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Date: Mon, 4 Aug 1997 19:41:24 -0400
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From: Dan Hoey <Hoey@aic.nrl.navy.mil>
To: Cube-Lovers@ai.mit.edu, ponder@austin.ibm.com
In-Reply-To: <199708020031.UAA28142@sun30.aic.nrl.navy.mil> (message from Dan
	Hoey on Fri, 1 Aug 1997 20:31:56 -0400)
Subject: Re: puzzle to be simulated

I've found out why the cube-plane groups related to the 1^2+2^2 square
torus are 1/6 the size we would expect.  It's the corners.  The group
has five corners {1,2,3,4,5} and five generators {A,B,C,D,E} that
operate on corners as

  5..CC/DDD`5    A: (1,2,4,3)
  EEE`1..DD/E    B: (2,3,5,4)
  .EE/AAA`2..    C: (3,4,1,5)
  B`3..AA/BBB    D: (4,5,2,1)
  B/CCC`4..BB    E: (5,1,3,2)
  5..CC/DDD`5

These generators do not generate the 120-element group S5, rather they
generate a 20-element subgroup known to GAP as

    5:4       = A split extension of C5 by C4      or equivalently
    H(2^2,5)  = <a,b ; a^5 = b^4 = a b a^2 b = 1 >.

Neither of these tells me a lot, except that the fact that this group
has index 6 in S5 means that there are six "orbits" of corner
permutations.

Dan
Hoey@AIC.NRL.Navy.Mil