Presentation for the Mathieu Group M24 from dedge superflip

Following on from a highly symmetric presentation I supplied for the miniature Rubiks cube group Presentation for the 2x2x2 Rubiks cube group I investigated whether the Mathieu Group M24 could be similarily presented taking full advantage of the plentiful symmetry inherent in the Rubiks Revenge cube (4x4x4). The answer was indeed yes - here is the presentation I found - again on three involutions:

< a,b,c | a2 = b2 = c2 = 1,

   (ab)6 = [(bc)6] = [(ca)6] = 1,

   bacabacacabacababacabac = 1,

   (ababacbc)3 = 1,

   bababcbcbcbabab = cacabacacabacac >

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Keep posting

This is fascinating to me, though much to digest. Please keep posting!

Thanks Tom...

BTW don't take this as ex cathedra - it hasn't been reviewed by anyone. M24 is normally thought of as the automorphism group of a Steiner S(5,8,24) system or the symmetry group of the extended binary (Golay) code.

As I've said at the end there is a presentation for a subgroup M22:2 yet to be added which make up the third part of this post - I'm just fine tuning it now and will post tomorrow.

Verifying the presentation

The presentation was verified computationally using the Monoid Automata Factory software on SourceForge. This is a suite of programs adapted from the KBMAG package of Derek Holt for solving the word problem.

Rather than computing a word acceptor for the whole group M24 itself an index 24 group was used, namely M23. The subgroup M23=<abcba,b,c> presentation was verified to yield a subgroup of order 10200960 (though other choices can be used, but with care). The recursive ordering option of the automata executable was found to work the best.

In the presentation the length 23 relator is an amalgam of the (abac)4 and (bacabac)3 relators. Also the length 30 one can be replaced a shorter one of length 27 to make total length 92. I speculate that it will be difficult to better this presentation length for M24 irrespective of choice of relators or number of relators!

Details added

The set of 24 edges of a 4x4x4 Rubiks cube void of centres & corners and orientation is acted on by M24 as well as the maximal subgroup M12:2. See the following diagram for how the edges are positioned on the 4x4x4 cube.

The dedges are a set of 12 1x1x2 edge blocks: {(10,16),(9,18),(6,13),(7,8),(12,19), (1,3),(4,15),(11,20),(14,24),(21,22), (2,23),(5,17)}.
The positions of the three hidden dedges (12,19), (14,24) and (21,22) can be found from the generators we are about to define but are respectively the BL, DL and DB edges in Singmaster notation (with light blue the U face, white the R face and green the F face)

Consider the the following permutation reps. in GAP:

a := (1,6)(3,13)(4,20)(5,9)(11,15)(12,22)(17,18)(19,21);
V := (1,12,11,4)(2,21,24,17)(3,15,20,19)(5,14,22,23)(6,9,10,8)(7,16,18,13);

The involution a interchanges only dedges, the quarter turn move V moves half the cube edges clockwise and half anti-clockwise around the same layers. Note that V2 is just the whole cube rotated 180 degrees. The order of this group is 190080. The command StructureDescription(Group(a,V)) tells us that this is the group M12:2 - considerably smaller than our intended group M24.

Further it is easy to check that we have a very neat presentation for this group in GAP.

< a,V | a2 = V4 = 1,

(aV)6 = 1,

(VaV-1a(VaV)2aV-1aV)3 = a >

Now dedge superflip expressed as a permutation is the following value read directly from the diagram

omega := (1,3)(2,23)(4,15)(5,17)(6,13)(7,8)(10,16)(9,18)(11,20)(12,19)(14,24)(21,22);

Using GAP for example the command f:=Factorization(Group(a,V),omega) and Length(f) it is also easy to show that length of dedge superflip (omega) in this metric is 37 and as it turns out the sole antipode of the Cayley graph for <a,V,V-1>. This can be established by iterating through all other values in Group(a,V) and obtaining the factorisation using the fact that groups are also domains. Now omega can be expressed as a word in the generating alphabet as     


Returning now to the generators for M24 in the presentation in permutation form we can derive b and c as involutions in M24 satisfying


and the conditions specified in the main presentation. Specifically in permutation form these are (again formatted for cutting and pasting into GAP)


For any presentation defining the group M24 with a finite set of relators R then Ri(a,b,c) = Ri(a,c,b) i.e. b and c can be interchanged in any relator. We also have the following notable properties for ω(=omega):   

aω = a


bω = c


It turns out that the diameter of the Cayley graph for M24=<a,b,c> is 47 and that dedge superflip is also that sole node on the diameter. Expressed as a word in the generating set this is the value

    ω =


cacbcb (abc)3 bac (bca)3 cbcbab


Distribution of word count in the Cayley graph for M24
Length  Count            Length  Count              Length  Count
0       1                16      55360              32      28119886
1       3                17      99438              33      26773374
2       6                18      177071             34      23250295
3       12               19      310844             35      18148253
4       24               20      536435             36      12422323
5       48               21      903424             37      7191138
6       93               22      1491155            38      3317480
7       180              23      2406705            39      1126025
8       349              24      3783161            40      252902
9       676              25      5760002            41      32337
10      1300             26      8472343            42      2202
11      2475             27      11957906           43      157
12      4709             28      16089013           44      16
13      8845             29      20460937           45      6
14      16454            30      24414879           46      3
15      30319            31      27202475           47      1

For such a Cayley graph on three generators the peak point is reached very early in the distribution - the tail is wagging!

There is a related presentation for M22:2 generated by a, B=bcacb, and C=cbabc (I chose upper case values to avoid confusion with the lower case b and c of our M24). This is as follows. :

< a,B,C | a2 = B2 = C2 = 1,

(BC)3 = (aBaC)4 = (aBCB)4 = 1,

(aBC)11 = 1,

(BaBaBCaCaC)2 = a >

The Cayley graph for this group is of diameter 32, with three involutions attaining this value in length - one of which is dedge superflip again.

Alternative model: Is is also possible to treat the quarter turn V as rotating the top half of the 4x4x4 cube clockwise once and the bottom half anti-clockwise once. The involution would then be altered accordingly. The value omega would then interchange every pair of antipodal edges (those furthest away from each other). The advantages of modelling it in this way is that the quarter turn is much less convoluted and that the Cayley antipode is more intuitive (being the equivalent of the unique central symmetry in the 48 symmetries of the cube as a Platonic solid). The presentations given would not change however, just the permutation values (in terms of the edges in the diagram).

To be added soon time permitting: A short note on "pure" edge superflip (an illegal position on the 4x4x4 cube with fixed centres and corners!) and a presentation for the Mathieu group M23