In a few

weeks I will give a _geometry 101_ course! It was decided that in

this course I should try to explain what rotations in $\mathbb{R}^3’$

are, so the classification of all finite rotation groups seemed like a

fun topic. Along the way I’ll have to introduce groups so bringing in a

little bit of GAP

may be a good idea. Clearly, the real power of GAP is lost on the

symmetry groups of the Platonic solids so I’ll do the traditional

computation of the transformation group of the Rubik’s cube. But

then I discovered that there is also a version of it on the dodecahedron

which is called megaminx so I couldn’t resist trying to work out the order of its

transformation group. Fortunately Coreyanne Rickwalt did already the

hard work giving a presentation as

a permutation group. So giving the generators to GAP

f1:=(1,3,5,7,9)(2,4,6,8,10)(20,31,42,53,64)(19,30,41,52,63)(18,29,40,51,62);

f2:=(12,14,16,18,20)(13,15,17,19,21)(1,60,73,84,31)(3,62,75,86,23)(2,61,74,85,32);

f3:=(23,25,27,29,31)(24,26,28,30,32)(82,95,42,3,16)(83,96,43,4,17)(84,97,34,5,18);

f4:=(34,36,38,40,42)(35,37,39,41,43)(27,93,106,53,5)(28,94,107,54,6)(29,95,108,45,7);

f5:=(45,47,49,51,53)(46,48,50,52,54)(38,104,117,64,7)(39,105,118,65,8),(40,106,119,56,9);

f6:=(56,58,60,62,64)(57,59,61,63,65)(49,115,75,20,9)(50,116,76,21,10),(51,117,67,12,1);

f7:=(67,69,71,73,75)(68,70,72,74,76)(58,113,126,86,12)(59,114,127,7,13),(60,115,128,78,14);

f8:=(78,80,82,84,86)(79,81,83,85,87)(71,124,97,23,14)(72,125,98,24,15),(73,126,89,25,16);

f9:=(89,91,93,95,97)(90,92,94,96,98)(80,122,108,34,25)(81,123,109,35,26),(82,124,100,36,27);

f10:=(100,102,104,106,108)(101,103,105,107,109)(91,130,119,45,36),(92,131,120,46,37)(93,122,111,47,38);

f11:=(111,113,115,117,119)(112,114,116,118,120)(102,128,67,56,47),(103,129,68,57,48)(104,130,69,58,49);

f12:=(122,124,126,128,130)(123,125,127,129,131)(100,89,78,69,111),(101,90,79,70,112)(102,91,80,71,113);

and defining the

megaminx group by

megaminx:=Group(f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12); Size(megaminx);

and asking for its order I was a bit surprised to get

after a couple of minutes the following awkward number

33447514567245635287940590270451862933763731665690149051478356761508167786224814946834370826

35992490654078818946607045276267204294704060929949240557194825002982480260628480000000000000

000000000000000

or if you prefer it is

$2^{115} 3^{58} 5^{28} 7^{19} 11^{10} 13^9 17^7 19^6 23^5 29^4 31^3

37^3 41^2 43^2 47^2 53^2 59^2 61 .67 .71. 73. 79 .83 .89 .97. 101 .103.

107 .109 .113$