Presentations of some simple groups

[Log of changes to this site]
© The research materials and data herein are the copyright of the Centre for Discrete Mathematics and Computing, The University of Queensland, 4 April, 2000. They may only be used with the permission of A/Professor George Havas and Dr Greg Gamble.
The following are the results of experiments producing efficient presentations of simple groups with trivial Schur multiplier. (Technically, these are not efficient, but the presentation that results when the first two order relators are concatenated, is.)

L2(8)

Order: 504 = 23.32.7

raw output (relatorword-000408b)
raw output (findrels-000415.g)

Presentation (syllable-length of 3rd relator: 15):

L2(8) = < x, y | x2, y3, xyxyxYxYxyxyxyxYxYxYxyxYxyxyxY >

L2(16)

Order: 4080 = 24.3.5.17

raw output

Presentation (syllable-length of 3rd relator: 17):

L2(16) = < x, y | x2, y3, xyxyxYxYxyxYxYxyxyxyxYxyxyxyxyxYxy >

L3(3)

Order: 5616 = 24.33.13

raw output

Presentation (syllable-length of 3rd relator: 23):

L3(3) = < a, b | a2, b3, ababaBababaBaBaBabababababababaBaBaBaBaBabaBaB >

Maximal subgroup: 32:2S4 = < ABaba, b >

M11

Order: 7920 = 24.32.5.11

raw output

Presentation (syllable-length of 3rd relator: 7):

M11 = < x, y | x2, y11, xyxy4xyxy5xY5xy5xY >

Maximal subgroup: M10 = < x, Y2xy3 >

L2(32)

Order: 32 736 = 25.3.11.31

raw output

Presentation (syllable-length of 3rd relator: 21):

L2(32) = < x, y | x2, y3, xyxyxYxYxyxyxYxyxYxyxyxyxyxyxYxyxYxyxyxYxY >

U3(4)

Order: 62 400 = 26.3.52.13

raw output

Presentation (syllable-length of 3rd relator: 21):

U3(4) = < a, b | a2, b3, ababaBaBababaBaBabaBababababaBaBababababab >

Maximal subgroup: 22 + 4:15 = < abababababa, ababaBabaBaBa, abaBaBabaBaB, abaBababaBaBab >

J1

Order: 175 560 = 23.3.5.7.11.19

raw output

Presentation (syllable-length of 3rd relator: 23):

J1 = < a, b | a2, b3, ababaBabababaBababaBaBaBababaBababaBababaBabab >

Maximal subgroup: L2(11) = < a, babababaBababaBaBaBaB >

L3(5)

Order: 372 000 = 25.3.53.31

raw output (findrels-000415.g)

Presentation (syllable-length of 3rd relator: 25):

L2(8) = < a, b | a2, b3, ababaBababababaBaBabaBaBabababaBaBabaBaBababababaB >

Maximal subgroup: 52:GL2(5) = < aBaba, baBaBabab >

Last update: 15 April, 2000.

Greg Gamble
http://www.csee.uq.edu.au/~gregg/