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PSL(2,7) embeds in the 2x2x2 cube group

Submitted by secondmouse on Wed, 01/05/2011 - 16:47.
I don't see any explicit reference to this but I've noticed that the simple group PSL(2,7) occurs naturally as a subgroup of the 2x2x2 cube group of order 3674160 (well - with a slight amount of wilful tinkering!). This is the model in which one of the the 8 cubelets stays fixed. One way of seeing how it is realised is to view to view the corner cubelets as a single block, i.e. suppose all three elements of each corner cubelet have the same colour. Then taking the following labellings where all of 1 could be coloured red, all of 2 yellow, etc. (UFR refers to the the cubelet in the "Up" "Front" "Right" position, etc). 
  UBL = 1
  UBR = 2
  UFL = 3
  UFR = 4
  DBL = the cubelet which does not move - just ignore this
  DBR = 5
  DFL = 6
  DFR = 7
Now take the generators which transpose two pairs of blocks: 
  a = (1,4)(2,5)
  b = (4,6)(1,3)
  c = (4,5)(6,7)
The group formed by these is of index 120 in the alternating group A_7 - the group of even permutations on seven points (even permutations are those comprised of an even number of disjoint transpositions). 
  G := Group(a,b,c);
  Order(G);
168
  IdSmallGroup(G);
[ 168, 42]
  IdSmallGroup(PSL(2,7));
[ 168, 42]
Admittedly this example of PSL(2,7) on these involutory generators is slightly contrived (but only slightly) as they have been deliberately chosen to be symmetry conjugates (it is easiest to draw a simple diagram to see the wedge-like symmetry in operation). This group PSL(2,7) might more commonly be thought of as the group of group of 2x2 matrices of determinant 1 with entries in GF(7) but it's also transitive on seven points. Are there any other simple ways of visualing any other transitive groups which embed inside the larger 3x3x3 cube group? I have a vague recollection of the sporadic Mathieu group M_12 being mentioned somewhere but it may be I was thinking of the so-called Rubik icosahedron. Comments on any aspect of this most welcome!
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Extending to PGL(2,7) and correction

Submitted by secondmouse on Wed, 01/12/2011 - 12:47.
Firstly sorry PSL(2,7) is of course of index 15 in A7 and not 120. It's also possible to extend this analysis to PGL(2,7) in the 2x2x2 cube with 8 corners that move. Defining the point 8 as the DBL cubelet from the previous example we see the 3 generators 

a = (1,4)(2,5)(7,8)
b = (4,6)(1,3)(2,8)
c = (4,5)(6,7)(3,8)
generate PGL(2,7) which is of index 120 in S8. Each generator is a diagonal planar reflection coupled with a single block "flip". Like the PSL(2,7) example though there are many other ways of doing this this in the 2x2x2 using non-symmetry conjugates.
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Some nice presentations for PGL(2,q)

Submitted by secondmouse on Sun, 02/13/2011 - 11:38.
This leads to the following highly symmetric presentation: 
PGL(2,7) = < a,b,c | a2 = b2 = (ab)4 = (bc)4 = 1,
                     acabaca = bcbabcb, babcbab = cacbcac >
One can also find a similar presentation for PGL(2,11) from the 12 centre blocks of a 3x3x3 cube. This is a subgroup of index 362880 in S12. 
PGL(2,11) = < a,b,c | a2 = b2 = (ab)6 = (bc)6 = 1,
                      abcbcba = bacacab, bcacacb = cbababc >
Each of these generators can be a symmetry conjugate e.g. 1/3 rotation with two antipodal cube corners remaining fixed as in the two cases above. Also we can find from the 24 outer edge blocks of a Professor's 5x5x5 cube that: 
PGL(2,23) = < a,b,c | a2 = b2 = (ab)6 = (bc)6 = 1,
                      acabaca = bcbabcb, babcbab = cacbcac >
From the presentation for PGL(2,11) above it is interesting to observe that by increasing o(ab) = o(bc) = o(ca) = 8 we have: 
PGL(2,17) = < a,b,c | a2 = b2 = (ab)8 = (bc)8 = (ca)8 = 1,
                      abcbcba = bacacab, bcacacb = cbababc >
It is interesting to observe the length 14 relations in the presentations for the PGL groups above. Compare this with the presentation for the symmetry group of the 2x2x2 cube on three involutions from an earlier thread Presentation for Rubik's cube . This required length 18 relations where both the LHS and the RHS were palindromic. I did some investigations and found that both of these pairs of length 14 relations can be used to construct presentations for other much larger PGL(2,q) groups, e.g. 
PGL(2,49) = < a,b,c | a2 = b2 = c2 = 1,
                      ((ab)4(abc)2)2 = 1,
                      ((bc)4(bca)2)2 = 1,
                      abcbcba = bacacab, bcacacb = cbababc >

PGL(2,103) = < a,b,c | a2 = b2 = c2 = 1,
                      ((ab)4(abc)2)2 = 1,
                      ((bc)4(bca)2)2 = 1,
                      acabaca = bcbabcb, babcbab = cacbcac >

PGL(2,113) = < a,b,c | a2 = b2 = c2 = 1,
                       ((ab)3(cb)3)2 = 1,
                       ((bc)3(ac)3)2 = 1,
                       acabaca = bcbabcb, babcbab = cacbcac >
Is there any reference to these in the literature - however obscure?
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Since the Mathieu M12 group c

Submitted by Bruce Norskog on Thu, 01/06/2011 - 00:03.

Since the Mathieu M12 group can be defined using permutations of 12 objects, you can use permutations of the edges to make such a group.

Generators:

  • U2 F2 R' D' R' B2 D' L' D' U2 R' B2 U' L2 U2
  • B R D' R' B2 F2 R2 D2 U2 R' D R' B'
  • B2 F2 L2 R2 U' B2 F2 L2 R2 U

You also might find this thread of interest.

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