Date: Sun, 23 Apr 95 23:29:00 -0400
From: Mark Longridge <mark.longridge@canrem.com >
~~~ Subject: Orders of Symmetry

Date: Wed, 8 Dec 93 16:28:29 EST
From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Message-Id: <9312082128.AA23718@Sun0.AIC.NRL.Navy.Mil>
To: mark.longridge@canrem.com (Mark Longridge), CRSO.Cube@canrem.com
Subject: Re: More corrections

* Hmmm, what are all the possible orders of symmetry? *

M has subgroups of order 48, 24, 16, 12, 8, 6, 4, 3, 2, 1. Some of
these subgroups (e.g., A, C) are not symmetry groups of any position,
so I can't be sure there are positions of all these symmetry orders.

Quite a while ago I asked Dan the question above, and I've thought

```So I decided to look at certain cube positions and I wrote a module
to perform
C and C + Sm
```

where
C = 24 rotations of the cube
Sm = Central Reflection

on any pattern I had in my database, and count how many different
patterns resulted from the 48 operations.

The following are some patterns which I found:

```Number of
different       Pattern
patterns        -------
---------

48              R1 U1
24              L2 U2
16              Mark's Pattern 1 (18 q+h, 22 q)
R2 U3 R1 D1 F1 B1 R3 L3 U1 D1 F3 U1 F3 U2 D3 B2 R2 U1
(Also 7 clockwise + 1 anticlockwise corner twist)
12              2 dot, 2 T, 2 ARM (sq group antipode, see p108)
8              6 Dot (a slice pattern)
6              2 DOT, 4 ARM (sq group antipode, see p99)
4              ????
3              4 Dot pattern (slice pattern)
2              6 H pattern type 2, T2 B2 L2 T2 D2 L2 F2 T2
1              Pons Asinorum (6 X order 2) or all edges flipped
```

It took a while to find a pattern which could be transformed 16
different ways. Still trying to find a pattern which will
result in 4 distinct ways, but I am not optimistic. A random walk
through the cube resulted in a pattern which would transform
48 ways in every case I tried.

A) What is the next most commutative element (the pancentre?)
after the 12-flip?

(presumably, start excluded as well)

i'll guess that these four conjugacy classes are tied for next.

```corner cycle structure:  (1+)(1+)(1+)(1+)(1+)(1+)(1+)(1-)
edge cycle structure:    (1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)
```

Here's a small followup to the pancentre question. The reason
why the 7 clockwise + 1 anticlockwise corner twist is the next
most commutative element after the 12-flip & start is because
it has the most number of cube elements (in this case corners)
the same as possible without all the elements being the same,
as with the 12-flip. It must be 7 clockwise + 1 anticlockwise
corner twist because the next most commutative element effecting
edges only would be the 10-flip and that would have 2 elements
not the same as the rest instead of just 1.

```-> Mark <-
```