The Atlas of Finite Group Representations has, among other things, generators in matrix form (over fields of positive characteristic, I think) for some groups.
(Example in MeatAxe text format. Example in MeatAxe binary format. I've used MeatAxe to multiply and print out matrices like these.)
Some groups, however, don't have explicit generators. What they sometimes have is a "black box algorithm" (in "computer readable format") to find generators.
Here's an example of such a computer readable program that finds generators of the Monster group. How can I run programs like this one? Is it a MeatAxe, or GAP or Magma program?
# Black box algorithm to find standard generators of the Fischer-Griess
# Monster group M.
set F 0
set G 0
set V 0
lbl SEMISTD
rand 1
ord 1 A
incr V
if V gt 1000 then timeout
if A notin 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 &
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 &
35 36 38 39 40 41 42 44 45 46 47 48 50 51 52 &
54 55 56 57 59 60 62 66 68 69 70 71 78 84 87 &
88 92 93 94 95 104 105 110 119 then fail
if F eq 0 then
if A in 34 38 50 54 62 68 94 104 110 then
div A 2 B
pwr B 1 2
set F 1
endif
endif
if G eq 0 then
if A in 9 18 27 36 45 54 then
div A 3 C
pwr C 1 3
set G 1
endif
endif
if F eq 0 then jmp SEMISTD
if G eq 0 then jmp SEMISTD
set X 0
lbl CONJUGATE
incr X
if X gt 1000 then timeout
rand 4
cjr 3 4
mu 2 3 5
ord 5 D
if D notin 3 6 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 &
23 24 25 26 27 28 29 30 31 32 33 34 35 36 &
38 39 40 41 42 45 46 48 50 51 55 56 60 62 &
66 68 69 70 71 78 84 88 94 104 105 then fail
if D noteq 29 then jmp CONJUGATE
oup 2 2 3
The answer, as always, is rtfm! :) If you read this page, the authors state that the algorithms are implemented in a language called BBOX. There's an interpreter for GAP on this page.