 (64,2,12) LP cyclic
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6
The polynomial used to create GF(27) is X3+2X+1. If x denotes a root, the field element a+bx+cx2 (a,b,c in 0,1,2) is encoded as integer a+3b+9c. An element of the semilinear group is denoted as a pair (A,i) where A is a matrix and i is an integer denoting the power of the field automorphism that is to be applied (i in 0,1,2).

BLT-sets of order 29

The classification of BLT sets of order 29 took around 30 hours CPU-time on an Intel Xeon desktop.
Line |Aut| induced kernel orbit structure common name comment
1 1461600 24360 60 (30) Linear
2 1800 1800 1 (30) Fi (D30 ×D30):C2
3 3 3 1 (310) LP
4 6 6 1 (64,32) LP
5 48 48 1 (24,6) LP
6 8 8 1 (83,4,2) LP
7 720 720 1 (30) LP Symmetric 6
8 60 60 1 (30) Mondello D60
9 24360 24360 1 (30) FTWKB PSL(2,29):C2
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6
BLT set #7
BLT set #8
BLT set #9

BLT-sets of order 31

The classification of BLT sets of order 31 took around 5 days CPU-time on an Intel Xeon desktop.
Line |Aut| induced kernel orbit structure common name comment
1 1904640 29760 64 (32) Linear
2 2048 2048 1 (32) Fi
3 96 96 1 (24,6,2) LP96
4 4 4 1 (47,22) LP4b elementary abelian
5 8 8 1 (83,42) LP8
6 4 4 1 (48) LP4a cyclic
7 10 10 1 (102,52,2) LP10
8 64 64 1 (32) Mondello
The BLT-sets LP4a (also denoted as X40) and LP4b (also denoted as X42) can be distinguished from their orbit structure. In addition, the group of LP4a is cyclic whereas LP4b has an elementary abelian group.
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6
BLT set #7
BLT set #8

BLT-sets of order 37

The classification of BLT sets of order 37 took around 200 days CPU-time. The computation was performed in parallel at the University of Kiel, Germany. I thank Dieter Betten for computational assistance.
Line |Aut| induced kernel orbit structure common name comment
1 3846816 50616 76 (38) Linear
2 2888 2888 1 (38) Fi
3 4 4 1 (49,2) LP4acyclic
4 4 4 1 (49,12) LP4bcyclic
5 4 4 1 (49,2) Newelementary abelian
6 72 72 1 (36,2) K2
7 72 72 1 (36,2) LP72
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6
BLT set #7

BLT-sets of order 41

The classification of BLT sets of order 41 took around 5 years CPU-time. The computation was performed in parallel at the University of Kiel, Germany. I thank Dieter Betten for computational assistance.
Line |Aut| induced kernel orbit structure common name comment
1 5785920 68880 84 (42) Linear kernel is GAP group [84,14]
2 3528 3528 1 (42) Fi
3 2 2 1 (221) New
4 3 3 1 (314) New
5 8 8 1 (85,2) LP D8
6 24 24 1 (24,12,6) LP Symmetric(4)
7 60 60 1 (30,12) LP Alternating(5)
8 84 84 1 (42) New S3 ×D14, GAP group [84,8]
9 84 84 1 (42) Mondello GAP group [84,14]
10 68880 68880 1 (42) FTWKB
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6
BLT set #7
BLT set #8
BLT set #9
BLT set #10

BLT-sets of order 43

The classification of BLT sets of order 43 took around 52 days CPU-time (with an improved search algorithm, so the time is much shorter than for instance the time it took to search for all BLT sets of order 41). The computation was performed in parallel on the Open Science Grid.
Line |Aut| induced kernel orbit structure common name comment
1 6992832 79464 88 (44) Linear
2 3872 3872 1 (44) Fi
3 2 2 1 (221,12) New
4 4 4 1 (410,22) New
5 4 4 1 (411) LP
6 84 84 1 (42,2) K
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6

BLT-sets of order 47

The classification of BLT sets of order 47 took around 13 days CPU-time (with a better choice of the parameter size-of-starters). The computation was performed in parallel on the Open Science Grid.
Line |Aut| induced kernel orbit structure common name comment
1 9962496 103776 96 (48) Linear
2 4608 4608 1 (48) Fi
3 2304 2304 1 (48) DCP
4 2 2 1 (223,12) New
5 3 3 1 (316) LP
6 8 8 1 (85,42) New
7 12 12 1 (123,62) New
8 24 24 1 (242) LP
9 92 92 1 (46,2) K
10 103776 103776 1 (48) FTWKB
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6
BLT set #7
BLT set #8
BLT set #9
BLT set #10

BLT-sets of order 49

Added July 1, 2009.
The classification of BLT sets of order 49 took around 11 days CPU-time and was performed in parallel on the Purdue condor pool. The isomorph rejection was done using resources of the Renaissance Computing Institute. The starter size was 5.
Line |Aut| induced kernel orbit structure common name comment
1 23520000 235200 100 (50) Linear
2 10000 10000 1 (50) Fi
3 940800 235200 4 (50) Kantor semifield
4 20 20 1 (202,52) LP
5 40 40 1 (40,10) LP
6 8 8 1 (86,2) New cyclic
7 8 8 1 (86,2) New Dihedral
8 200 200 1 (50) Mondello
The polynomial used to create GF(49) is X2+6X+3. If x denotes a root, the field element a+bx (a,b in 0,1,2,...,6) is encoded as integer a+7b. An element of the semilinear group is denoted as a pair (A,i) where A is a matrix and i is an integer denoting the power of the field automorphism that is to be applied (i in 0,1).
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6
BLT set #7
BLT set #8

BLT-sets of order 53

Added July 7, 2009.
The classification of BLT sets of order 53 took around 85 days CPU-time and was performed in parallel on the Purdue condor pool. The isomorph rejection was done using resources of the Renaissance Computing Institute. The starter size was 5.
Line |Aut| induced kernel orbit structure common name comment
1 16072992 148824 108 (54) Linear
2 5832 5832 1 (54) Fisher
3 3 3 1 (318) New
4 12 12 1 (124,6) LP A4
5 24 24 1 (242,6) LP S4
6 8 8 1 (86,4,2) New D8
7 104 104 1 (52,2) K2
8 148824 148824 1 (54) FTWKB
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6
BLT set #7
BLT set #8

BLT-sets of order 59

Added July 17, 2009.
The classification of BLT sets of order 59 took around 15 months CPU-time and was performed in parallel on the Open Science Grid. The isomorph rejection was done using resources of the Renaissance Computing Institute. The starter size was 5.
Line |Aut| induced kernel orbit structure common name comment
1 24638400 205320 120 (60) Linear
2 7200 7200 1 (60) Fisher
3 8 8 1 (87,4) New Dihedral
4 3 3 1 (320) New
5 120 120 1 (60) LP S5
6 120 120 1 (60) Mondello Dihedral
7 5 5 1 (512) LP
8 24 24 1 (242,12) LP S4
9 205320 205320 1 (60) FTWKB
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6
BLT set #7
BLT set #8
BLT set #9

BLT-sets of order 61

Added July 27, 2009.
The classification of BLT sets of order 61 took around 2.5 years CPU-time and was performed in parallel on the Open Science Grid and on the Teragrid. The isomorph rejection was done using resources of the Renaissance Computing Institute. The starter size was 5.
Line |Aut| induced kernel orbit structure common name comment
1 28138080 226920 124 (62) Linear
2 7688 7688 1 (62) Fisher
3 4 4 1 (415,2) New el.ab.
4 4 4 1 (415,12) New cyclic
5 124 124 1 (62) Mondello Dihedral
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5

BLT-sets of order 67

Added November 21, 2009.
The classification of BLT sets of order 67 took around 16 years CPU-time and was performed in parallel on the Open Science Grid and on the Teragrid. The isomorph rejection was done using resources of the Renaissance Computing Institute. The isomorph rejection took 2 weeks. The starter size was 5.
Line |Aut| induced kernel orbit structure common name comment
1 40894656 300696 136 (68) Linear
2 9248 9248 1 (68) Fisher
3 4 4 1 (416,2) New el.ab.
4 4 4 1 (417) New cyclic
5 68 68 1 (68) New
6 132 132 1 (66,2) K2
More detailed information can be found here:
BLT set #1
BLT set #2
BLT set #3
BLT set #4
BLT set #5
BLT set #6

Coordinates in Machine Readable Format

Added July 27, 2009.
Using the form x02+x1x2+x3x4
The following file contains all BLT sets of order 3-61:
coordinates.tar.gz

Abbreviations used

Cn = Cyclic group of degree n
An = Alternating group of degree n
Sn = Symmetric group of degree n
Dn = Dihedral group of order n
Fi = J. Chris Fisher, J.A. Thas: Flocks in PG(3,2), Math. Z. 169(1): 1-11, 1979.
DCH = De Clerck, Herssens: Flocks of the quadratic cone in PG(3,q), for q small, The CAGe reports 8, University of Ghent, 1992
DCP = De Clerck, Penttila
PR = Penttila, Royle 1998
K1 = Kantor 1982.
K2 = Kantor: Some generalized quadrangles with parameters q2,q Math. Z. 192(1):45-50, 1986
K3 = GJT / Kantor 1982
FTWKB = Fisher / Thas / Kantor / Walker / Betten
LP = M. Law, T. Penttila: Construction of BLT-sets over small fields, European J. Comb. 1-22, 2004
Mondello = Penttila 1998
G = M. Ganley: Central weak nucleus semifields, Europ. J. Combin. 2(4) 339-347, 1981.
GJT = H. Geveart, N.L. Johnson, J.A. Thas: Spreads covered by reguli. Simon Stevin 62(1): 51-62, 1988.
Kantor 1982 = Kantor: On point transitive affine planes, Isreal J. Math. 42, 227-234.

File translated from TEX by TTH, version 3.81.
On 21 Nov 2009, 23:20.